1.
Jacobi ellipsoid
–
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by any of the two following properties, every planar cross section is either an ellipse, or is empty, or is reduced to a single point. It is bounded, which means that it may be enclosed in a large sphere. An ellipsoid has three perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, if the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, and the axes are uniquely defined. If two of the axes have the length, then the ellipsoid is an ellipsoid of revolution. In this case, the ellipsoid is invariant under a rotation around the third axis, if the third axis is shorter, the ellipsoid is an oblate spheroid, if it is longer, it is prolate spheroid. If the three axes have the length, the ellipsoid is a sphere. The points, and lie on the surface, the line segments from the origin to these points are called the semi-principal axes of the ellipsoid, because a, b, c are half the length of the principal axes. They correspond to the axis and semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid, if a = b < c, one has a prolate spheroid, if a = b = c, one has a sphere. It is easy to check, The intersection of a plane, remark, The contour of an ellipsoid, seen from a point outside the ellipsoid or from infinity, is in any case a plane section, hence an ellipse. The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is x = a cos cos , y = b cos sin , z = c sin and these parameters may be interpreted as spherical coordinates. More precisely, π /2 − θ is the polar angle, and φ is the azimuth angle of the point of the ellipsoid. More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the x to the equation T A =1. The eigenvectors of A define the axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes
Jacobi ellipsoid
–
Tri-axial ellipsoid with distinct semi-axis lengths
Jacobi ellipsoid
–
Artist's conception of
Haumea, a Jacobi-ellipsoid
dwarf planet, with its two moons
2.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
Sphere
–
Circumscribed cylinder to a sphere
Sphere
–
A two-dimensional
perspective projection of a sphere
Sphere
Sphere
–
Deck of playing cards illustrating engineering instruments, England, 1702.
King of spades: Spheres
3.
Spheroid
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A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes, in other words, an ellipsoid with two equal semi-diameters. If the ellipse is rotated about its axis, the result is a prolate spheroid. If the ellipse is rotated about its axis, the result is an oblate spheroid. If the generating ellipse is a circle, the result is a sphere, because of the combined effects of gravity and rotation, the Earths shape is not quite a sphere but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere, the current World Geodetic System model uses a spheroid whose radius is 6,378.137 km at the equator and 6,356.752 km at the poles. The semi-major axis a is the radius of the spheroid. There are two cases, c < a, oblate spheroid c > a, prolate spheroid The case of a = c reduces to a sphere. An oblate spheroid with c < a has surface area S o b l a t e =2 π a 2 where e 2 =1 − c 2 a 2. The oblate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. A prolate spheroid with c > a has surface area S p r o l a t e =2 π a 2 where e 2 =1 − a 2 c 2. The prolate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis c and semi-minor axis a and these formulas are identical in the sense that the formula for Soblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with the eccentricity, both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse. The volume inside a spheroid is 4π/3a2c ≈4. 19a2c, if A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is π/6A2C ≈0. 523A2C. Both of these curvatures are always positive, so every point on a spheroid is elliptic. These are just two of different parameters used to define an ellipse and its solid body counterparts. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate and oblate spheroidal, deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects. An extreme example of a planet in science fiction is Mesklin, in Hal Clements novel Mission of Gravity. The prolate spheroid is the shape of the ball in several sports, several moons of the Solar system approximate prolate spheroids in shape, though they are actually triaxial ellipsoids
Spheroid
–
oblate spheroid
4.
Affine transformation
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In geometry, an affine transformation, affine map or an affinity is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation, an affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. If X and Y are affine spaces, then every affine transformation f, X → Y is of the form x ↦ M x + b, unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear, all Euclidean spaces are affine, but there are affine spaces that are non-Euclidean. In affine coordinates, which include Cartesian coordinates in Euclidean spaces, another way to deal with affine transformations systematically is to select a point as the origin, then, any affine transformation is equivalent to a linear transformation followed by a translation. An affine map f, A → B between two spaces is a map on the points that acts linearly on the vectors. In symbols, f determines a linear transformation φ such that and we can interpret this definition in a few other ways, as follows. If an origin O ∈ A is chosen, and B denotes its image f ∈ B, the conclusion is that, intuitively, f consists of a translation and a linear map. In other words, f preserves barycenters, as shown above, an affine map is the composition of two functions, a translation and a linear map. Ordinary vector algebra uses matrix multiplication to represent linear maps, using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. If A is a matrix, = is equivalent to the following y → = A x → + b →, the above-mentioned augmented matrix is called an affine transformation matrix, or projective transformation matrix. This representation exhibits the set of all affine transformations as the semidirect product of K n and G L. This is a group under the operation of composition of functions, ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending the additional coordinate 1 to every vector, one considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the coordinate is 1. Thus the origin of the space can be found at. A translation within the space by means of a linear transformation of the higher-dimensional space is then possible
Affine transformation
–
An image of a fern-like
fractal that exhibits affine
self-similarity. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation.
5.
Quadric surface
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In mathematics, a quadric or quadric surface, is a generalization of conic sections. It is an hypersurface in a space, and is defined as the zero set of an irreducible polynomial of degree two in D +1 variables. When the defining polynomial is not absolutely irreducible, the set is generally not considered as a quadric. The values Q, P and R are often taken to be real numbers or complex numbers. A quadric is an algebraic variety, or, if it is reducible. Quadrics may also be defined in spaces, see Quadric. Quadrics in the Euclidean plane are those of dimension D =1, in this case, one talks of conic sections, or conics. In three-dimensional Euclidean space, quadrics have dimension D =2 and they are classified and named by their orbits under affine transformations. More precisely, if an affine transformation maps a quadric onto another one, they belong to the same class, each of these 17 normal forms correspond to a single orbit under affine transformations. In three cases there are no points, ε1 = ε2 =1, ε1 =0, ε2 =1. In one case, the cone, there is a single point. If ε4 =1, one has a line, for ε4 =0, one has a double plane. For ε4 =1, one has two intersecting planes and it remains nine true quadrics, a cone and three cylinders and five non-degenerated quadrics, which are detailed in the following table. In a three-dimensional Euclidean space there are 17 such normal forms, of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all, the quadrics can be treated in a uniform manner by introducing homogeneous coordinates on a Euclidean space, thus effectively regarding it as a projective space. Thus if the coordinates on RD+1 are one introduces new coordinates on RD+2 related to the original coordinates by x i = X i / X0. In the new variables, every quadric is defined by an equation of the form Q = ∑ i j a i j X i X j =0 where the coefficients aij are symmetric in i and j. Regarding Q =0 as an equation in projective space exhibits the quadric as an algebraic variety
Quadric surface
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Ellipse (e = 1/2), parabola (e =1) and hyperbola (e = 2) with fixed
focus F and directrix.
6.
Polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
Polynomial
–
The
graph of a polynomial function of degree 3
7.
