1.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
2.
Quadric
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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
3.
Point reflection
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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
4.
Parabola
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A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
5.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
6.
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
7.
Almost all
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In mathematics, the phrase almost all has a number of specialised uses which extend its intuitive meaning. Almost all is used synonymously with all but finitely many or all but a countable set. A simple example is that almost all numbers are odd. Perversely, if we allow almost all to all but a countable set, then it follows that almost all prime numbers are even. When speaking about the reals, sometimes it means all reals, in this sense almost all reals are not a member of the Cantor set even though the Cantor set is uncountable. More generally, almost all is used in the sense of almost everywhere in measure theory. Thus, almost all positive integers are composite, however there are still a number of primes. Generic property Sufficiently large Weisstein, Eric W
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
9.
Hyperbola
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In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other, the hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, each branch of the hyperbola has two arms which become straighter further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the point about which each branch reflects to form the other branch. In the case of the curve f =1 / x the asymptotes are the two coordinate axes, hyperbolas share many of the ellipses analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term, many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids, hyperboloids, hyperbolic geometry, hyperbolic functions, and gyrovector spaces. The word hyperbola derives from the Greek ὑπερβολή, meaning over-thrown or excessive, hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have coined by Apollonius of Perga in his definitive work on the conic sections. The rectangle could be applied to the segment, be shorter than the segment or exceed the segment, the midpoint M of the line segment joining the foci is called the center of the hyperbola. 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, C2 is called the director circle of the hyperbola. In order to get the branch of the hyperbola, one has to use the director circle related to F1. This property should not be confused with the definition of a hyperbola 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 hyperbola if the condition is fulfilled 2 + y 2 −2 + y 2 = ±2 a. Remove the square roots by suitable squarings and use the relation b 2 = c 2 − a 2 to obtain the equation of the hyperbola, the shape parameters a, b are called the semi major axis and semi minor axis or conjugate axis. As opposed to an ellipse, a hyperbola has two vertices
10.
Cylinder
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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
11.
Implicit surface
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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
12.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
13.
Complex conjugate
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In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign. For example, the conjugate of 3 + 4i is 3 − 4i. In polar form, the conjugate of ρ e i ϕ is ρ e − i ϕ and this can be shown using Eulers formula. Complex conjugates are important for finding roots of polynomials, according to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients, so is its conjugate. The complex conjugate of a number z is written as z ¯ or z ∗. The first notation avoids confusion with the notation for the transpose of a matrix. The second is preferred in physics, where dagger is used for the conjugate transpose, If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the conjugate of a previous known number is abbreviated as c. c. A significant property of the conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero. The conjugate of the conjugate of a number z is z. The ultimate relation is the method of choice to compute the inverse of a number if it is given in rectangular coordinates. Exp = exp ¯ log = log ¯ if z is non-zero If p is a polynomial with real coefficients, thus, non-real roots of real polynomials occur in complex conjugate pairs. In general, if ϕ is a function whose restriction to the real numbers is real-valued. The map σ = z ¯ from C to C is a homeomorphism and antilinear, even though it appears to be a well-behaved function, it is not holomorphic, it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension C / R and this Galois group has only two elements, σ and the identity on C. Thus the only two field automorphisms of C that leave the real numbers fixed are the identity map and complex conjugation. Similarly, for a fixed complex unit u = exp, the equation z − z 0 z ¯ − z 0 ¯ = u determines the line through z 0 in the direction of u
14.
Maxima and minima
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Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. The value of the function at a point is called the maximum value of the function. Similarly, the function has a minimum point at x∗ if f ≤ f for all x in X within distance ε of x∗. A similar definition can be used when X is a space, since the definition just given can be rephrased in terms of neighbourhoods. Note that a maximum point is always a local maximum point. In both the global and local cases, the concept of a strict extremum can be defined, note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. A continuous real-valued function with a compact domain always has a maximum point, an important example is a function whose domain is a closed interval of real numbers. Finding global maxima and minima is the goal of mathematical optimization, if a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum either must be a maximum in the interior of the domain. So a method of finding a maximum is to look at all the local maxima in the interior, and also look at the maxima of the points on the boundary. Local extrema of functions can be found by Fermats theorem. For any function that is defined piecewise, one finds a maximum by finding the maximum of each piece separately, the function x2 has a unique global minimum at x =0. The function x3 has no global minima or maxima, although the first derivative is 0 at x =0, this is an inflection point. The function x x has a global maximum at x = e. The function x−x has a global maximum over the positive real numbers at x = 1/e. The function x3/3 − x has first derivative x2 −1, setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative we can see that −1 is a local maximum, note that this function has no global maximum or minimum
15.
Coordinate system
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The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in the x-coordinate. The coordinates are taken to be real numbers in elementary mathematics, the use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa, this is the basis of analytic geometry. The simplest example of a system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O is chosen on a given line. The coordinate of a point P is defined as the distance from O to P. Each point is given a unique coordinate and each number is the coordinate of a unique point. The prototypical example of a system is the Cartesian coordinate system. In the plane, two lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three perpendicular planes are chosen and the three coordinates of a point are the distances to each of the planes. This can be generalized to create n coordinates for any point in n-dimensional Euclidean space, depending on the direction and order of the coordinate axis the system may be a right-hand or a left-hand system. This is one of many coordinate systems, another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis, for a given angle θ, there is a single line through the pole whose angle with the polar axis is θ. Then there is a point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates there is a single point, for example, and are all polar coordinates for the same point. The pole is represented by for any value of θ, there are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system, a z-coordinate with the meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple. Spherical coordinates take this a further by converting the pair of cylindrical coordinates to polar coordinates giving a triple. A point in the plane may be represented in coordinates by a triple where x/z and y/z are the Cartesian coordinates of the point
16.
Hyperboloid
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In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. An hyperboloid is a surface that may be obtained from a paraboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation. A hyperboloid 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, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, a hyperboloid has also three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry. Both of these surfaces are asymptotic to the cone of equation x 2 a 2 + y 2 b 2 − z 2 c 2 =0, one has an hyperboloid of revolution if and only if a 2 = b 2. It is a ruled surface. In case of a = b the hyperboloid is a surface of revolution, the more common generation of a hyperboloid of revolution is rotating a hyperbola around its semi-minor axis. Remark, A hyperboloid of two sheets is projectively equivalent to a hyperbolic paraboloid, for simplicity the plane sections of the unit hyperboloid with equation H1, x 2 + y 2 − z 2 =1 are considered. Because a hyperboloid in general position is an image of the unit hyperboloid. The hyperboloid of two sheets does not contain lines, the discussion of plane sections can be performed for the unit hyperboloid of two sheets with equation H2, x 2 + y 2 − z 2 = −1. Remark, A hyperboloid of two sheets is projectively equivalent to a sphere, whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry. More generally, an arbitrarily oriented hyperboloid, centered at v, is defined by the equation T A =1, where A is a matrix and x, v are vectors. The eigenvectors of A define the directions of the hyperboloid. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue, the two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues. Imaginary hyperboloids are frequently found in mathematics of higher dimensions, for example, in a pseudo-Euclidean space one has the use of a quadratic form, q = −, k < n. When c is any constant, then the part of the space given by is called a hyperboloid, the degenerate case corresponds to c =0. As an example, consider the following passage, however, the term quasi-sphere is also used in this context since the sphere and hyperboloid have some commonality
17.
