1.
Orientation (vector space)
–
In linear algebra, the notion of orientation makes sense in arbitrary finite dimension. In this setting, the orientation of a basis is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple rotation. As a result, in the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed, the orientation on a real vector space is the arbitrary choice of which ordered bases are positively oriented and which are negatively oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, a vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called unoriented. Let V be a real vector space and let b1. It is a result in linear algebra that there exists a unique linear transformation A, V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation if A has positive determinant, the property of having the same orientation defines an equivalence relation on the set of all ordered bases for V. If V is non-zero, there are two equivalence classes determined by this relation. An orientation on V is an assignment of +1 to one equivalence class, every ordered basis lives in one equivalence class or another. Thus any choice of an ordered basis for V determines an orientation. For example, the basis on Rn provides a standard orientation on Rn. Any choice of an isomorphism between V and Rn will then provide an orientation on V. The ordering of elements in a basis is crucial, two bases with a different ordering will differ by some permutation. They will have the same/opposite orientations according to whether the signature of this permutation is ±1 and this is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Similarly, let A be a linear mapping of vector space Rn to Rn. This mapping is orientation-preserving if its determinant is positive, a zero-dimensional vector space has only a single point, the zero vector. Consequently, the basis of a zero-dimensional vector space is the empty set ∅. Therefore, there is an equivalence class of ordered bases, namely
2.
Rigid body
–
In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, in classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules are often seen as rigid bodies, the position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, if the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, however, typically a different, mathematically more convenient, but equivalent approach is used. Thus, the position of a body has two components, linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a body, such as linear and angular velocity, acceleration, momentum, impulse. This reference point may define the origin of a coordinate system fixed to the body, there are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix. In general, when a body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position. Velocity and angular velocity are measured with respect to a frame of reference, the linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a point fixed to the body. During purely translational motion, all points on a body move with the same velocity. However, when motion involves rotation, the velocity of any two points on the body will generally not be the same. Two points of a body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a quantity that describes the angular speed at which the orientation of the rigid body is changing. All points on a rigid body experience the same velocity at all times
3.
Rotation (mathematics)
–
Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a space that preserves at least one point. It can describe, for example, the motion of a body around a fixed point. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude, mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group, for example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations, the rotation group is a Lie group of rotations about a fixed point. This fixed point is called the center of rotation and is identified with the origin. The rotation group is a point stabilizer in a group of motions. For a particular rotation, The axis of rotation is a line of its fixed points and they exist only in n >2. The plane of rotation is a plane that is invariant under the rotation, unlike the axis, its points are not fixed themselves. The axis and the plane of a rotation are orthogonal, a representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory, rotations of spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations, whereas the latter are vector rotations, see the article below for details. A motion of a Euclidean space is the same as its isometry, but a rotation also has to preserve the orientation structure. The improper rotation term refers to isometries that reverse the orientation, in the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point, there are no non-trivial rotations in one dimension. In two dimensions, only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group. The rotation is acting to rotate an object counterclockwise through an angle θ about the origin, composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute
4.
Frame of reference
–
In physics, a frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements. In n dimensions, n+1 reference points are sufficient to define a reference frame. Using rectangular coordinates, a frame may be defined with a reference point at the origin. In Einsteinian relativity, reference frames are used to specify the relationship between an observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes observational frame of reference, a relativistic reference frame includes the coordinate time, which does not correspond across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent, the need to distinguish between the various meanings of frame of reference has led to a variety of terms. For example, sometimes the type of system is attached as a modifier. Sometimes the state of motion is emphasized, as in rotating frame of reference, sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference, in this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a system may be employed for many purposes where the state of motion is not the primary concern. For example, a system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors and it seems useful to divorce the various aspects of a reference frame for the discussion below. A coordinate system is a concept, amounting to a choice of language used to describe observations. Consequently, an observer in a frame of reference can choose to employ any coordinate system to describe observations made from that frame of reference. A change in the choice of coordinate system does not change an observers state of motion. This viewpoint can be found elsewhere as well, which is not to dispute that some coordinate systems may be a better choice for some observations than are others. Choice of what to measure and with what observational apparatus is a separate from the observers state of motion. D. Norton, The discussion is taken beyond simple space-time coordinate systems by Brading, extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity
5.