Cross section (geometry)
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In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections, conic sections – circles, ellipses, parabolas, and hyperbolas – are formed by cross-sections of a cone at various different angles, as seen in the diagram at left. Any planar cross-section passing through the center of an ellipsoid forms an ellipse on its surface, a cross-section of a cylinder is a circle if the cross-section is parallel to the cylinders base, or an ellipse with non-zero eccentricity if it is neither parallel nor perpendicular to the base. If the cross-section is perpendicular to the base it consists of two line segments unless it is just tangent to the cylinder, in which case it is a single line segment. A cross section of a polyhedron is a polygon, if instead the cross section is taken for a fixed value of the density, the result is an iso-density contour. For the normal distribution, these contours are ellipses, a cross section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown at left. In economics, a function f specifies the output that can be produced by various quantities x and y of inputs, typically labor. The production function of a firm or a society can be plotted in three-dimensional space, also in economics, a cardinal or ordinal utility function u gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. Cross sections are used in anatomy to illustrate the inner structure of an organ. A cross section of a trunk, as shown at left, reveals growth rings that can be used to find the age of the tree. Cavalieris principle states that solids with corresponding sections of equal areas have equal volumes. The cross-sectional area of an object when viewed from an angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has A ′ = π r 2 when viewed along its central axis, a sphere of radius r has A ′ = π r 2 when viewed from any angle. For a convex body, each ray through the object from the viewers perspective crosses just two surfaces, descriptive geometry Exploded view drawing Graphical projection Plans
Cross section (geometry)
–
Pinus taeda cross section showing annual rings,
Cheraw, South Carolina.
8.
Ellipse
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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin
Ellipse
–
Drawing an ellipse with two pins, a loop, and a pen
Ellipse
–
An ellipse obtained as the intersection of a
cone with an inclined plane.
9.
Bounded set
–
Bounded and boundary are distinct concepts, for the latter see boundary. A circle in isolation is a bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, conversely, a set which is not bounded is called unbounded. The word bounded makes no sense in a topological space without a metric. A set S of real numbers is called bounded from above if there is a number k such that k ≥ s for all s in S. The number k is called a bound of S. The terms bounded from below and lower bound are similarly defined, a set S is bounded if it has both upper and lower bounds. Therefore, a set of numbers is bounded if it is contained in a finite interval. M is a metric space if M is bounded as a subset of itself. For subsets of Rn the two are equivalent, a metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space Rn is compact if and only if it is closed and bounded, in topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces. A set of numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any ordered set. Note that this general concept of boundedness does not correspond to a notion of size. A subset S of an ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called a bound of S. The concepts of bounded below and lower bound are defined similarly, a subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval
Bounded set
–
An
artist's impression of a bounded set (top) and of an unbounded set (bottom). The set at the bottom continues forever towards the right.
10.
Perpendicular
–
In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects, a line is said to be perpendicular to another line if the two lines intersect at a right angle. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, for this reason, we may speak of two lines as being perpendicular without specifying an order. Perpendicularity easily extends to segments and rays, in symbols, A B ¯ ⊥ C D ¯ means line segment AB is perpendicular to line segment CD. A line is said to be perpendicular to an if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines, two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. Perpendicularity is one instance of the more general mathematical concept of orthogonality, perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the perpendicular is sometimes used to describe much more complicated geometric orthogonality conditions. The word foot is used in connection with perpendiculars. This usage is exemplified in the top diagram, above, the diagram can be in any orientation. The foot is not necessarily at the bottom, step 2, construct circles centered at A and B having equal radius. Let Q and R be the points of intersection of two circles. Step 3, connect Q and R to construct the desired perpendicular PQ, to prove that the PQ is perpendicular to AB, use the SSS congruence theorem for and QPB to conclude that angles OPA and OPB are equal. Then use the SAS congruence theorem for triangles OPA and OPB to conclude that angles POA, to make the perpendicular to the line g at or through the point P using Thales theorem, see the animation at right. The Pythagorean Theorem can be used as the basis of methods of constructing right angles, for example, by counting links, three pieces of chain can be made with lengths in the ratio 3,4,5. These can be out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, the chains can be used repeatedly whenever required. If two lines are perpendicular to a third line, all of the angles formed along the third line are right angles
Perpendicular
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The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.
11.
Rotational symmetry
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
Rotational symmetry
–
The starting position in
shogi
Rotational symmetry
–
The
triskelion appearing on the
Isle of Man flag.
12.
Central symmetry
–
Not to be confused with inversive geometry, in which inversion is through a circle instead of a point. In geometry, a point reflection or inversion in a point is a type of isometry of Euclidean space, point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has one fixed point. It is equivalent to a transformation with scale factor equal to -1. The point of inversion is called homothetic center. The term reflection is loose, and considered by some an abuse of language, with preferred, however. Such maps are involutions, meaning that they have order 2 – they are their own inverse, in dimension 1 these coincide, as a point is a hyperplane in the line. In terms of algebra, assuming the origin is fixed. Reflection in a hyperplane has a single −1 eigenvalue, while point reflection has only the −1 eigenvalue. The term inversion should not be confused with inversive geometry, where inversion is defined with respect to a circle In two dimensions, a point reflection is the same as a rotation of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation composed with reflection across a plane perpendicular to the axis of rotation, in dimension n, point reflections are orientation-preserving if n is even, and orientation-reversing if n is odd. Given a vector a in the Euclidean space Rn, the formula for the reflection of a across the point p is R e f p =2 p − a, in the case where p is the origin, point reflection is simply the negation of the vector a. In Euclidean geometry, the inversion of a point X with respect to a point P is a point X* such that P is the midpoint of the segment with endpoints X. In other words, the vector from X to P is the same as the vector from P to X*, the formula for the inversion in P is x*=2a−x where a, x and x* are the position vectors of P, X and X* respectively. This mapping is an isometric involutive affine transformation which has one fixed point. When the inversion point P coincides with the origin, point reflection is equivalent to a case of uniform scaling. This is an example of linear transformation, when P does not coincide with the origin, point reflection is equivalent to a special case of homothetic transformation, homothety with homothetic center coinciding with P, and scale factor = -1. This is an example of non-linear affine transformation), the composition of two point reflections is a translation
Central symmetry
–
A point reflection in 2 dimensions is the same as a 180° rotation.
13.
Line segment
–
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
Line segment
–
historical image – create a line segment (1699)
14.
Surface of revolution
–
A surface of revolution is a surface in Euclidean space created by rotating a curve around an axis of rotation. Examples of surfaces of revolution generated by a line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. The sections of the surface of revolution made by planes through the axis are called meridional sections, any meridional section can be considered to be the generatrix in the plane determined by it and the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles, some special cases of hyperboloids and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular and this formula is the calculus equivalent of Pappuss centroid theorem. The quantity 2 +2 comes from the Pythagorean theorem and represents a segment of the arc of the curve. The quantity 2πx is the path of this segment, as required by Pappus theorem. Likewise, when the axis of rotation is the x-axis and provided that y is never negative and these come from the above formula. For example, the surface with unit radius is generated by the curve y = sin, x = cos. Its area is therefore A =2 π ∫0 π sin 2 +2 d t =2 π ∫0 π sin d t =4 π. A basic problem in the calculus of variations is finding the curve between two points that produces this surface of revolution. There are only two minimal surfaces of revolution, the plane and the catenoid, geodesics on a surface of revolution are governed by Clairauts relation. A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid, for example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus, the use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to surface area without the use of measuring the length
Surface of revolution
–
A portion of the curve x =2+cos z rotated around the z axis
15.