Ruled surface
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In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the surface of a cylinder or cone. Other examples are a surface with elliptical directrix, the right conoid, the helicoid. A ruled surface can always be described as the set of points swept by a straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle, a surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces, the plane is the only surface which contains at least three distinct lines through each of its points. The properties of being ruled or doubly ruled are preserved by projective maps, thus, for example, if one uses p = r = one obtains a ruled surface that contains the Möbius strip. Alternatively, a surface can be parametrized as S = p + u q. In particular, when p and q move with constant speed along two lines, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet. A developable surface is a surface that can be unrolled onto a plane without tearing or stretching it. If a developable surface lies in three-dimensional Euclidean space, and is complete, then it is necessarily ruled, for instance, the cylinder and cone are developable, but the general hyperboloid of one sheet is not. More generally, any surface in three dimensions is part of a complete ruled surface, and so itself must be locally ruled. There are developable surfaces embedded in four dimensions which are not ruled. In algebraic geometry, ruled surfaces were originally defined as projective surfaces in projective space containing a line through any given point. This is equivalent to saying that they are birational to the product of a curve, sometimes a ruled surface is defined to be one satisfying the stronger condition that it has a fibration over a curve with fibers that are projective lines. This excludes the projective plane, which has a line though every point. Ruled surfaces appear in the Enriques classification of complex surfaces. Every minimal projective ruled surface other than the plane is the projective bundle of a 2-dimensional vector bundle over some curve
18.
Saddle
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The saddle is a supportive structure for a rider or other load, fastened to an animals back by a girth. The most common type is the equestrian saddle designed for a horse, today, modern saddles come in a wide variety of styles, each designed for a specific equestrianism discipline, and require careful fit to both the rider and the horse. Proper saddle care can extend the life of a saddle. The word saddle originates from the Proto-Germanic language *sathulaz, with cognates in various other Indo-European languages, there is evidence, though disputed, that humans first began riding the horse not long after domestication, possibly as early as 4000 BC. The earliest known saddle-like equipment were fringed cloths or pads used by Assyrian cavalry around 700 BC and these were held on with a girth or surcingle that included breast straps and cruppers. From the earliest depictions, saddles became status symbols and these were located in Pazyryk burials finds. These saddles, found in the Ukok Plateau, Siberia were dated to 500-400 BC, the Scythians also developed an early saddle that included padding and decorative embellishments. Though they had neither a solid tree nor stirrups, these early treeless saddles and pads provided protection and comfort to the rider, with a slight increase in security. The Sarmatians also used a padded treeless early saddle, possibly as early as the century, BC. Early solid-treed saddles were made of felt that covered a wooden frame, Asian designs appeared during the Han dynasty approximately 200 BC. One of the earliest solid-treed saddles in the west was the four horn design, the invention of the solid saddle tree also allowed development of the true stirrup as it is known today. Without a solid tree, the weight in the stirrups creates abnormal pressure points. Thermography studies on treeless and flexible tree saddle designs have found there is considerable friction across the center line of a horses back. The stirrup was one of the milestones in saddle development, the first stirrup-like object was invented in India in the 2nd century BC, and consisted of a simple leather strap in which the riders toe was placed. It offered very little support, however, the stirrup appeared to be in widespread use across China by 477 AD. which then spread to Europe. This invention gave great support for the rider, and was essential in later warfare, Saddles were improved upon during the Middle Ages, as knights needed saddles that were stronger and offered more support. The resulting saddle had a higher cantle and pommel and was built on a tree that supported more weight from a rider with armor. This saddle, a predecessor to the modern Western saddle, was padded with wool or horsehair
19.
Paraboloid
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In geometry, a paraboloid is a quadric surface that has one axis of symmetry and no center of symmetry. The term paraboloid is derived from parabola, which refers to a section that has the same property of symmetry. In a suitable coordinate system with three axes x, y, and z, it can be represented by the equation z = x 2 a 2 + y 2 b 2. Where a and b are constants that dictate the level of curvature in the xz, in this position, the elliptic paraboloid opens upward. A hyperbolic paraboloid is a ruled surface shaped like a saddle. In a suitable system, a hyperbolic paraboloid can be represented by the equation z = y 2 b 2 − x 2 a 2. In this position, the hyperbolic paraboloid opens down along the x-axis, obviously both the paraboloids contain a lot of parabolas. But there are differences, too, an elliptic paraboloid contains ellipses. With a = b an elliptic paraboloid is a paraboloid of revolution and this shape is also called a circular paraboloid. This also works the way around, a parallel beam of light incident on the paraboloid parallel to its axis is concentrated at the focal point. This applies also for other waves, hence parabolic antennas, for a geometrical proof, click here. The hyperbolic paraboloid is a ruled surface, it contains two families of mutually skew lines. The lines in each family are parallel to a common plane, hence the hyperbolic paraboloid is a conoid. This property makes easy to realize a hyperbolic paraboloid with concrete, the widely sold fried snack food Pringles potato crisps resemble a truncated hyperbolic paraboloid. The distinctive shape of these allows them to be stacked in sturdy tubular containers. Examples in architecture St. a point, if the plane is a tangent plane, remark, an elliptic paraboloid is projectively equivalent to a sphere. Remarks, A hyperbolic paraboloid is a surface, but not developable. The Gauss curvature at any point is negative, hence it is a saddle surface
20.
Parabolic reflector
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A parabolic reflector is a reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a paraboloid, that is. The parabolic reflector transforms an incoming plane wave traveling along the axis into a spherical wave converging toward the focus, conversely, a spherical wave generated by a point source placed in the focus is reflected into a plane wave propagating as a collimated beam along the axis. Parabolic reflectors are used to collect energy from a distant source, since the principles of reflection are reversible, parabolic reflectors can also be used to focus radiation from an isotropic source into a narrow beam. In optics, parabolic mirrors are used to light in reflecting telescopes and solar furnaces, and project a beam of light in flashlights, searchlights, stage spotlights. In acoustics, parabolic microphones are used to record faraway sounds such as calls, in sports reporting. Strictly, the shape of the reflector is called a paraboloid. A parabola is the two-dimensional figure, however, in informal language, the word parabola and its associated adjective parabolic are often used in place of paraboloid and paraboloidal. Correspondingly, the dimensions of a symmetrical paraboloidal dish are related by the equation,4 F D = R2, where F is the length, D is the depth of the dish. All units must be the same, if two of these three quantities are known, this equation can be used to calculate the third. A more complex calculation is needed to find the diameter of the dish measured along its surface. This is sometimes called the diameter, and equals the diameter of a flat, circular sheet of material, usually metal. Two intermediate results are useful in the calculation, P =2 F and Q = P2 + R2, where F, D, and R are defined as above. The diameter of the dish, measured along the surface, is given by, R Q P + P ln . The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom, is given by 12 π R2 D, where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder, a hemisphere (23 π R2 D, {\displaystyle \scriptstyle, and a cone. π R2 is the area of the dish, the area enclosed by the rim. The area of the surface of the dish can be found using the area formula for a surface of revolution which gives A = π R6 D2
21.