Geometry
–
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
6.
Line (geometry)
–
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
7.
Plane (geometry)
–
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
8.
Euclidean space
–
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
9.
Rotation
–
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object always rotates around a line called a rotation axis. If the axis passes through the center of mass, the body is said to rotate upon itself. A rotation about a point, e. g. the Earth about the Sun, is called a revolution or orbital revolution. The axis is called a pole, mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two, a rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion, the axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit, there is no fundamental difference between a “rotation” and an “orbit” and or spin. The key distinction is simply where the axis of the rotation lies and this distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a rotation around the same point/axis. The reverse of a rotation is also a rotation, thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis and that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the rotations are known as yaw, pitch. This terminology is used in computer graphics. In astronomy, rotation is an observed phenomenon. Stars, planets and similar bodies all spin around on their axes, the rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features and this rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator
10.
Translation (geometry)
–
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time
11.
Euler's rotation theorem
–
It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a structure, known as a rotation group. The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry, the axis of rotation is known as an Euler axis, typically represented by a unit vector e ^. Its product by the angle is known as an axis-angle. The extension of the theorem to kinematics yields the concept of instant axis of rotation, in linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems, Euler states the theorem as follows, Theorema. Quomodocunque sphaera circa centrum suum conuertatur, semper assignari potest diameter, or, When a sphere is moved around its centre it is always possible to find a diameter whose direction in the displaced position is the same as in the initial position. Eulers original proof was made using spherical geometry and therefore whenever he speaks about triangles they must be understood as spherical triangles, to arrive at a proof, Euler analyses what the situation would look like if the theorem were true. Then he considers a great circle that does not contain O, and its image after rotation. He labels a point on their intersection as point A, now A is on the initial circle, so its image will be on the transported circle. He labels that image as point a, since A is also on the transported circle, it is the image of another point that was on the initial circle and he labels that preimage as ɑ. Then he considers the two arcs joining ɑ and a to A and these arcs have the same length because arc ɑA is mapped onto arc Aa. Also, since O is a point, triangle ɑOA is mapped onto triangle AOa, so these triangles are isosceles. Lets construct a point that could be invariant using the previous considerations and we start with the blue great circle and its image under the transformation, which is the red great circle as in the Figure 1. Let point A be a point of intersection of those circles, otherwise we label A’s image as a and its preimage as ɑ, and connect these two points to A with arcs ɑA and Aa. These arcs have the same length, then since ɑA = Aa and O is on the bisector of angle ɑAa, we also have ɑO = aO. Now lets suppose that O is the image of O, then we know angle ɑAO = angle AaO and orientation is preserved*, so O must be interior to angle ɑAa. Now AO is transformed to aO, so AO = aO, since AO is also the same length as aO, angle AaO = angle aAO
12.
Quaternions and spatial rotation
–
Unit quaternions, also known as versors, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock, compared to rotation matrices they are more compact, more numerically stable, and may be more efficient. Quaternions have applications in graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites. When used to represent rotation, unit quaternions are also called rotation quaternions, when used to represent an orientation, they are called orientation quaternions or attitude quaternions. The Euler axis is represented by a unit vector u→. Therefore. A Euclidean vector such as or can be rewritten as 2 i + 3 j + 4 k or ax i + ay j + az k, where i, j, k are unit vectors representing the three Cartesian axes. A rotation through an angle of θ around the axis defined by a vector u → = = u x i + u y j + u z k can be represented by a quaternion. In a programmatic implementation, this is achieved by constructing a quaternion whose vector part is p and real part equals zero, the vector part of the resulting quaternion is the desired vector p. The rotation is clockwise if our line of points in the same direction as u→. In this instance, q is a unit quaternion and q −1 = e − θ2 = cos θ2 − sin θ2, the scalar component of the result is necessarily zero. The quaternion inverse of a rotation is the rotation, since q −1 q = v →. The square of a rotation is a rotation by twice the angle around the same axis. More generally qn is a rotation by n times the angle around the axis as q. This can be extended to arbitrary real n, allowing for smooth interpolation between spatial orientations, see Slerp. Two rotation quaternions can be combined into one equivalent quaternion by the relation, thus, an arbitrary number of rotations can be composed together and then applied as a single rotation. Conjugating p by q refers to the operation p ↦ q p q−1, consider the rotation f around the axis v → = i + j + k, with a rotation angle of 120°, or 2π/3 radians. α =2 π3 The length of v→ is √3, the angle is π/3 with cosine 1/2. Lets show how we reached the previous result, as we can see, such computations are relatively long and tedious if done manually, however, in a computer program, this amounts to calling the quaternion multiplication routine twice
13.