Rotation
–
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object always rotates around a line called a rotation axis. If the axis passes through the center of mass, the body is said to rotate upon itself. A rotation about a point, e. g. the Earth about the Sun, is called a revolution or orbital revolution. The axis is called a pole, mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two, a rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion, the axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit, there is no fundamental difference between a “rotation” and an “orbit” and or spin. The key distinction is simply where the axis of the rotation lies and this distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a rotation around the same point/axis. The reverse of a rotation is also a rotation, thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis and that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the rotations are known as yaw, pitch. This terminology is used in computer graphics. In astronomy, rotation is an observed phenomenon. Stars, planets and similar bodies all spin around on their axes, the rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features and this rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator
Rotation
–
Star trails caused by the
Earth's rotation during the
camera's long exposure time.
Rotation
–
A
sphere rotating about an axis
16.
Oblate spheroid
–
A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes, in other words, an ellipsoid with two equal semi-diameters. If the ellipse is rotated about its axis, the result is a prolate spheroid. If the ellipse is rotated about its axis, the result is an oblate spheroid. If the generating ellipse is a circle, the result is a sphere, because of the combined effects of gravity and rotation, the Earths shape is not quite a sphere but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere, the current World Geodetic System model uses a spheroid whose radius is 6,378.137 km at the equator and 6,356.752 km at the poles. The semi-major axis a is the radius of the spheroid. There are two cases, c < a, oblate spheroid c > a, prolate spheroid The case of a = c reduces to a sphere. An oblate spheroid with c < a has surface area S o b l a t e =2 π a 2 where e 2 =1 − c 2 a 2. The oblate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. A prolate spheroid with c > a has surface area S p r o l a t e =2 π a 2 where e 2 =1 − a 2 c 2. The prolate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis c and semi-minor axis a and these formulas are identical in the sense that the formula for Soblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with the eccentricity, both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse. The volume inside a spheroid is 4π/3a2c ≈4. 19a2c, if A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is π/6A2C ≈0. 523A2C. Both of these curvatures are always positive, so every point on a spheroid is elliptic. These are just two of different parameters used to define an ellipse and its solid body counterparts. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate and oblate spheroidal, deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects. An extreme example of a planet in science fiction is Mesklin, in Hal Clements novel Mission of Gravity. The prolate spheroid is the shape of the ball in several sports, several moons of the Solar system approximate prolate spheroids in shape, though they are actually triaxial ellipsoids
Oblate spheroid
–
oblate spheroid
17.
Prolate spheroid
–
A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes, in other words, an ellipsoid with two equal semi-diameters. If the ellipse is rotated about its axis, the result is a prolate spheroid. If the ellipse is rotated about its axis, the result is an oblate spheroid. If the generating ellipse is a circle, the result is a sphere, because of the combined effects of gravity and rotation, the Earths shape is not quite a sphere but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere, the current World Geodetic System model uses a spheroid whose radius is 6,378.137 km at the equator and 6,356.752 km at the poles. The semi-major axis a is the radius of the spheroid. There are two cases, c < a, oblate spheroid c > a, prolate spheroid The case of a = c reduces to a sphere. An oblate spheroid with c < a has surface area S o b l a t e =2 π a 2 where e 2 =1 − c 2 a 2. The oblate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. A prolate spheroid with c > a has surface area S p r o l a t e =2 π a 2 where e 2 =1 − a 2 c 2. The prolate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis c and semi-minor axis a and these formulas are identical in the sense that the formula for Soblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with the eccentricity, both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse. The volume inside a spheroid is 4π/3a2c ≈4. 19a2c, if A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is π/6A2C ≈0. 523A2C. Both of these curvatures are always positive, so every point on a spheroid is elliptic. These are just two of different parameters used to define an ellipse and its solid body counterparts. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate and oblate spheroidal, deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects. An extreme example of a planet in science fiction is Mesklin, in Hal Clements novel Mission of Gravity. The prolate spheroid is the shape of the ball in several sports, several moons of the Solar system approximate prolate spheroids in shape, though they are actually triaxial ellipsoids
Prolate spheroid
–
oblate spheroid
18.
Cartesian coordinate system
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
Cartesian coordinate system
–
The
right hand rule.
Cartesian coordinate system
–
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinate system
–
3D Cartesian Coordinate Handedness
19.
Implicit surface
–
In mathematics an implicit surface is a surface in Euclidean space defined by an equation F =0 An implicit surface is the set of zeros of a function of 3 variables. Implicit means, that the equation is not solved for x or y or z, the graph of a function is usually described by an equation z = f and is called an explicit representation. The change of representations is usually simple only, when the explicit representation z = f is given, examples, plane x +2 y −3 z +1 =0. Sphere x 2 + y 2 + z 2 −4 =0, surface of genus 2,2 y +2 − =0. Surface of revolution x 2 + y 2 −2 −0.02 =0, for a plane, a sphere and a torus there exist simple parametric representations. This is not true for the 4, the implicit function theorem describes conditions, under which an equation F =0 can be solved for x, y or z. But in general the solution may not be feasible and this theorem is the key to the computation of essential geometric features of a surface, tangent planes, surface normals, curvatures. But they have a drawback, their visualization is difficult. If F is polynomial in x, y and z, the surface is called algebraic, despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically and practically interesting surfaces. Throughout the following considerations the implicit surface is represented by an equation F =0 where function F meets the conditions of differentiability. The partial derivatives of F are F x, a surface point is called regular, if ≠, otherwise the point is singular. The equation of the tangent plane at a point is F x + F y + F z =0. The proof of this relies on the implicit function theorem. As in the case of implicit curves it is a task to generate implicit surfaces with desired shapes by applying algebraic operations on simple primitives. The electrical potential of a point charge q i at point p i = generates at point p = the potential F i = q i ∥ p − p i ∥. The equipotential surface for the value c is the implicit surface F i − c =0 which is a sphere with center at point p i. For the picture the four charges equal 1 and are located at the points, the displayed surface is the equipotential surface F −2.8 =0. A Cassini oval can be defined as the point set for which the product of the distances to two points is constant
Implicit surface
–
implicit surface torus (R=40, a=15)
20.
Real number
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
Real number
–
A symbol of the set of real numbers (ℝ)
21.
Semi-major axis
–
In geometry, the major axis of an ellipse is its longest diameter, a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the axis, and thus runs from the centre, through a focus. Essentially, it is the radius of an orbit at the two most distant points. For the special case of a circle, the axis is the radius. One can think of the axis as an ellipses long radius. The semi-major axis of a hyperbola is, depending on the convention, thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction. Thus a and b tend to infinity, a faster than b, the semi-minor axis is a line segment associated with most conic sections that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve, in an ellipse, the one, in a hyperbola. The semi-major axis is the value of the maximum and minimum distances r max and r min of the ellipse from a focus — that is. In astronomy these extreme points are called apsis, the semi-minor axis of an ellipse is the geometric mean of these distances, b = r max r min. The eccentricity of an ellipse is defined as e =1 − b 2 a 2 so r min = a, r max = a. Now consider the equation in polar coordinates, with one focus at the origin, the mean value of r = ℓ / and r = ℓ /, for θ = π and θ =0 is a = ℓ1 − e 2. In an ellipse, the axis is the geometric mean of the distance from the center to either focus. The semi-minor axis of an ellipse runs from the center of the ellipse to the edge of the ellipse, the semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the axis that connects two points on the ellipses edge. The semi-minor axis b is related to the axis a through the eccentricity e. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction
Semi-major axis
–
The semi-major and semi-minor axis of an ellipse
22.