Mirror
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This is different from other light-reflecting objects that do not preserve much of the original wave signal other than color and diffuse reflected light. The most familiar type of mirror is the mirror, which has a flat screen surface. Curved mirrors are used, to produce magnified or diminished images or focus light or simply distort the reflected image. Mirrors are commonly used for personal grooming or admiring oneself, decoration, Mirrors are also used in scientific apparatus such as telescopes and lasers, cameras, and industrial machinery. Most mirrors are designed for light, however, mirrors designed for other wavelengths of electromagnetic radiation are also used. The first mirrors used by people were most likely pools of dark, still water, the requirements for making a good mirror are a surface with a very high degree of flatness, and a surface roughness smaller than the wavelength of the light. The earliest manufactured mirrors were pieces of polished stone such as obsidian, examples of obsidian mirrors found in Anatolia have been dated to around 6000 BC. Mirrors of polished copper were crafted in Mesopotamia from 4000 BC, polished stone mirrors from Central and South America date from around 2000 BC onwards. In China, bronze mirrors were manufactured from around 2000 BC, some of the earliest bronze, Mirrors made of other metal mixtures such as copper and tin speculum metal may have also been produced in China and India. Mirrors of speculum metal or any precious metal were hard to produce and were owned by the wealthy. Stone mirrors often had poor reflectivity compared to metals, yet metals scratch or tarnish easily, depending upon the color, both often yielded reflections with poor color rendering. The poor image quality of ancient mirrors explains 1 Corinthians 13s reference to seeing as in a mirror, glass was a desirable material for mirrors. Because the surface of glass is smooth, it produces reflections with very little blur. In addition, glass is very hard and scratch resistant, however, glass by itself has little reflectivity, so people began coating it with metals to increase the reflectivity. According to Pliny, the people of Sidon developed a technique for creating crude mirrors by coating glass with molten lead. Glass mirrors backed with gold leaf are mentioned by Pliny in his Natural History and these circular mirrors were typically small, from only a fraction of an inch to as much as eight inches in diameter. These small mirrors produced distorted images, yet were prized objects of high value and these ancient glass mirrors were very thin, thus very fragile, because the glass needed to be extremely thin to prevent cracking when coated with a hot, molten metal. Due to the quality, high cost, and small size of these ancient glass mirrors
22.
Antenna (radio)
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In radio and electronics, an antenna, or aerial, is an electrical device which converts electric power into radio waves, and vice versa. It is usually used with a transmitter or radio receiver. In reception, an antenna intercepts some of the power of a wave in order to produce a tiny voltage at its terminals. Antennas are essential components of all equipment that uses radio, typically an antenna consists of an arrangement of metallic conductors, electrically connected to the receiver or transmitter. These time-varying fields radiate away from the antenna into space as a transverse electromagnetic field wave. Antennas can be designed to transmit and receive radio waves in all directions equally. The first antennas were built in 1888 by German physicist Heinrich Hertz in his experiments to prove the existence of electromagnetic waves predicted by the theory of James Clerk Maxwell. Hertz placed dipole antennas at the point of parabolic reflectors for both transmitting and receiving. He published his work in Annalen der Physik und Chemie, the words antenna and aerial are used interchangeably. Occasionally the term aerial is used to mean a wire antenna, however, note the important international technical journal, the IEEE Transactions on Antennas and Propagation. In the United Kingdom and other areas where British English is used, the origin of the word antenna relative to wireless apparatus is attributed to Italian radio pioneer Guglielmo Marconi. In the summer of 1895, Marconi began testing his wireless system outdoors on his fathers estate near Bologna, Marconi discovered that by raising the aerial wire above the ground and connecting the other side of his transmitter to ground, the transmission range was increased. Soon he was able to transmit signals over a hill, a distance of approximately 2.4 kilometres, in Italian a tent pole is known as lantenna centrale, and the pole with the wire was simply called lantenna. Until then wireless radiating transmitting and receiving elements were simply as aerials or terminals. Because of his prominence, Marconis use of the word spread among wireless researchers. In common usage, the antenna may refer broadly to an entire assembly including support structure, enclosure. Especially at microwave frequencies, an antenna may include not only the actual electrical antenna. An antenna, in converting radio waves to electrical signals or vice versa, is a form of transducer, Antennas are required by any radio receiver or transmitter to couple its electrical connection to the electromagnetic field
23.
Liquid mirror telescope
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Liquid mirror telescopes are telescopes with mirrors made with a reflective liquid. The most common liquid used is mercury, but other liquids will work as well, the rotating liquid assumes the paraboloidal shape regardless of the containers shape. To reduce the amount of metal needed, and thus weight. Liquid mirrors can be a low cost alternative to large telescopes. Compared to a glass mirror that must be cast, ground, and polished. A very large telescope already exists at the South Pole, currently, the mercury mirror of the Large Zenith Telescope in Canada is the largest liquid metal mirror in operation. It has a diameter of six meters, and rotates at a rate of about 8.5 revolutions per minute, the force diagram represents a snapshot of the forces acting on the parcel, in a non-rotating frame of reference. The direction of each arrow shows the direction of a force, the red arrow represents the weight of the parcel, caused by gravity and directed vertically downward. The green arrow shows the force exerted on the parcel by the bulk of the liquid. Since, in equilibrium, the liquid cannot exert a force parallel with its surface, the short blue arrow shows the net force on the parcel. It is the sum of the forces of weight and buoyancy. It is the force that constantly accelerates the parcel toward the axis. Therefore, the arrow is tilted from the vertical by an angle whose tangent is the quotient of these forces. The equation of the paraboloid in terms of its length can be written as,4 f h = r 2 where f is the focal length. Note that no other variables are involved, the density of the liquid, for example, has no effect on the focal length of the paraboloid. The units must be consistent, e. g. f may be in metres, ω in radians per second, the angle unit in ω must be radians. If we write F for the value of the focal length in metres. The rotation speed is still in RPM and these are made of liquid stored in a cylindrical container made of a composite material, such as Kevlar
24.
Rotating furnace
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A rotating furnace is a device for making solid objects which have concave surfaces that are segments of axially symmetrical paraboloids. Usually, the objects are made of glass, paraboloids can be used in various ways, including as primary mirrors in reflecting telescopes and solar cookers. The furnace includes a mechanism that rotates an open-topped container at constant speed around a vertical axis, when the rotation is stopped, the glass is solid, so the paraboloidal shape of its top surface is preserved. This process is called spin casting, the same process can be used to make a lens with a concave paraboloidal surface. The other surface is shaped by the container holds the molten glass acting as a mould. Lenses made this way are used as objectives in refracting telescopes. The word spin is frequently used in this context without implying that the rotation is rapid, making a mirror with a focal length of five metres, for example, requires a rotation speed less than ten revolutions per minute. Since the speed is low, accurate dynamic balancing of the components is not needed. The axis of rotation becomes the axis of the paraboloid and it is not necessary for this axis to be in the centre of the container of glass, or even for it to pass through the container. By placing the container away from the axis, off-axis paraboloidal segments can be cast and this is done in the making of very large telescopes which have mirrors consisting of several segments. They must be in compatible units so, for example, f can be in metres, ω in radians per second, the angle unit in ω must be radians. 1 radian per second is about 9.55 rotations per minute, on the Earths surface, g is about 9.81 metres per second-squared. Putting these numbers into the equation produces the approximation, f s 2 ≈447, where f is the length in metres. Generally, a spin-cast paraboloid is not sufficiently accurate to permit its use as a telescope mirror or lens. The amount of grinding done, and the mass of material wasted, are much less than would have been required without spinning. Spin casting every paraboloid that is made would be too slow and costly, so the prototype is simply copied relatively quickly and cheaply, liquid mirror telescopes have rotating mirrors that consist of a liquid metal such as mercury or a low-melting alloy of gallium. The mirrors do not solidify, but are used while liquid, the rotation shapes them into paraboloids that are accurate enough to be used as primary reflectors in telescopes. No correction of the shape is necessary, spin-cast glass mirrors need correction because of distortions that arise during and after solidification
25.