Euler angles
–
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They can also represent the orientation of a frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra. Any orientation can be achieved by composing three elemental rotations, i. e. rotations about the axes of a coordinate system, Euler angles can be defined by three of these rotations. They can also be defined by geometry and the geometrical definition demonstrates that three rotations are always sufficient to reach any frame. The three elemental rotations may be extrinsic, or intrinsic, Euler angles are typically denoted as α, β, γ, or φ, θ, ψ. Different authors may use different sets of rotation axes to define Euler angles, therefore, any discussion employing Euler angles should always be preceded by their definition. Tait–Bryan angles are also called Cardan angles, nautical angles, heading, elevation, and bank, or yaw, pitch, sometimes, both kinds of sequences are called Euler angles. In that case, the sequences of the first group are called proper or classic Euler angles, the axes of the original frame are denoted as x, y, z and the axes of the rotated frame are denoted as X, Y, Z. The geometrical definition begins defining the line of nodes as the intersection of the planes xy, using it, the three Euler angles can be defined as follows, α is the angle between the x axis and the N axis. β is the angle between the z axis and the Z axis, γ is the angle between the N axis and the X axis. Euler angles between two frames are defined only if both frames have the same handedness. Intrinsic rotations are elemental rotations occur about the axes of a coordinate system XYZ attached to a moving body. Therefore, they change their orientation after each elemental rotation, the XYZ system rotates, while xyz is fixed. Starting with XYZ overlapping xyz, a composition of three intrinsic rotations can be used to any target orientation for XYZ. Euler angles can be defined by intrinsic rotations, the rotated frame XYZ may be imagined to be initially aligned with xyz, before undergoing the three elemental rotations represented by Euler angles. Hence, N can be simply denoted x’, moreover, since the third elemental rotation occurs about Z, it does not change the orientation of Z. Extrinsic rotations are elemental rotations occur about the axes of the fixed coordinate system xyz. The XYZ system rotates, while xyz is fixed, starting with XYZ overlapping xyz, a composition of three extrinsic rotations can be used to reach any target orientation for XYZ
14.
Rotation matrix
–
In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix R = rotates points in the xy-Cartesian plane counter-clockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix R, the position of each point must be represented by a vector v. A rotated vector is obtained by using the matrix multiplication Rv, Rotation matrices also provide a means of numerically representing an arbitrary rotation of the axes about the origin, without appealing to angular specification. These coordinate rotations are a way to express the orientation of a camera, or the attitude of a spacecraft. The examples in this article apply to active rotations of vectors counter-clockwise in a coordinate system by pre-multiplication. If any one of these is changed, then the inverse of the matrix should be used. Since matrix multiplication has no effect on the vector, rotation matrices can only be used to describe rotations about the origin of the coordinate system. Rotation matrices provide a description of such rotations, and are used extensively for computations in geometry, physics. Rotation matrices are matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1, that is, in some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with determinant −1. These combine proper rotations with reflections, in other cases, where reflections are not being considered, the label proper may be dropped. This convention is followed in this article, the set of all orthogonal matrices of size n with determinant +1 forms a group known as the special orthogonal group SO. The most important special case is that of the rotation group SO, the set of all orthogonal matrices of size n with determinant +1 or -1 forms the orthogonal group O. In two dimensions, every rotation matrix has the form, R =. This rotates column vectors by means of the matrix multiplication. So the coordinates of the point after rotation are x ′ = x cos θ − y sin θ, y ′ = x sin θ + y cos θ. The direction of rotation is counterclockwise if θ is positive
15.