Semi-minor axis
–
In geometry, the major axis of an ellipse is its longest diameter, a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the axis, and thus runs from the centre, through a focus. Essentially, it is the radius of an orbit at the two most distant points. For the special case of a circle, the axis is the radius. One can think of the axis as an ellipses long radius. The semi-major axis of a hyperbola is, depending on the convention, thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction. Thus a and b tend to infinity, a faster than b, the semi-minor axis is a line segment associated with most conic sections that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve, in an ellipse, the one, in a hyperbola. The semi-major axis is the value of the maximum and minimum distances r max and r min of the ellipse from a focus — that is. In astronomy these extreme points are called apsis, the semi-minor axis of an ellipse is the geometric mean of these distances, b = r max r min. The eccentricity of an ellipse is defined as e =1 − b 2 a 2 so r min = a, r max = a. Now consider the equation in polar coordinates, with one focus at the origin, the mean value of r = ℓ / and r = ℓ /, for θ = π and θ =0 is a = ℓ1 − e 2. In an ellipse, the axis is the geometric mean of the distance from the center to either focus. The semi-minor axis of an ellipse runs from the center of the ellipse to the edge of the ellipse, the semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the axis that connects two points on the ellipses edge. The semi-minor axis b is related to the axis a through the eccentricity e. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction
Semi-minor axis
–
The semi-major axis (in red) and semi-minor axis (in blue) of an ellipse.
23.
Spherical coordinates
–
It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal angle, the use of symbols and the order of the coordinates differs between sources. In both systems ρ is often used instead of r, other conventions are also used, so great care needs to be taken to check which one is being used. A number of different spherical coordinate systems following other conventions are used outside mathematics, in a geographical coordinate system positions are measured in latitude, longitude and height or altitude. There are a number of different celestial coordinate systems based on different fundamental planes, the polar angle is often replaced by the elevation angle measured from the reference plane. Elevation angle of zero is at the horizon, the spherical coordinate system generalises the two-dimensional polar coordinate system. It can also be extended to spaces and is then referred to as a hyperspherical coordinate system. To define a coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, and an origin point in space. These choices determine a plane that contains the origin and is perpendicular to the zenith. The spherical coordinates of a point P are then defined as follows, the inclination is the angle between the zenith direction and the line segment OP. The azimuth is the angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane. The sign of the azimuth is determined by choosing what is a sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate systems definition, the elevation angle is 90 degrees minus the inclination angle. If the inclination is zero or 180 degrees, the azimuth is arbitrary, if the radius is zero, both azimuth and inclination are arbitrary. In linear algebra, the vector from the origin O to the point P is often called the vector of P. Several different conventions exist for representing the three coordinates, and for the order in which they should be written. The use of to denote radial distance, inclination, and azimuth, respectively, is common practice in physics, and is specified by ISO standard 80000-2,2009, and earlier in ISO 31-11
Spherical coordinates
–
Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (
theta), and azimuthal angle φ (
phi). The symbol ρ (
rho) is often used instead of r.
24.
Euclidean vector
–
In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years, about a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence, working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
Euclidean vector
25.
Eigenvector
–
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it. This condition can be written as the equation T = λ v, there is a correspondence between n by n square matrices and linear transformations from an n-dimensional vector space to itself. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. Geometrically an eigenvector, corresponding to a real eigenvalue, points in a direction that is stretched by the transformation. If the eigenvalue is negative, the direction is reversed, Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen for proper, inherent, own, individual, special, specific, peculiar, or characteristic. In essence, an eigenvector v of a linear transformation T is a vector that. Applying T to the eigenvector only scales the eigenvector by the scalar value λ and this condition can be written as the equation T = λ v, referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar, for example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex. The Mona Lisa example pictured at right provides a simple illustration, each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping, the vectors pointing to each point in the original image are therefore tilted right or left and made longer or shorter by the transformation. Notice that points along the horizontal axis do not move at all when this transformation is applied, therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation because the mapping does not change its direction. Moreover, these all have an eigenvalue equal to one because the mapping does not change their length. Linear transformations can take different forms, mapping vectors in a variety of vector spaces. Alternatively, the transformation could take the form of an n by n matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix, the set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T. If the set of eigenvectors of T form a basis of the domain of T, Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms, in the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes
Eigenvector
–
In this
shear mapping the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it doesn't change direction, and since its length is unchanged, its eigenvalue is 1.
26.
Eigenvalue
–
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it. This condition can be written as the equation T = λ v, there is a correspondence between n by n square matrices and linear transformations from an n-dimensional vector space to itself. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. Geometrically an eigenvector, corresponding to a real eigenvalue, points in a direction that is stretched by the transformation. If the eigenvalue is negative, the direction is reversed, Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen for proper, inherent, own, individual, special, specific, peculiar, or characteristic. In essence, an eigenvector v of a linear transformation T is a vector that. Applying T to the eigenvector only scales the eigenvector by the scalar value λ and this condition can be written as the equation T = λ v, referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar, for example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex. The Mona Lisa example pictured at right provides a simple illustration, each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping, the vectors pointing to each point in the original image are therefore tilted right or left and made longer or shorter by the transformation. Notice that points along the horizontal axis do not move at all when this transformation is applied, therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation because the mapping does not change its direction. Moreover, these all have an eigenvalue equal to one because the mapping does not change their length. Linear transformations can take different forms, mapping vectors in a variety of vector spaces. Alternatively, the transformation could take the form of an n by n matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix, the set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T. If the set of eigenvectors of T form a basis of the domain of T, Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms, in the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes
Eigenvalue
–
In this
shear mapping the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it doesn't change direction, and since its length is unchanged, its eigenvalue is 1.
27.
Linear transformation
–
In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE
Linear transformation
–
"Linear transformation" redirects here. For fractional linear transformations, see
Möbius transformation.
28.
Singular value decomposition
–
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix to any m × n matrix via an extension of polar decomposition and it has many useful applications in signal processing and statistics. The diagonal entries σ i of Σ are known as the values of M. The columns of U and the columns of V are called the left-singular vectors and right-singular vectors of M, the singular value decomposition can be computed using the following observations, The left-singular vectors of M are a set of orthonormal eigenvectors of MM∗. The right-singular vectors of M are a set of eigenvectors of M∗M. The non-zero singular values of M are the roots of the non-zero eigenvalues of both M∗M and MM∗. Suppose M is a m × n matrix whose entries come from the field K, V∗ is the conjugate transpose of the n × n unitary matrix, V, thus also unitary. The diagonal entries σi of Σ are known as the values of M. A common convention is to list the singular values in descending order, in this case, the diagonal matrix, Σ, is uniquely determined by M. Thus the expression UΣV∗ can be interpreted as a composition of three geometrical transformations, a rotation or reflection, a scaling, and another rotation or reflection. For instance, the figure above explains how a matrix can be described as such a sequence. If the rotation is done first, M = PR, then R is the same and P = UΣU∗ has the same eigenvalues and this shows that the SVD is a generalization of the eigenvalue decomposition of pure stretches in orthogonal directions to arbitrary matrices which both stretch and rotate. As shown in the figure, the values can be interpreted as the semiaxes of an ellipse in 2D. This concept can be generalized to n-dimensional Euclidean space, with the values of any n × n square matrix being viewed as the semiaxes of an n-dimensional ellipsoid. Since U and V∗ are unitary, the columns of each of them form a set of orthonormal vectors, the matrix M maps the basis vector Vi to the stretched unit vector σi Ui. By the definition of a matrix, the same is true for their conjugate transposes U∗ and V. In short, the columns of U, U∗, V, and V∗ are orthonormal bases. Consider the 4 ×5 matrix M = A singular value decomposition of this matrix is given by UΣV∗ U = Σ = V ∗ = Notice Σ is zero outside of the diagonal and one diagonal element is zero
Singular value decomposition
Singular value decomposition
29.