Focus (optics)
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In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent and this non-ideal focusing may be caused by aberrations of the imaging optics. In the absence of significant aberrations, the smallest possible blur circle is the Airy disc, aberrations tend to get worse as the aperture diameter increases, while the Airy circle is smallest for large apertures. An image, or image point or region, is in focus if light from object points is converged almost as much as possible in the image, the border between these is sometimes defined using a circle of confusion criterion. A principal focus or focal point is a focus, For a lens, or a spherical or parabolic mirror. Since light can pass through a lens in either direction, a lens has two focal points—one on each side, the distance in air from the lens or mirrors principal plane to the focus is called the focal length. Elliptical mirrors have two points, light that passes through one of these before striking the mirror is reflected such that it passes through the other. The focus of a mirror is either of two points which have the property that light from one is reflected as if it came from the other. Diverging lenses and convex mirrors do not focus a beam to a point. Instead, the focus is the point from which the light appears to be emanating, a convex elliptical mirror will reflect light directed towards one focus as if it were radiating from the other focus, both of which are behind the mirror. A convex hyperbolic mirror will reflect rays emanating from the point in front of the mirror as if they were emanating from the focal point behind the mirror. Conversely, it can focus rays directed at the point that is behind the mirror towards the focal point that is in front of the mirror as in a Cassegrain telescope. Autofocus Cardinal point Defocus aberration Depth of field Depth of focus Far point Focus Fixed focus Bokeh Focus stacking Focal Plane
26.
Parabolic antenna
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A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. The most common form is shaped like a dish and is called a dish antenna or parabolic dish. The main advantage of an antenna is that it has high directivity. It functions similarly to a searchlight or flashlight reflector to direct the waves in a narrow beam. Parabolic antennas have some of the highest gains, that is, they can produce the narrowest beamwidths and they are also used in radio telescopes. The other large use of antennas is for radar antennas, in which there is a need to transmit a narrow beam of radio waves to locate objects like ships, airplanes. With the advent of satellite television receivers, parabolic antennas have become a common feature of the landscapes of modern countries. The parabolic antenna was invented by German physicist Heinrich Hertz during his discovery of radio waves in 1887 and he used cylindrical parabolic reflectors with spark-excited dipole antennas at their focus for both transmitting and receiving during his historic experiments. Conversely, a plane wave parallel to the axis will be focused to a point at the focal point. A typical parabolic antenna consists of a parabolic reflector with a small feed antenna suspended in front of the reflector at its focus. The reflector is a surface formed into a paraboloid of revolution. In a transmitting antenna, radio frequency current from a transmitter is supplied through a transmission line cable to the feed antenna, the radio waves are emitted back toward the dish by the feed antenna and reflect off the dish into a parallel beam. The reflector can be of metal, metal screen, or wire grill construction. Large dishes often require a supporting structure behind them to provide the required stiffness. A reflector made of a grill of parallel wires or bars oriented in one direction acts as a polarizing filter as well as a reflector and it only reflects linearly polarized radio waves, with the electric field parallel to the grill elements. This type is used in radar antennas. Combined with a linearly polarized feed horn, it helps filter out noise in the receiver, the feed antenna at the reflectors focus is typically a low-gain type such as a half-wave dipole or more often a small horn antenna called a feed horn. The feed antenna is connected to the associated radio-frequency transmitting or receiving equipment by means of a cable transmission line or waveguide
27.
Skew lines
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In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar, if four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines. After the first three points have been chosen, the point will define a non-skew line if, and only if. However, the plane through the first three points forms a subset of measure zero of the cube, and the probability that the point lies on this plane is zero. If it does not, the lines defined by the points will be skew, similarly, in three-dimensional space a very small perturbation of any two parallel or intersecting lines will almost certainly turn them into skew lines. Therefore, any four points in general position always form skew lines, in this sense, skew lines are the usual case, and parallel or intersecting lines are special cases. If each line in a pair of lines is defined by two points that it passes through, then these four points must not be coplanar, so they must be the vertices of a tetrahedron of nonzero volume. Conversely, any two pairs of points defining a tetrahedron of nonzero volume also define a pair of skew lines, therefore, a test of whether two pairs of points define skew lines is to apply the formula for the volume of a tetrahedron in terms of its four vertices. To calculate the distance between two lines the lines may be expressed using vectors, x = a + λ b, y = c + μ d. The cross product of b and d is perpendicular to the lines, the distance between the lines is then d = | n ⋅ |. N = d 1 × d 2 The plane formed by the translations of Line 2 along n contains the point p 2 and is perpendicular to n 2 = d 2 × n, a configuration of skew lines is a set of lines in which all pairs are skew. The number of configurations of n lines in R3, starting at n =1, is 1,1,2,3,7,19,74. If one rotates a line L around another line M skew but not perpendicular to it, for instance, the three hyperboloids visible in the illustration can be formed in this way by rotating a line L around the central white vertical line M. The two reguli display the hyperboloid as a ruled surface, a third type of ruled surface is the hyperbolic paraboloid. Like the hyperboloid of one sheet, the hyperbolic paraboloid has two families of lines, in each of the two families the lines are parallel to a common plane although not to each other. Any three skew lines in R3 lie on exactly one ruled surface of one of these types, if three skew lines all meet three other skew lines, any transversal of the first set of three meets any transversal of the second set
28.
Conoid
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In geometry a conoid is a ruled surface, whose rulings fulfill the additional conditions All rulings are parallel to a plane, the directrix plane. All rulings intersect a fixed line, the axis, the conoid is a right conoid, if its axis is perpendicular to its directrix plane. Hence all rulings are perpendicular to the axis, the planarity of the vectors r can be represented by det =0. If the directrix is a circle the conoid is called circular conoid, the term conoid was already used by Archimedes in his treatise On conoids and spheroides. Its axis is the line x ∈ R, special features, The intersection with a horizontal plane is an ellipse. 2 − y 2 z 02 =0 is an implicit representation, hence the right circular conoid is a surface of degree 4. Keplers rule gives for a right circular conoid with radius r and height h the exact volume, the implicit representation is fulfilled by the points of the line, too. For these points there exist no tangent planes, the parametric representation x = + v =, u, v ∈ R, describes a parabolic conoid with the equation z = − x y 2. The conoid has a parabola as directrix, the y-axis as axis and it is used by architects as roof surface. The parabolic conoid has no singular points, hyperbolic paraboloid Plücker conoid Whitney Umbrella helicoid There are a lot of conoids with singular points, which are investigated in algebraic geometry. Like other ruled surfaces conoids are of high interest with architects, right conoids can be manufactured easily, one threads bars onto an axis such that they can be rotated around this axis, only. Afterwards one deflects the bars by a directrix and generates a conoid, rovenskii, Geometry of curves and surfaces with MAPLE
29.