Strike and dip
–
Strike and dip refer to the orientation or attitude of a geologic feature. The strike line of a bed, fault, or other feature, is a line representing the intersection of that feature with a horizontal plane. On a geologic map, this is represented with a straight line segment oriented parallel to the strike line. One technique is to take the strike so the dip is 90° to the right of the strike. The map symbol is a line attached and at right angles to the strike symbol pointing in the direction which the planar surface is dipping down. The angle of dip is generally included on a map without the degree sign. Beds that are dipping vertically are shown with the dip symbol on both sides of the strike, and beds that are flat are shown like the vertical beds, both vertical and flat beds do not have a number written with them. Another way of representing strike and dip is by dip and dip direction, the dip direction is the azimuth of the direction the dip as projected to the horizontal, which is 90° off the strike angle. For example, a bed dipping 30° to the South, would have an East-West strike, but would be written as 30/180 using the dip and dip direction method. Strike and dip are determined in the field with a compass, compass-clinometers which measure dip and dip direction in a single operation are often called stratum or Klar compasses after a German professor. Smartphone apps are also now available, that use of the internal accelerometer to provide orientation measurements. Combined with the GPS functionality of devices, this allows readings to be recorded. Any planar feature can be described by strike and dip and this includes sedimentary bedding, faults and fractures, cuestas, igneous dikes and sills, metamorphic foliation and any other planar feature in the Earth. Linear features are measured very similar methods, where plunge is the dip angle. Apparent dip is the name of any dip measured in a plane that is not perpendicular to the strike line. True dip can be calculated from apparent dip using trigonometry if you know the strike, Geologic cross sections use apparent dip when they are drawn at some angle not perpendicular to strike. New York, J. Wiley and Sons, tarbuck, Edward J. Lutgens, Frederick K. 0-13-092025-8 Earth, An Introduction to Physical Geology. Upper Saddle River, N. J. Pearson Prentice Hall, digital Cartographic Standard for Geologic Map Symbolization
16.
Grade (slope)
–
The grade of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of tilt, often slope is calculated as a ratio of rise to run, or as a fraction in which run is the horizontal distance and rise is the vertical distance. The grades or slopes of existing physical features such as canyons and hillsides, stream and river banks, grades are typically specified for new linear constructions. The grade may refer to the slope or the perpendicular cross slope. There are several ways to express slope, as an angle of inclination to the horizontal, as a percentage, the formula for which is 100 rise run which could also be expressed as the tangent of the angle of inclination times 100. In the U. S. this percentage grade is the most commonly used unit for communicating slopes in transportation, surveying, construction, and civil engineering. As a per mille figure, the formula for which is 1000 rise run which could also be expressed as the tangent of the angle of inclination times 1000 and this is commonly used in Europe to denote the incline of a railway. As a ratio of one part rise to so many parts run, for example, a slope that has a rise of 5 feet for every 100 feet of run would have a slope ratio of 1 in 20. This is generally the method used to describe railway grades in Australia, any of these may be used. Grade is usually expressed as a percentage, but this is converted to the angle α from horizontal or the other expressions. Slope may still be expressed when the run is not known. This is not the way to specify slope, it follows the sine function rather than the tangent function. But in practice the way to calculate slope is to measure the distance along the slope and the vertical rise. When the angle of inclination is small, using the slope length rather than the horizontal displacement makes only an insignificant difference, Railway gradients are usually expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In any case, the identity holds for all inclinations up to 90 degrees, tan α = sin α1 − sin 2 α In Europe. Grades are related using the equations with symbols from the figure at top
17.
Cartesian coordinate system
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
18.
Rotational symmetry
–
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
19.
Line segment
–
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
20.
Euclidean vector
–
In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years, about a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence, working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
21.
Direction cosines
–
In analytic geometry, the direction cosines of a vector are the cosines of the angles between the vector and the three coordinate axes. Equivalently, they are the contributions of each component of the basis to a vector in that direction. Direction cosines are an extension of the usual notion of slope to higher dimensions. It follows that by squaring each equation and adding the results cos 2 a + cos 2 b + cos 2 c =1. Here α, β and γ are the direction cosines and the Cartesian coordinates of the unit vector v/|v|, more generally, direction cosine refers to the cosine of the angle between any two vectors. They are useful for forming direction cosine matrices that express one set of basis vectors in terms of another set. Spiegel, M. R. Lipschutz, S. Spellman, an introduction to tensor analysis for engineers and applied scientists. Mathematical Methods for Engineers and Scientists
22.