Volume
–
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
Volume
–
A
measuring cup can be used to measure volumes of
liquids. This cup measures volume in units of
cups,
fluid ounces, and
millilitres.
30.
Circumscribed
–
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Circumscribed
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
31.
Elliptic cylinder
–
In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
Elliptic cylinder
–
Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section
Elliptic cylinder
–
A right circular cylinder with radius r and height h.
Elliptic cylinder
–
In
projective geometry, a cylinder is simply a cone whose
apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
32.
Volumes
–
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
Volumes
–
A
measuring cup can be used to measure volumes of
liquids. This cup measures volume in units of
cups,
fluid ounces, and
millilitres.
33.
Inscribed
–
In geometry, an inscribed planar shape or solid is one that is enclosed by and fits snugly inside another geometric shape or solid. To say that figure F is inscribed in figure G means precisely the same thing as figure G is circumscribed about figure F, a circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, a circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a polygon. The inradius or filling radius of a given outer figure is the radius of the circle or sphere. The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces. For an alternative usage of the term inscribed, see the inscribed square problem, every circle has an inscribed triangle with any three given angle measures, and every triangle can be inscribed in some circle. Every triangle has a circle, called the incircle. Every circle has a regular polygon of n sides, for any n≥3. Every regular polygon has a circle, and every circle can be inscribed in some regular polygon of n sides. Not every polygon with more than three sides has a circle, those polygons that do are called tangential polygons. Not every polygon with more than three sides is a polygon of a circle, those polygons that are so inscribed are called cyclic polygons. Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangles centroid, every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides, every acute triangle has three inscribed squares. In a right triangle two of them are merged and coincide with other, so there are only two distinct inscribed squares. An obtuse triangle has an inscribed square, with one side coinciding with part of the triangles longest side. A Reuleaux triangle, or more generally any curve of constant width, circumconic and inconic Cyclic quadrilateral Inscribed and circumscribed figures
Inscribed
–
Inscribed circles of various polygons
34.
Rectangular cuboid
–
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. By Eulers formula the numbers of faces F, of vertices V, in the case of a cuboid this gives 6 +8 =12 +2, that is, like a cube, a cuboid has 6 faces,8 vertices, and 12 edges. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, in a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a rectangular prism, and the terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms rectangular prism and oblong prism, however, are ambiguous, the square cuboid, square box, or right square prism is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol ×, and its symmetry is doubled from to, the cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol, and its symmetry is raised from, to, if the dimensions of a rectangular cuboid are a, b and c, then its volume is abc and its surface area is 2. The length of the diagonal is d = a 2 + b 2 + c 2. Cuboid shapes are used for boxes, cupboards, rooms, buildings. Cuboids are among those solids that can tessellate 3-dimensional space, the shape is fairly versatile in being able to contain multiple smaller cuboids, e. g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building. A cuboid with integer edges as well as integer face diagonals is called an Euler brick, a perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists, the number of different nets for a simple cube is 11, however this number increases significantly to 54 for a rectangular cuboid of 3 different lengths. Hyperrectangle Trapezohedron Weisstein, Eric W. Cuboid, rectangular prism and cuboid Paper models and pictures
Rectangular cuboid
–
Rectangular cuboid
35.
Surface area
–
The surface area of a solid object is a measure of the total area that the surface of the object occupies. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces and this definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of area was sought by Henri Lebesgue. Their work led to the development of measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface, while the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function S ↦ A which assigns a real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the area is its additivity. More rigorously, if a surface S is a union of many pieces S1, …, Sr which do not overlap except at their boundaries. Surface areas of polygonal shapes must agree with their geometrically defined area. Since surface area is a notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface. This means that surface area is invariant under the group of Euclidean motions and these properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of many pieces that can be represented in the parametric form S D, r → = r →, ∈ D with a continuously differentiable function r →. The area of a piece is defined by the formula A = ∬ D | r → u × r → v | d u d v. Thus the area of SD is obtained by integrating the length of the vector r → u × r → v to the surface over the appropriate region D in the parametric uv plane. The area of the surface is then obtained by adding together the areas of the pieces. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f and surfaces of revolution. It was demonstrated by Hermann Schwarz that already for the cylinder, various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a natural notion of surface area, if a surface is very irregular, or rough
Surface area
–
The inner membrane of the
mitochondrion has a large surface area due to infoldings, allowing higher rates of
cellular respiration (electron
micrograph).
Surface area
–
A
sphere of radius has surface area
36.
Elliptic integral
–
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler, in general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this rule are when P has repeated roots. However, with the reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions. Besides the Legendre form given below, the integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals, incomplete elliptic integrals are functions of two arguments, complete elliptic integrals are functions of a single argument. These arguments are expressed in a variety of different but equivalent ways, most texts adhere to a canonical naming scheme, using the following naming conventions. Thus, they can be used interchangeably, the other argument can likewise be expressed as φ, the amplitude, or as x or u, where x = sin φ = sn u and sn is one of the Jacobian elliptic functions. Specifying the value of any one of these determines the others. Note that u also depends on m, some additional relationships involving u include cos φ = cn u, and 1 − m sin 2 φ = dn u. The latter is called the delta amplitude and written as Δ = dn u. Sometimes the literature refers to the complementary parameter, the complementary modulus. These are further defined in the article on quarter periods, the incomplete elliptic integral of the first kind F is defined as F = F = F = ∫0 φ d θ1 − k 2 sin 2 θ. This is the form of the integral, substituting t = sin θ and x = sin φ, one obtains Jacobis form. Equivalently, in terms of the amplitude and modular angle one has, in this notation, the use of a vertical bar as delimiter indicates that the argument following it is the parameter, while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude, with x = sn one has, F = u, thus, the Jacobian elliptic functions are inverses to the elliptic integrals. There are still other conventions for the notation of elliptic integrals employed in the literature, the notation with interchanged arguments, F, is often encountered, and similarly E for the integral of the second kind. e
Elliptic integral
–
Plot of the complete elliptic integral of the first kind
37.
Eccentricity (mathematics)
–
In mathematics, the eccentricity, denoted e or ε, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular, in particular, The eccentricity of a circle is zero. The eccentricity of an ellipse which is not a circle is greater than zero, the eccentricity of a parabola is 1. The eccentricity of a hyperbola is greater than 1, the eccentricity of a line is infinite. Furthermore, two sections are similar if and only if they have the same eccentricity. Any conic section can be defined as the locus of points whose distances to a point and that ratio is called eccentricity, commonly denoted as e. The eccentricity can also be defined in terms of the intersection of a plane, for β =0 the plane section is a circle, for β = α a parabola. The linear eccentricity of an ellipse or hyperbola, denoted c, is the distance between its center and either of its two foci, the eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a, that is, e = c a. The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity, the eccentricity is also sometimes called numerical eccentricity. In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation, three notational conventions are in common use, e for the eccentricity and c for the linear eccentricity. ε for the eccentricity and e for the linear eccentricity, E or ϵ for the eccentricity and f for the linear eccentricity. This article uses the first notation, where, for the ellipse and the hyperbola, a is the length of the semi-major axis and b is the length of the semi-minor axis. The eccentricity of an ellipse is strictly less than 1, for any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis. The eccentricity is also the ratio of the axis a to the distance d from the center to the directrix. The eccentricity can be expressed in terms of the g, e = g. Define the maximum and minimum radii r max and r min as the maximum and minimum distances from either focus to the ellipse, the eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a hyperbola is 2. The eccentricity of a quadric is the eccentricity of a designated section of it
Eccentricity (mathematics)
–
All types of conic sections, arranged with increasing eccentricity. Note that curvature decreases with eccentricity, and that none of these curves intersect.