Architecture
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Architecture is both the process and the product of planning, designing, and constructing buildings and other physical structures. Architectural works, in the form of buildings, are often perceived as cultural symbols. Historical civilizations are often identified with their surviving architectural achievements, Architecture can mean, A general term to describe buildings and other physical structures. The art and science of designing buildings and nonbuilding structures, the style of design and method of construction of buildings and other physical structures. A unifying or coherent form or structure Knowledge of art, science, technology, the design activity of the architect, from the macro-level to the micro-level. The practice of the architect, where architecture means offering or rendering services in connection with the design and construction of buildings. The earliest surviving work on the subject of architecture is De architectura. According to Vitruvius, a building should satisfy the three principles of firmitas, utilitas, venustas, commonly known by the original translation – firmness, commodity. An equivalent in modern English would be, Durability – a building should stand up robustly, utility – it should be suitable for the purposes for which it is used. Beauty – it should be aesthetically pleasing, according to Vitruvius, the architect should strive to fulfill each of these three attributes as well as possible. Leon Battista Alberti, who elaborates on the ideas of Vitruvius in his treatise, De Re Aedificatoria, saw beauty primarily as a matter of proportion, for Alberti, the rules of proportion were those that governed the idealised human figure, the Golden mean. The most important aspect of beauty was, therefore, an inherent part of an object, rather than something applied superficially, Gothic architecture, Pugin believed, was the only true Christian form of architecture. The 19th-century English art critic, John Ruskin, in his Seven Lamps of Architecture, Architecture was the art which so disposes and adorns the edifices raised by men. That the sight of them contributes to his health, power. For Ruskin, the aesthetic was of overriding significance and his work goes on to state that a building is not truly a work of architecture unless it is in some way adorned. For Ruskin, a well-constructed, well-proportioned, functional building needed string courses or rustication, but suddenly you touch my heart, you do me good. I am happy and I say, This is beautiful, le Corbusiers contemporary Ludwig Mies van der Rohe said Architecture starts when you carefully put two bricks together. The notable 19th-century architect of skyscrapers, Louis Sullivan, promoted an overriding precept to architectural design, function came to be seen as encompassing all criteria of the use, perception and enjoyment of a building, not only practical but also aesthetic, psychological and cultural
30.
Constructivism (art)
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Constructivism was an artistic and architectural philosophy that originated in Russia beginning in 1913 by Vladimir Tatlin. This was a rejection of the idea of autonomous art, the movement was in favour of art as a practice for social purposes. Constructivism had an effect on modern art movements of the 20th century, influencing major trends such as the Bauhaus. Its influence was pervasive, with effects upon architecture, graphic design, industrial design, theatre, film, dance, fashion. The term Construction Art was first used as a term by Kazimir Malevich to describe the work of Alexander Rodchenko in 1917. Constructivism first appears as a term in Naum Gabos Realistic Manifesto of 1920. Aleksei Gan used the word as the title of his book Constructivism, Constructivism was a post-World War I development of Russian Futurism, and particularly of the counter reliefs of Vladimir Tatlin, which had been exhibited in 1915. Constructivism as theory and practice was derived largely from a series of debates at INKhUK in Moscow, later the definition would be extended to designs for two-dimensional works such as books or posters, with montage and factography becoming important concepts. As much as involving itself in designs for industry, the Constructivists worked on public festivals, perhaps the most famous of these was in Vitebsk, where Malevichs UNOVIS Group painted propaganda plaques and buildings. Inspired by Vladimir Mayakovskys declaration the streets our brushes, the squares our palettes, artists, in addition some Constructivists were heavily involved in the ROSTA Windows, a Bolshevik public information campaign of around 1920. Some of the most famous of these were by the poet-painter Vladimir Mayakovsky, the constructivists tried to create works that would make the viewer an active viewer of the artwork. These theories were tested in theatre, particularly with the work of Vsevolod Meyerhold, Meyerhold developed a biomechanical acting style, which was influenced both by the circus and by the scientific management theories of Frederick Winslow Taylor. Meanwhile, the sets by the likes of Vesnin, Popova and Stepanova tested Constructivist spatial ideas in a public form. A more populist version of this was developed by Alexander Tairov, with sets by Aleksandra Ekster. These ideas would influence German directors like Bertolt Brecht and Erwin Piscator, Gabo publicly criticized Tatlins design saying, Either create functional houses and bridges or create pure art, not both. This had already caused a controversy in the Moscow group in 1920 when Gabo. This was opposed to the utilitarian and adaptable version of Constructivism held by Tatlin, the tower was never built, however, due to a lack of money following the revolution. A Constructivist international was formed, which met with Dadaists and De Stijl artists in Germany in 1922, participants in this short-lived international included Lissitzky, Hans Richter, and László Moholy-Nagy
31.
Pringles
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Pringles is a brand of potato and wheat-based stackable snack chips owned by Kelloggs. The snack was developed by Procter & Gamble, who first sold the product in 1967. P&G sold the brand to Kelloggs in 2012, Pringles were initially sold in 1967 and became nationally distributed across the US in 1975, and internationally from 1991 onwards. P&G wanted to create a perfect chip to address complaints about broken, greasy. The task was assigned to chemist Fredric Baur, who, from 1956 to 1958, created Pringles’ saddle shape from fried dough, Baur could not figure out how to make the chips taste good and he eventually was pulled off the Pringles job to work on another brand. In the mid-1960s, another P&G researcher, Alexander Liepa of Montgomery, Ohio, restarted Baur’s work, and set out to improve on the Pringles taste, while Baur was the true inventor of the Pringles chip, Liepas name is on the patent. Gene Wolfe, a mechanical engineer-author known for science fiction and fantasy novels and their consistent saddle shape is mathematically known as a hyperbolic paraboloid. Their designers reportedly used supercomputers to ensure that the chips aerodynamics would keep them in place during packaging, there are several theories behind the origin of the name Pringles. One theory refers to Mark Pringle, who filed a US Patent 2,286,644 titled Method, Pringles work was cited by Procter & Gamble in filing their own patent for improving the taste of dehydrated processed potatoes. Another theory suggested two Procter advertising employees lived on Pringle Drive in Finneytown, and the name paired well with potato, another theory says that P&G chose the Pringles name from a Cincinnati telephone book. They were originally known as Pringles Newfangled Potato Chips, but other snack manufacturers objected, faced with such an unpalatable appellation, Pringles eventually opted to rename their product potato crisps instead of chips. This later led to issues in the United Kingdom, where the term potato crisps refers to the product Americans call potato chips. In April 2011, P&G agreed to the $2.35 billion sale of the brand to Diamond Foods of California, however, the deal fell through in February 2012 after a year-long delay due to issues over Diamonds accounts. On 31 May 2012, Kellogg Company officially acquired Pringles for $2.695 billion as part of a plan to grow its international snacks business, the acquisition of Pringles makes Kellogg the second-largest savory snacks company in the world. Pringles are manufactured in factories in Jackson, Tennessee, Mechelen, Belgium, Johor, Malaysia, Kutno, Poland, Fujian, Pringles have about 42% potato content, the remainder being wheat starch and flours combined with vegetable oils, an emulsifier, salt, and seasoning. Pringles varieties vary in their ingredients, in July 2008 in the London High Court, P&G lawyers successfully argued that Pringles were not crisps as the potato content was only 42% and their shape, P&G stated, is not found in nature. This ruling, against a United Kingdom VAT and Duties Tribunal decision to the contrary, exempted Pringles from the then 17. 5% VAT for potato crisps, in May 2009, the Court of Appeal reversed the earlier decision. A spokesman for P&G stated it had been paying the VAT proactively, standard flavors include original, salt and vinegar, sour cream and onion, cheddar cheese, ranch dressing, barbecue, hot and spicy, and loaded baked potato
32.