Normal (geometry)
–
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a normal, or simply normal. The word normal is used as an adjective, a line normal to a plane, the normal component of a force. The concept of normality generalizes to orthogonality, the concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at a point P is the set of the vectors which are orthogonal to the tangent space at P, in the case of differential curves, the curvature vector is a normal vector of special interest. For a convex polygon, a surface normal can be calculated as the cross product of two edges of the polygon. For a plane given by the equation a x + b y + c z + d =0, the vector is a normal. For a hyperplane in n+1 dimensions, given by the equation r = a 0 + α1 a 1 + ⋯ + α n a n, where a0 is a point on the hyperplane and ai for i =1. N are non-parallel vectors lying on the hyperplane, a normal to the hyperplane is any vector in the space of A where A is given by A =. That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. If a surface S is parameterized by a system of coordinates x, with s and t real variables. For a surface S given explicitly as a function f of the independent variables x, y, the first one is obtaining its implicit form F = z − f =0, from which the normal follows readily as the gradient ∇ F. The second way of obtaining the normal follows directly from the gradient of the form, ∇ f, by inspection, ∇ F = k ^ − ∇ f. Note that this is equal to ∇ F = k ^ − ∂ f ∂ x i ^ − ∂ f ∂ y j ^, if a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base, however, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous, a normal to a surface does not have a unique direction, the vector pointing in the opposite direction of a surface normal is also a surface normal. For an oriented surface, the normal is usually determined by the right-hand rule
23.
Two-dimensional space
–
In physics and mathematics, two-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called length and width, both directions lie in the same plane. A sequence of n numbers can be understood as a location in n-dimensional space. When n =2, the set of all locations is called two-dimensional space or bi-dimensional space. Each reference line is called an axis or just axis of the system. The coordinates can also be defined as the positions of the projections of the point onto the two axes, expressed as signed distances from the origin. The idea of system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided and this was known as the complex plane. The complex plane is called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin and they are usually labeled x and y. Another widely used system is the polar coordinate system, which specifies a point in terms of its distance from the origin. In two dimensions, there are infinitely many polytopes, the polygons, the first few regular ones are shown below, The Schläfli symbol represents a regular p-gon. The regular henagon and regular digon can be considered degenerate regular polygons and they can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus. There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers and they are called star polygons and share the same vertex arrangements of the convex regular polygons
24.
Geometric shape
–
A geometric shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object. That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the shape as the original. Objects that have the shape as each other are said to be similar. If they also have the scale as each other, they are said to be congruent. Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, such shapes are called polygons and include triangles, squares, and pentagons. Other shapes may be bounded by such as the circle or the ellipse. Such shapes are called polyhedrons and include cubes as well as such as tetrahedrons. Other three-dimensional shapes may be bounded by curved surfaces, such as the ellipsoid, a shape is said to be convex if all of the points on a line segment between any two of its points are also part of the shape
25.
Rotation formalisms in three dimensions
–
In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a rotational motion. According to Eulers rotation theorem the rotation of a body is described by a single rotation about some axis. Such a rotation may be described by a minimum of three real parameters. However, for reasons, there are several ways to represent it. Many of these representations use more than the minimum of three parameters, although each of them still has only three degrees of freedom. An example where rotation representation is used is in computer vision, lets consider a rigid body, with three orthogonal unit vectors fixed to its body. The basic problem is to specify the orientation of three unit vectors, and hence the rigid body, with respect to the observers coordinate system. Rotation formalisms are focused on proper motions of the Euclidean space with one fixed point, although physical motions with a fixed point are an important case, this approach creates a knowledge about all motions. Any proper motion of the Euclidean space decomposes to a rotation around the origin, whichever the order of their composition will be, the pure rotation component wouldnt change, uniquely determined by the complete motion. One can also understand pure rotations as linear maps in a space equipped with Euclidean structure. In other words, a rotation formalism captures only the part of a motion. The above-mentioned triad of unit vectors is called a basis. Specifying the coordinates of vectors of this basis in its current position, in terms of the coordinate axes. The three unit vectors, u ^, v ^ and w ^, that form the basis each consist of 3 coordinates. These parameters can be written as the elements of a 3×3 matrix A, the rotation matrix has the following properties, A is a real, orthogonal matrix, hence each of its rows or columns represents a unit vector. The eigenvalues of A are = where i is the imaginary unit with the property i2 = −1 The determinant of A is +1. The trace of A is 1 +2 cos , equivalent to the sum of its eigenvalues, the angle θ which appears in the eigenvalue expression corresponds to the angle of the Euler axis and angle representation
26.