38.
Earth ellipsoid
–
An Earth ellipsoid is a mathematical figure approximating the shape of the Earth, used as a reference frame for computations in geodesy, astronomy and the geosciences. Various different ellipsoids have been used as approximations and it is an ellipsoid of revolution, whose short axis is approximately aligned with the rotation axis of the Earth. The ellipsoid is defined by the axis a and the polar axis b. Additional parameters are the mass function J2, the correspondent gravity formula, many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks. Amongst the different set of used in national surveys are several of special importance, the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924. A data set describes the global average of the Earths surface curvature is called the mean Earth Ellipsoid. It refers to a theoretical coherence between the latitude and the meridional curvature of the geoid. The latter is close to the sea level, and therefore an ideal Earth ellipsoid has the same volume as the geoid. While the mean Earth ellipsoid is the basis of global geodesy. Another reason is a one, the coordinates of millions of boundary stones should remain fixed for a long period. If their reference surface changes, the coordinates themselves also change, however, for international networks, GPS positioning, or astronautics, these regional reasons are less relevant. As knowledge of the Earths figure is accurate, the International Geoscientific Union IUGG usually adapts the axes of the Earth ellipsoid to the best available data. High precision land surveys can be used to determine the distance between two places at nearly the same longitude by measuring a base line and a chain of triangles. The distance Δ along the meridian from one end point to a point at the latitude as the second end point is then calculated by trigonometry. The surface distance Δ is reduced to Δ, the distance at mean sea level. The intermediate distances to points on the meridian at the same latitudes as other stations of the survey may also be calculated. The geographic latitudes of both end points, φs and φf and possibly at other points are determined by astrogeodesy, if latitudes are measured at end points only, the radius of curvature at the midpoint of the meridian arc can be calculated from R = Δ/. A second meridian arc will allow the derivation of two parameters required to specify a reference ellipsoid, longer arcs with intermediate latitude determinations can completely determine the ellipsoid
Earth ellipsoid
–
Geodesy
39.
Reference ellipsoid
–
In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Current practice uses the word alone in preference to the full term oblate ellipsoid of revolution or the older term oblate spheroid. In the rare instances where a more general shape is required as a model the term used is triaxial ellipsoid. A great many ellipsoids have been used with various sizes and centres, the shape of an ellipsoid is determined by the shape parameters of that ellipse which generates the ellipsoid when it is rotated about its minor axis. The semi-major axis of the ellipse, a, is identified as the radius of the ellipsoid. For the Earth, f is around 1/300 corresponding to a difference of the major and minor semi-axes of approximately 21 km, some precise values are given in the table below and also in Figure of the Earth. A great many other parameters are used in geodesy but they can all be related to one or two of the set a, b and f, a primary use of reference ellipsoids is to serve as a basis for a coordinate system of latitude, longitude, and elevation. For this purpose it is necessary to identify a zero meridian, for other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the crater Airy-0. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid, the longitude measures the rotational angle between the zero meridian and the measured point. By convention for the Earth, Moon, and Sun it is expressed in degrees ranging from −180° to +180° For other bodies a range of 0° to 360° is used. The latitude measures how close to the poles or equator a point is along a meridian, and is represented as an angle from −90° to +90°, the common or geodetic latitude is the angle between the equatorial plane and a line that is normal to the reference ellipsoid. Depending on the flattening, it may be different from the geocentric latitude. For non-Earth bodies the terms planetographic and planetocentric are used instead, see geodetic system for more detail. If these coordinates, i. e. N is the radius of curvature in the prime vertical, in contrast, extracting φ, λ and h from the rectangular coordinates usually requires iteration. A straightforward method is given in an OSGB publication and also in web notes, more sophisticated methods are outlined in geodetic system. Currently the most common reference used, and that used in the context of the Global Positioning System, is the one defined by WGS84. Traditional reference ellipsoids or geodetic datums are defined regionally and therefore non-geocentric, modern geodetic datums are established with the aid of GPS and will therefore be geocentric, e. g. WGS84. Reference ellipsoids are also useful for mapping of other planetary bodies including planets, their satellites, asteroids
Reference ellipsoid
–
Flattened sphere
40.
Rigid body
–
In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, in classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules are often seen as rigid bodies, the position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, if the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, however, typically a different, mathematically more convenient, but equivalent approach is used. Thus, the position of a body has two components, linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a body, such as linear and angular velocity, acceleration, momentum, impulse. This reference point may define the origin of a coordinate system fixed to the body, there are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix. In general, when a body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position. Velocity and angular velocity are measured with respect to a frame of reference, the linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a point fixed to the body. During purely translational motion, all points on a body move with the same velocity. However, when motion involves rotation, the velocity of any two points on the body will generally not be the same. Two points of a body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a quantity that describes the angular speed at which the orientation of the rigid body is changing. All points on a rigid body experience the same velocity at all times
Rigid body
–
The position of a rigid body is determined by the position of its center of mass and by its
attitude (at least six parameters in total).
41.
Ellipsoidal reflector floodlight
–
In stage lighting, an ellipsoidal reflector floodlight, better known as a scoop, is a large, simple lighting fixture with a dome-like reflector, large high-wattage lamp and no lens. It consists almost entirely of a lamp in the center of a big curved metal dome that acts as a reflector, the result is a wide, soft-edged pool of light good for general lighting. However, since scoop lights do not have a mechanism for cutting down the size of their beam, many theaters use scoop lights for worklights, rehearsals, non-performance times, and certain performance times. Scoops can be used to mimic the effect of a striplight to illuminate a cyclorama and they are easy to set up and take down, are relatively inexpensive, typically have long lamp life. When used as worklights, scoop lights frequently dont require the use of the board to operate. Scoop lights are used for flooding a stage with downlight. However, fresnel lanterns are more used for this task. Most scoops use a PS52 mogul screw incandescent lamp
Ellipsoidal reflector floodlight
–
Scoop
42.