Saddle roof
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A saddle roof is a roof form which follows a convex curve about one axis and a concave curve about the other. The hyperbolic paraboloid form has been used for roofs at various times since it is constructed from straight sections of lumber, steel. The term is used because the form resembles the shape of a saddle, sometimes referred to as a hypar, the saddle roof may also be formed as a tensegrity structure. Mathematically, a saddle shape contains at least one saddle point, the historical meaning is a synonym for a gable roof particularly a dual-pitched roof on a tower, also called a pack-saddle roof. Scandinavium Xavier University Metro San Lázaro Kansas State Historical Society newsletter featuring house with hyperbolic paraboloid roof
33.
St. Mary's Cathedral, Tokyo
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St. Marys Cathedral is the seat of the Roman Catholic Archdiocese of Tokyo. It is located in the Sekiguchi neighborhood of Bunkyo, Tokyo, the original wooden structure, constructed in 1899 in the Gothic style, was destroyed during the air raids on Tokyo during World War II. The present church, designed by Kenzo Tange, was completed in 1964, the parabolas open upwards to form a cross of light, which continues vertically along the length of the four facades. To this rhomboid volume other secondary constructions are added, including the baptistry, the rectangular shapes contrast with the symbolic character of the cathedral. The bell tower is 61.6 m high, standing a distance away from the main building. The exterior cladding is made of stainless steel, in 2004 a large organ built by Italian firm Mascioni was installed. Official website Photographs of exterior and interior of church
34.
Scotiabank Saddledome
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Scotiabank Saddledome is a multi-use indoor arena in Calgary, Alberta, Canada. The facility also hosts concerts, conferences and other sporting championships and it underwent a major renovation in 1994–95 and sold its naming rights, during which its original name of Olympic Saddledome was changed to Canadian Airlines Saddledome. The facility was given the name Pengrowth Saddledome in 2000, after Pengrowth Management Ltd. signed a ten-year agreement and it adopted its current name in October 2010 as Scotiabank signed on as title sponsor. The Saddledome is owned by the City of Calgary who leased it to the Saddledome Foundation and it is currently managed by the Calgary Flames. The Saddledome was damaged during the 2013 Alberta floods but was repaired and reopened in time for the 2013–14 NHL season, Calgary had been served for 30 years by the Stampede Corral when the Calgary Flames arrived in 1980. With a total capacity of 8,700, the Corral was the largest arena in Canada west of Toronto in 1950, but had fallen below major league standards by the 1970s. The Corral was deemed insufficient for the National Hockey League in 1977, Calgarys bid to host the 1988 Winter Olympics, coupled with the arrival of the Flames, drove the need to build a new arena. City Council debated the merits of several locations for the citys new Olympic Coliseum, two other sites, one on the west end of downtown, and a late bid by several businessmen pushing to build the arena in the northern suburb of Airdrie were also considered. The Victoria Park Community Association fought the bid to build the arena in their neighborhood, City Council voted on March 3,1981 to build the proposed 20,000 seat arena on the Stampede grounds, immediately east of the Corral and south of Victoria Park. In a bid to end the battle, Mayor Ralph Klein asked the government in July 1981 to take over the land designated for the arena to bypass the appeals process. The province supported the city amidst protests by community associations and invoked rarely used powers to overrule planning regulations, the following day, on July 29,1981, builders began construction of the arena. The facility was designed by Graham McCourt Architects, while they set out to design a unique building, the idea of a western theme never occurred to Barry Graham or his team. When the design was unveiled, the roof was immediately referred to as being saddle-shaped, of 1,270 entries submitted in a contest to name the arena,735 involved the word Saddle. The winning name in the contest, Olympic Saddledome, was drawn from a hat filled with several similar saddle-themed names, as of 2008, the Saddledome was still reported as the world record holder for the longest spanning hyperbolic paraboloid concrete shell. The Saddledome was featured on the cover of Time Magazine on September 27,1987, for an article discussing the city of Calgary and the upcoming 1988 Olympics. The location of the Saddledome within Stampede Park allows for access to Calgarys C-Train light rail transit system via the Victoria Park/Stampede station that stands parallel to Macleod Trail. The C-Train station, BMO Centre, Stampede Corral and Saddledome are all connected via a Plus 15 pedestrian skyway, direct vehicle access is gained from the north via 5th Street East or Olympic Way. The arena was initially projected to cost $60 million to build, attempts to fast track construction resulted in a $16 million cost overrun, resulting in a final cost of $97.7 million and an eight-month delay in its completion
35.
Lee Valley VeloPark
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Lee Valley VeloPark is a cycling centre on Queen Elizabeth Olympic Park, East London. It is owned and managed by Lee Valley Regional Park Authority, the facility was one of the permanent venues for the 2012 Olympic and Paralympic Games. Lee Valley VeloPark is at the end of Queen Elizabeth Olympic Park. It has a velodrome and BMX racing track, which have used for the Games. The park replaces the Eastway Cycle Circuit demolished to make way for it, the facilities built for the Olympics were constructed between 2009 and 2011. The first event in the Velopark was the London round of the 2011 UCI BMX Supercross World Cup series, in February 2005 plans were announced for a £22 million VeloPark. Sport England would invest £10.5 million, Lee Valley Regional Park Authority £6 million, the site was to be 34 hectares on the northern end of the proposed Olympic Park, next to the A12. The park would include a velodrome seating 1,500, which could be increased to 6,000 if Londons bid for the 2012 Olympic and Paralympic Games were successful. The site would also have an international competition BMX circuit, a BMX freestyle park, cyclo-cross/cross-country course mountain bike course, the facilities would be used by internationals as well as those learning to ride. It was estimated that the park would attract 88,000 users a year, Eastway Cycle Circuit opened in 1975, it was the first purpose built road cycling venue in Britain. The facility closed in September 2006 to make way for Londons VeloPark, the velodrome is the third 250 m covered track in Great Britain. In September 2008 plans for the VeloPark were revealed, which were chosen with help from Chris Hoy, however, by March 2007, the VeloPark was revealed to be only a third of its original size, rescaled from 34 to 10 hectares. The decrease in the size of the led to users of the Eastway cycle circuit to protest to the Mayor of London. The Velopark was scheduled to be completed by the contractor, ISG, in 2004, during Londons Olympic and Paralympic bid, the estimated cost was £37 million, including £20 million for the velodrome. In 2009, at the work began on the construction of the velodrome. Work on the velodrome was completed in February 2011, and was the first Olympic Park venue to be completed, the roof is designed to reflect the geometry of cycling as well as being lightweight and efficient reflecting a bike. There is also a 360-degree concourse level with windows allowing people views of the Olympic Park, the velodrome is energy efficient—rooflights reduce the need for artificial lights, and natural ventilation reduces the need for air condition. Rain water is collected, which reduces the amount of water used from the municipal water system
36.