Ellipsoidal reflector spotlight
–
The optics of an ERS instrument are roughly similar to those of a 35 mm slide projector. There are many types of ERS that are designed for the applications found in the entertainment industry. ERS instruments come in all shapes and sizes, each particular model of ERS has its own set of characteristics. Generally, ERS instruments are the most varied and utilized type of stage lighting instrument, ERS may also be referred to as Profile Spotlights because the beam can be shaped to the profile of an object. Ellipsoidal reflectors are used for their strong, well-defined light and their versatility, leko and Source Four are brand names which are often, but inaccurately, used to refer to any sort of ellipsoidal. Characteristics of a typical ellipsoidal lighting unit include, An ellipsoidal reflector An adjustable lens tube containing the lens or lens train. Adjusting the tube by pushing it further in or pulling it out of the unit allows changes to the focus of the beam of light projected by the unit. This results from changing the distance between the reflector and the lens train, Zoom ERS instruments can vary the size of beam as well as the focus One or two Plano-Convex lenses within the lens tube to create the lens train. The Plano-Convex lenses, named for having one side and one convex side, have their convex sides facing each other within the tube. Most modern units include two slots that allow for combining different accessories A series of four shutters mounted at the focal point of the unit. These allow for precise shaping and sizing of the beam as lines. Additionally, an iris may be present to size the beam circularly, the filament of the lamp is at one focal point of the ellipsoidal reflector and the gate with the shutters and gobo are at the other focal point. Ellipsoidals are supplied with a certain size lens or lenses that determines the field angle, field angle is the angle of the beam of light where it reaches 10% of the intensity of the center of the beam. Most manufacturers now use field angle to indicate the fixtures spread typically in this series, older fixtures are described by the width of the lens x focal length of the instrument. For example, a 6x9 ellipsoidal would have a 6 diameter lens, 6x9 Instruments have a field angle of approximately 37°. 6x12 instruments have an angle of approximately 27°. As the field angle narrows, the instrument can be used further from the stage, variable focus instruments with two lenses that move in and out from the lamp housing are also available, allowing the user to manually adjust to the desired focal length within a certain range. Ellipsoidals can be used for any job but their function is to illuminate a specific proximity
Ellipsoidal reflector spotlight
–
A Colortran ERS.
Ellipsoidal reflector spotlight
–
An Elipsoidal Reflector from a Leko
Ellipsoidal reflector spotlight
–
Source Four ERS
Ellipsoidal reflector spotlight
–
Niethammer Enizoom ERS with variable focus (Zoom).
43.
Mass
–
In physics, mass is a property of a physical body. It is the measure of a resistance to acceleration when a net force is applied. It also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram, Mass is not the same as weight, even though mass is often determined by measuring the objects weight using a spring scale, rather than comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity and this is because weight is a force, while mass is the property that determines the strength of this force. In Newtonian physics, mass can be generalized as the amount of matter in an object, however, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy. In addition, matter is a defined term in science. There are several distinct phenomena which can be used to measure mass, active gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force, according to Newtons second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A bodys mass also determines the degree to which it generates or is affected by a gravitational field and this is sometimes referred to as gravitational mass. The standard International System of Units unit of mass is the kilogram, the kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the prototype kilogram. As of January 2013, there are proposals for redefining the kilogram yet again. In this context, the mass has units of eV/c2, the electronvolt and its multiples, such as the MeV, are commonly used in particle physics. The atomic mass unit is 1/12 of the mass of a carbon-12 atom, the atomic mass unit is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include, the slug is an Imperial unit of mass, the pound is a unit of both mass and force, used mainly in the United States
Mass
–
Depiction of early
balance scales in the
Papyrus of Hunefer (dated to the
19th dynasty, ca. 1285 BC). The scene shows
Anubis weighing the heart of Hunefer.
Mass
–
The kilogram is one of the seven
SI base units and one of three which is defined ad hoc (i.e. without reference to another base unit).
Mass
–
Galileo Galilei (1636)
Mass
–
Distance traveled by a freely falling ball is proportional to the square of the elapsed time
44.
Moment of Inertia
–
It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change the bodys rotation. It is a property, the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. One of its definitions is the moment of mass with respect to distance from an axis r, I = ∫ Q r 2 d m. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about a perpendicular to the plane. When a body is rotating, or free to rotate, around an axis, the amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared in SI units, moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, for a point-like mass, the moment of inertia about some axis is given by mr2, where r is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a shape and uniform density. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, the term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Eulers second law. Comparison of this frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, and in Newtons laws of motion for a rigid body as a physical parameter that combines its shape. There is a difference in the way moment of inertia appears in planar. The moment of inertia of a flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, if the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their arms or divers curl their bodies into a tuck position during a dive. For a simple pendulum, this yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as. Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation
Moment of Inertia
–
Tightrope walker Samuel Dixon using the long rod's moment of inertia for balance while crossing the
Niagara River in 1890.
Moment of Inertia
–
Flywheels have large moments of inertia to smooth out mechanical motion. This example is in a Russian museum.
Moment of Inertia
–
Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to
conservation of angular momentum.
Moment of Inertia
–
Pendulums used in Mendenhall
gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.
45.
Haumea
–
Haumea, minor-planet designation 136108 Haumea, is a dwarf planet located beyond Neptunes orbit. On September 17,2008, it was recognized as a planet by the International Astronomical Union and named after Haumea. Haumeas mass is about one-third that of Pluto, and 1/1400 that of Earth, although its shape has not been directly observed, calculations from its light curve indicate that it is a triaxial ellipsoid, with its major axis twice as long as its minor. Its gravity is thought to be sufficient for it to have relaxed into hydrostatic equilibrium, two teams claim credit for the discovery of Haumea. Mike Brown and his team at Caltech discovered Haumea in December 2004 on images they had taken on May 6,2004, on July 20,2005, they published an online abstract of a report intended to announce the discovery at a conference in September 2005. At around this time, José Luis Ortiz Moreno and his team at the Instituto de Astrofísica de Andalucía at Sierra Nevada Observatory in Spain found Haumea on images taken on March 7–10,2003. Ortiz emailed the Minor Planet Center with their discovery on the night of July 27,2005, Ortiz later admitted he had accessed the Caltech observation logs but denied any wrongdoing, stating he was merely verifying whether they had discovered a new object. However, the IAU announcement on September 17,2008, that Haumea had been accepted as a dwarf planet, did not mention a discoverer. Until it was given a permanent name, the Caltech discovery team used the nickname Santa among themselves, because they had discovered Haumea on December 28,2004, the Spanish team were the first to file a claim for discovery to the Minor Planet Center, in July 2005. On July 29,2005, Haumea was given the provisional designation 2003 EL61, on September 7,2006, it was numbered and admitted into the official minor planet catalogue as 2003 EL61. The names were proposed by David Rabinowitz of the Caltech team, Haumea is the matron goddess of the island of Hawaiʻi, where the Mauna Kea Observatory is located. The two known moons, also believed to have formed in this manner, are named after two of Haumeas daughters, Hiʻiaka and Nāmaka. The proposal by the Ortiz team, Ataecina, did not meet IAU naming requirements, Haumea is a plutoid, a dwarf planet beyond Neptunes orbit. Haumea appears to have an ellipsoid shape resulting its rapid rotation complicated by tidal interactions with its moons. This contrasts with the simpler oblate shape typically assumed by less rapidly rotating astronomical bodies such as the Earth, in other words, Haumea is spinning so fast that if it spun much faster these bulges would distort into a dumbbell shape and split the planet in two. Haumea was initially listed as a classical Kuiper belt object in 2006 by the Minor Planet Center, the nominal trajectory suggests that it is in the weak 7,12 resonance with Neptune, because its perihelion distance of 35 AU is near the limit of stability with Neptune. There are precovery images of Haumea dating back to March 22,1955 from the Palomar Mountain Digitized Sky Survey, further observations of the orbit will be required to verify its dynamic status. Haumea has a period of 284 Earth years, a perihelion of 35 AU
Haumea
–
Keck image of Haumea and its two moons.