Indian Institute of Technology Delhi
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The Indian Institute of Technology Delhi is a public engineering institution located in Delhi, India. It was declared to be Institute of National Importance by Government of India under Institutes of Technology Act, IIT Delhi is one of the two educational institutes in India which have been listed in Quacquarelli Symonds’ list of top 200 universities globally in 2015. The Government of India negotiated with the British Government for collaboration in setting up an Institute of Technology at Delhi, the British Government agreed in principle to such a collaboration, but were inclined initially to start in a modest way. It was therefore agreed that a College of Engineering & Technology should be established at Delhi with their assistance, a trust called the Delhi Engineering College Trust was established with the help of the UK Government and the Federation of British Industries in London. Prince Philips, Duke of Edinburgh, during his visit to India, the first admissions were made in 1961. The College was affiliated to the University of Delhi, the school was later accorded the status of a University and was renamed as Indian Institute of Technology Delhi. IIT Delhi is located in Hauz Khas, South Delhi, the campus of 325 acres is surrounded by the beautiful Hauz Khas area and monuments such as the Qutub Minar and Lotus Temple. The inside of the campus resembles a city, with gardens, lawns, residential complexes, the campus has its own water supply and backup electricity supply along with shopping complexes to cater to the daily needs of residents. The student residential zone is divided into two main sectors—one for boys hostels and another for girls hostels, There are 13 hostels, There are also apartments for married students. All the hostels are named after mountain ranges in India, each Hostel has its distinct culture of sports and cultural activities. Hostels compete in inter hostel events to bring home various trophies which include the BRCA trophy for cultural activities, BRCA trophy was won by Kumaon in 2016 when it was reinstated after being non-competitive for the previous 2 years. The GC was won by Jwalamukhi hostel in 2016 bringing an end to the 5 year reign of Kumaon hostel, the Student Activity Center or SAC is a part of the Student Recreation Zone in IIT Delhi. The SAC is for the activities of the students. The SAC also has an Open Air Theatre where concerts are hosted, students can use the radio broadcasting facility in the SAC although its use has declined over the years. There are a number of student bodies in IIT Delhi, each with its own set of responsibilities, the highest student body in IIT Delhi is the Student Affairs Council. The BSW shall provide financial aid to the students as per the decided rules. BSW has the responsibility of organising Speranza, the youth festival of IIT Delhi. Each club has a hostel representative working under the aegis of the cultural secretary
37.
Tangent space
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The elements of the tangent space are called tangent vectors at x. This is a generalization of the notion of a vector in a Euclidean space. The dimension of all the tangent spaces of a manifold is the same as that of the manifold. More generally, if a manifold is thought of as an embedded submanifold of Euclidean space one can picture a tangent space in this literal fashion. This was the approach to defining parallel transport, and used by Dirac. More strictly this defines a tangent space, distinct from the space of tangent vectors described by modern terminology. In algebraic geometry, in contrast, there is a definition of tangent space at a point P of a variety V. The points P at which the dimension is exactly that of V are called the non-singular points, for example, a curve that crosses itself doesnt have a unique tangent line at that point. The singular points of V are those where the test to be a manifold fails, once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold. A vector field attaches to every point of the manifold a vector from the tangent space at that point, all the tangent spaces can be glued together to form a new differentiable manifold of twice the dimension of the original manifold, called the tangent bundle of the manifold. The informal description above relies on a manifold being embedded in a vector space Rm. However, it is convenient to define the notion of tangent space based on the manifold itself. There are various equivalent ways of defining the tangent spaces of a manifold, while the definition via velocities of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below, in the embedded manifold picture, a tangent vector at a point x is thought of as the velocity of a curve passing through the point x. We can therefore take a tangent vector to be a class of curves passing through x while being tangent to each other at x. Suppose M is a Ck manifold and x is a point in M. Pick a chart φ, U → Rn, where U is an open subset of M containing x. Suppose two curves γ1, → M and γ2, → M with γ1 = γ2 = x are given such that φ ∘ γ1, then γ1 and γ2 are called equivalent at 0 if the ordinary derivatives of φ ∘ γ1 and φ ∘ γ2 at 0 coincide. This defines a relation on such curves, and the equivalence classes are known as the tangent vectors of M at x
38.
Gauss curvature
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In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point, K = κ1 κ2. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane, the Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is a measure of curvature, depending only on distances that are measured on the surface. This is the content of the Theorema egregium, Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827. At any point on a surface, we can find a vector that is at right angles to the surface. The intersection of a plane and the surface will form a curve called a normal section. For most points on most surfaces, different normal sections will have different curvatures, the Gaussian curvature is the product of the two principal curvatures Κ = κ1 κ2. The sign of the Gaussian curvature can be used to characterise the surface, if both principal curvatures are of the same sign, κ1κ2 >0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane, all sectional curvatures will have the same sign. If the principal curvatures have different signs, κ1κ2 <0, then the Gaussian curvature is negative, at such points, the surface will be saddle shaped. If one of the principal curvatures is zero, κ1κ2 =0, the Gaussian curvature is zero, most surfaces will contain regions of positive Gaussian curvature and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line. When a surface has a constant zero Gaussian curvature, then it is a developable surface, when a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry. When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface, in differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point. We represent the surface by the implicit function theorem as the graph of a function, f, then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f. This definition allows one immediately to grasp the distinction between cup/cap versus saddle point and it is also given by K = ⟨ e 1, e 2 ⟩ det g, where ∇ i = ∇ e i is the covariant derivative and g is the metric tensor. At a point p on a surface in R3, the Gaussian curvature is also given by K = det. A useful formula for the Gaussian curvature is Liouvilles equation in terms of the Laplacian in isothermal coordinates, the surface integral of the Gaussian curvature over some region of a surface is called the total curvature
39.
Saddle point
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The saddle point will always occur at a relative minimum along one axial direction and where the crossing axis is a relative maximum. For example, the Hessian matrix of the function z = x 2 − y 2 at the point is the matrix which is indefinite. Therefore, this point is a saddle point and this criterion gives only a sufficient condition. For example, the point is a point for the function z = x 4 − y 4, but the Hessian matrix of this function at the origin is the null matrix. In one dimension, a point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum, a saddle surface is a smooth surface containing one or more saddle points. Classical examples of saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid z = x 2 − y 2. The Pringles potato chip or crisp is an example of a hyperbolic paraboloid shape. Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature, a classical third-order saddle surface is the monkey saddle. Then a saddle point is a periodic point whose stable and unstable manifolds have a dimension that is not zero. In a two-player zero sum game defined on a continuous space, a saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row. For a second-order linear autonomous system, a point is a saddle point if the characteristic equation has one positive. Advanced calculus, New York, Dover Publications, p.128, ISBN 0-486-66103-2 Agarwal, study on the Nash Equilibrium Hilbert, David, Cohn-Vossen, Stephan. CS1 maint, Multiple names, authors list
40.
Gaussian curvature
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In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point, K = κ1 κ2. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane, the Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is a measure of curvature, depending only on distances that are measured on the surface. This is the content of the Theorema egregium, Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827. At any point on a surface, we can find a vector that is at right angles to the surface. The intersection of a plane and the surface will form a curve called a normal section. For most points on most surfaces, different normal sections will have different curvatures, the Gaussian curvature is the product of the two principal curvatures Κ = κ1 κ2. The sign of the Gaussian curvature can be used to characterise the surface, if both principal curvatures are of the same sign, κ1κ2 >0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane, all sectional curvatures will have the same sign. If the principal curvatures have different signs, κ1κ2 <0, then the Gaussian curvature is negative, at such points, the surface will be saddle shaped. If one of the principal curvatures is zero, κ1κ2 =0, the Gaussian curvature is zero, most surfaces will contain regions of positive Gaussian curvature and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line. When a surface has a constant zero Gaussian curvature, then it is a developable surface, when a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry. When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface, in differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point. We represent the surface by the implicit function theorem as the graph of a function, f, then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f. This definition allows one immediately to grasp the distinction between cup/cap versus saddle point and it is also given by K = ⟨ e 1, e 2 ⟩ det g, where ∇ i = ∇ e i is the covariant derivative and g is the metric tensor. At a point p on a surface in R3, the Gaussian curvature is also given by K = det. A useful formula for the Gaussian curvature is Liouvilles equation in terms of the Laplacian in isothermal coordinates, the surface integral of the Gaussian curvature over some region of a surface is called the total curvature
41.
Mean curvature
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The concept was introduced by Sophie Germain in her work on elasticity theory. Let p be a point on the surface S, each plane through p containing the normal line to S cuts S in a curve. Fixing a choice of unit normal gives a signed curvature to that curve, as the plane is rotated by an angle θ that curvature can vary. The maximal curvature κ1 and minimal curvature κ2 are known as the principal curvatures of S, the mean curvature at p ∈ S is then the average of the signed curvature over all angles θ, H =12 π ∫02 π κ d θ. By applying Eulers theorem, this is equal to the average of the principal curvatures, more generally, for a hypersurface T the mean curvature is given as H =1 n ∑ i =1 n κ i. More abstractly, the curvature is the trace of the second fundamental form divided by n. A surface is a surface if and only if the mean curvature is zero. Furthermore, a surface which evolves under the curvature of the surface S, is said to obey a heat-type equation called the mean curvature flow equation. The sphere is the embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the embedded surface is weakened to immersed surface. For a surface defined in 3D space, the curvature is related to a unit normal of the surface,2 H = − ∇ ⋅ n ^ where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal, the formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated. Mean Curvature may also be calculated 2 H = Trace where I and II denote first and second quadratic form matrices, in particular at a point where ∇ S =0, the mean curvature is half the trace of the Hessian matrix of S. The mean curvature of a surface specified by an implicit equation F =0 can be calculated by using the gradient ∇ F = and the Hessian matrix Hess =. The mean curvature is given by, H = ∇ F Hess ∇ F T − | ∇ F |2 Trace 2 | ∇ F |3 Another form is as the divergence of the unit normal. A unit normal is given by ∇ F | ∇ F |, an alternate definition is occasionally used in fluid mechanics to avoid factors of two, H f =. A minimal surface is a surface which has zero mean curvature at all points, classic examples include the catenoid, helicoid and Enneper surface. Recent discoveries include Costas minimal surface and the Gyroid, an extension of the idea of a minimal surface are surfaces of constant mean curvature
42.
Right-hand rule
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In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation conventions for vectors in three dimensions. Most of the left and right-hand rules arise from the fact that the three axes of 3-dimensional space have two possible orientations. This can be seen by holding your hands outward and together, palms up, with the fingers curled. If the curl of your fingers represents a movement from the first or X axis to the second or Y axis then the third or Z axis can point either along your left thumb or right thumb. Left and right-hand rules arise when dealing with co-ordinate axes, rotation, spirals, electromagnetic fields, mirror images and enantiomers in mathematics and chemistry. For right-handed coordinates your right thumb points along the Z axis in a positive Z-direction, when viewed from the top or Z axis the system is counter-clockwise. When viewed from the top or Z axis the system is clockwise, interchanging the labels of any two axes reverses the handedness. Reversing the direction of one axis also reverses the handedness, reversing two axes amounts to a 180° rotation around the remaining axis. In mathematics a rotating body is represented by a vector along the axis of rotation. This allows some easy calculations using the cross product. Note that no part of the body is moving in the direction of the axis arrow, by coincidence, if your thumb points north the earth rotates according to the right-hand rule. This causes the sun and stars to appear to revolve according to the left-hand rule, a helix, to use a more accurate term than spiral, is basically a circular curve that advances along the z-axis while rotating in the x-y plane. Helices are either right- or left-handed, curled fingers giving the direction of rotation, the two types are mirror images of each other, physically distinct and cannot be transformed into each other by any physical operation such as turning them over. The threads on a right-handed screw are a right-handed helix and they are basically a long inclined plane wrapped around a cylinder such that turning the screw advances the screw back and forth along the z-axis. From the point of view of the threads, turning the screw forces the screw up or down the inclined plane. If a screw is right-handed the rule is this, point your right thumb in the direction you want the screw to go and turn the screw in the direction of your curled right fingers. Viewed from the earth, the path of an object moving in a straight line appears to bend to the right in the northern hemisphere. This causes low-pressure areas in the northern hemispheres to rotate according to the right-hand rule, handedness is not obvious here but it is clear in the underlying mathematics
43.
Nomogram
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Nomograms use a parallel coordinate system invented by dOcagne rather than standard Cartesian coordinates. A nomogram consists of a set of n scales, one for each variable in an equation. Knowing the values of n-1 variables, the value of the variable can be found, or by fixing the values of some variables. The result is obtained by laying a straightedge across the values on the scales. The virtual or drawn line created by the straightedge is called a line or isopleth. Nomograms flourished in different contexts for roughly 75 years because they allowed quick. Results from a nomogram are obtained very quickly and reliably by simply drawing one or more lines, the user does not have to know how to solve algebraic equations, look up data in tables, use a slide rule, or substitute numbers into equations to obtain results. The user does not even need to know the underlying equation the nomogram represents, in addition, nomograms naturally incorporate implicit or explicit domain knowledge into their design. For example, to create larger nomograms for greater accuracy the nomographer usually includes only scale ranges that are reasonable, many nomograms include other useful markings such as reference labels and colored regions. All of these provide useful guideposts to the user, while the slide rule is intended to be a general-purpose device, a nomogram is designed to perform a specific calculation, with tables of values effectively built into the construction of the scales. Nomograms are typically used in applications where the level of accuracy they offer is sufficient, alternatively, a nomogram can be used to check an answer obtained from another, more exact but possibly error-prone calculation. Other types of graphical calculators such as charts, trilinear diagrams. These do not meet the definition of a nomogram as a graphical calculator whose solution is found by the use of one or more linear isopleths. A nomogram for a three-variable equation typically has three scales, although there exist nomograms in which two or even all three scales are common, here two scales represent known values and the third is the scale where the result is read off. The simplest such equation is u1 + u2 + u3 =0 for the three variables u1, u2 and u3, an example of this type of nomogram is shown on the right, annotated with terms used to describe the parts of a nomogram. More complicated equations can sometimes be expressed as the sum of functions of the three variables. For example, the nomogram at the top of this article could be constructed as a parallel-scale nomogram because it can be expressed as such a sum after taking logarithms of both sides of the equation. The scale for the variable can lie between the other two scales or outside of them
44.
Multiplication table
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In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an part of elementary arithmetic around the world. Many educators believe it is necessary to memorize the table up to 9 ×9, the oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60, the oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during Chinas Warring States period. The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras and it is also called the Table of Pythagoras in many languages, sometimes in English. In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a table up to 99 ×99. Leslie also recommended that young pupils memorize the table up to 25 ×25. The illustration below shows a table up to 12 ×12, there is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below, Figure 1 is used for multiples of 1,3,7, Figure 2 is used for the multiples of 2,4,6, and 8. These patterns can be used to memorize the multiples of any number from 0 to 10, as you would start on the number you are multiplying, when you multiply by 0, you stay on 0. For example, to all the multiples of 7, Look at the 7 in the first picture. The next number in the direction of the arrow is 4, so think of the next number after 7 that ends with 4, which is 14. The next number in the direction of the arrow is 1, so think of the next number after 14 that ends with 1, which is 21. After coming to the top of column, start with the bottom of the next column. So think of the number after 21 that ends with 8. Proceed in the way until the last number,3. Next, use the 0 at the bottom, then, start again with the 7. This time it will correspond to 77, tables can also define binary operations on groups, fields, rings, and other algebraic systems