Hiʻiaka is above Haumea (center), and
Namaka is directly below.
Haumea
Haumea
–
Artist's conception of Haumea with its moons
Hiʻiaka and
Namaka. The moons are much more distant than depicted here.
Haumea
–
Asteroid belt
46.
Dwarf planet
–
A dwarf planet is a planetary-mass object that is neither a planet nor a natural satellite. The International Astronomical Union currently recognizes five dwarf planets, Ceres, Pluto, Haumea, Makemake, another hundred or so known objects in the Solar System are suspected to be dwarf planets. Individual astronomers recognize several of these, and in August 2011 Mike Brown published a list of 390 candidate objects, Stern states that there are more than a dozen known dwarf planets. Only two of these bodies, Ceres and Pluto, have observed in enough detail to demonstrate that they actually fit the IAUs definition. The IAU accepted Eris as a dwarf planet because it is more massive than Pluto and they subsequently decided that unnamed trans-Neptunian objects with an absolute magnitude brighter than +1 are to be named under the assumption that they are dwarf planets. The classification of bodies in other systems with the characteristics of dwarf planets has not been addressed. Starting in 1801, astronomers discovered Ceres and other bodies between Mars and Jupiter which were for some decades considered to be planets. Between then and around 1851, when the number of planets had reached 23, astronomers started using the asteroid for the smaller bodies. With the discovery of Pluto in 1930, most astronomers considered the Solar System to have nine planets and it was roughly one-twentieth the mass of Mercury, which made Pluto by far the smallest planet. Although it was more than ten times as massive as the largest object in the asteroid belt, Ceres. In the 1990s, astronomers began to find objects in the region of space as Pluto. Many of these shared several of Plutos key orbital characteristics, and Pluto started being seen as the largest member of a new class of objects and this led some astronomers to stop referring to Pluto as a planet. Several terms, including subplanet and planetoid, started to be used for the now known as dwarf planets. By 2005, three trans-Neptunian objects comparable in size to Pluto had been reported and it became clear that either they would also have to be classified as planets, or Pluto would have to be reclassified. Astronomers were also confident that more objects as large as Pluto would be discovered, Eris was discovered in January 2005, it was thought to be slightly larger than Pluto, and some reports informally referred to it as the tenth planet. As a consequence, the became a matter of intense debate during the IAU General Assembly in August 2006. The IAUs initial draft proposal included Charon, Eris, and Ceres in the list of planets, dropping Charon from the list, the new proposal also removed Pluto, Ceres, and Eris, because they have not cleared their orbits. The IAUs final Resolution 5A preserved this three-category system for the bodies orbiting the Sun
Dwarf planet
Dwarf planet
47.
Cuboid
–
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. By Eulers formula the numbers of faces F, of vertices V, in the case of a cuboid this gives 6 +8 =12 +2, that is, like a cube, a cuboid has 6 faces,8 vertices, and 12 edges. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, in a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a rectangular prism, and the terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms rectangular prism and oblong prism, however, are ambiguous, the square cuboid, square box, or right square prism is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol ×, and its symmetry is doubled from to, the cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol, and its symmetry is raised from, to, if the dimensions of a rectangular cuboid are a, b and c, then its volume is abc and its surface area is 2. The length of the diagonal is d = a 2 + b 2 + c 2. Cuboid shapes are used for boxes, cupboards, rooms, buildings. Cuboids are among those solids that can tessellate 3-dimensional space, the shape is fairly versatile in being able to contain multiple smaller cuboids, e. g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building. A cuboid with integer edges as well as integer face diagonals is called an Euler brick, a perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists, the number of different nets for a simple cube is 11, however this number increases significantly to 54 for a rectangular cuboid of 3 different lengths. Hyperrectangle Trapezohedron Weisstein, Eric W. Cuboid, rectangular prism and cuboid Paper models and pictures
Cuboid
–
Rectangular cuboid
48.
Moment of inertia
–
It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change the bodys rotation. It is a property, the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. One of its definitions is the moment of mass with respect to distance from an axis r, I = ∫ Q r 2 d m. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about a perpendicular to the plane. When a body is rotating, or free to rotate, around an axis, the amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared in SI units, moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, for a point-like mass, the moment of inertia about some axis is given by mr2, where r is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a shape and uniform density. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, the term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Eulers second law. Comparison of this frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, and in Newtons laws of motion for a rigid body as a physical parameter that combines its shape. There is a difference in the way moment of inertia appears in planar. The moment of inertia of a flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, if the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their arms or divers curl their bodies into a tuck position during a dive. For a simple pendulum, this yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as. Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation
Moment of inertia
–
Tightrope walker Samuel Dixon using the long rod's moment of inertia for balance while crossing the
Niagara River in 1890.
Moment of inertia
–
Flywheels have large moments of inertia to smooth out mechanical motion. This example is in a Russian museum.
Moment of inertia
–
Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to
conservation of angular momentum.
Moment of inertia
–
Pendulums used in Mendenhall
gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.
49.
Tidal locking
–
Tidal locking occurs when, over the course of an orbit, there is no net transfer of angular momentum between an astronomical body and its gravitational partner. This state can result from the gradient between two co-orbiting bodies, acting over a sufficiently long period of time. In the case where the orbital eccentricity is zero, tidal locking results in one hemisphere of the revolving object constantly facing its partner. For example, the side of the Moon always faces the Earth. A tidally locked body in synchronous rotation takes just as long to rotate around its own axis as it does to revolve around its partner, usually, only the satellite is tidally locked to the larger body. However, if both the difference between the two bodies and the distance between them are relatively small, each may be tidally locked to the other, this is the case for Pluto. This effect is employed to stabilize some artificial satellites, the possibility of lifeforms existing on tidally-locked planets has been debated. The change in rotation rate necessary to lock a body B to a larger body A is caused by the torque applied by As gravity on bulges it has induced on B by tidal forces. These distortions are known as tidal bulges, when B is not yet tidally locked, the bulges travel over its surface, with one of the two high tidal bulges traveling close to the point where body A is overhead. Smaller bodies also experience distortion, but this distortion is less regular, the material of B exerts resistance to this periodic reshaping caused by the tidal force. In effect, some time is required to reshape B to the equilibrium shape. Seen from a point in space, the points of maximum bulge extension are displaced from the axis oriented toward A. Because the bulges are now displaced from the A–B axis, As gravitational pull on the mass in them exerts a torque on B. The torque on the A-facing bulge acts to bring Bs rotation in line with its period, whereas the back bulge. However, the bulge on the A-facing side is closer to A than the bulge by a distance of approximately Bs diameter. The net resulting torque from both bulges, then, is always in the direction that acts to synchronize Bs rotation with its orbital period and this results in a raising of Bs orbit about A in tandem with its rotational slowdown. For the other case where B starts off rotating too slowly, the tidal locking effect is also experienced by the larger body A, but at a slower rate because Bs gravitational effect is weaker due to Bs smaller mass. For example, Earths rotation is gradually being slowed by the Moon, current estimations are that this has helped lengthen the Earth day from about 6 hours to the current 24 hours
Tidal locking
–
If rotational frequency is larger than orbital frequency, a small torque counteracting the rotation arises, eventually locking the frequencies (situation depicted in green)
Tidal locking
Tidal locking
–
Because the
Moon is 1:1 tidally locked, only
one side is visible from
Earth.