Leonhard Euler
–
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
27.
Unit vector
–
In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is denoted by a lowercase letter with a circumflex, or hat. The term direction vector is used to describe a unit vector being used to represent spatial direction, two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are equivalent numerically to points on the unit circle, the same construct is used to specify spatial directions in 3D. As illustrated, each direction is equivalent numerically to a point on the unit sphere. The normalized vector or versor û of a vector u is the unit vector in the direction of u, i. e. u ^ = u ∥ u ∥ where ||u|| is the norm of u. The term normalized vector is used as a synonym for unit vector. Unit vectors are often chosen to form the basis of a vector space, every vector in the space may be written as a linear combination of unit vectors. By definition, in a Euclidean space the dot product of two vectors is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the product of two arbitrary unit vectors is a 3rd vector orthogonal to both of them having length equal to the sine of the smaller subtended angle. Unit vectors may be used to represent the axes of a Cartesian coordinate system and they are often denoted using normal vector notation rather than standard unit vector notation. In most contexts it can be assumed that i, j, the notations, or, with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity. When a unit vector in space is expressed, with Cartesian notation, as a combination of i, j, k. The value of each component is equal to the cosine of the angle formed by the vector with the respective basis vector. This is one of the used to describe the orientation of a straight line, segment of straight line, oriented axis. It is important to note that ρ ^ and φ ^ are functions of φ, when differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. For a more complete description, see Jacobian matrix, to minimize degeneracy, the polar angle is usually taken 0 ≤ θ ≤180 ∘. It is especially important to note the context of any ordered triplet written in spherical coordinates, here, the American physics convention is used
28.
Eigenvalues and eigenvectors
–
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it. This condition can be written as the equation T = λ v, there is a correspondence between n by n square matrices and linear transformations from an n-dimensional vector space to itself. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. Geometrically an eigenvector, corresponding to a real eigenvalue, points in a direction that is stretched by the transformation. If the eigenvalue is negative, the direction is reversed, Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen for proper, inherent, own, individual, special, specific, peculiar, or characteristic. In essence, an eigenvector v of a linear transformation T is a vector that. Applying T to the eigenvector only scales the eigenvector by the scalar value λ and this condition can be written as the equation T = λ v, referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar, for example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex. The Mona Lisa example pictured at right provides a simple illustration, each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping, the vectors pointing to each point in the original image are therefore tilted right or left and made longer or shorter by the transformation. Notice that points along the horizontal axis do not move at all when this transformation is applied, therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation because the mapping does not change its direction. Moreover, these all have an eigenvalue equal to one because the mapping does not change their length. Linear transformations can take different forms, mapping vectors in a variety of vector spaces. Alternatively, the transformation could take the form of an n by n matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix, the set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T. If the set of eigenvectors of T form a basis of the domain of T, Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms, in the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes
29.
Symmetry
–
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
30.
Orthogonal group
–
Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O is the orthogonal group, denoted SO. This group is called the rotation group, because, in dimensions 2 and 3. In low dimension, these groups have been studied, see SO, SO and SO. This is a subgroup of the linear group GL given by O = where QT is the transpose of Q and I is the identity matrix. This article mainly discusses the groups of quadratic forms that may be expressed over some bases as the dot product, over the reals. Over the reals, for any quadratic form, there is a basis. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O, for details, see indefinite orthogonal group. The derived subgroup Ω of O is an often studied object because, the Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form. The determinant of any orthogonal matrix is either 1 or −1, the orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O known as the special orthogonal group SO, consisting of all proper rotations. By analogy with GL–SL, the group is sometimes called the general orthogonal group and denoted GO. The term rotation group can be used to either the special or general orthogonal group. When this distinction is to be emphasized, the groups may be denoted O and O, reserving n for the dimension of the space. The letters p or r are also used, indicating the rank of the corresponding Lie algebra, in odd dimension the corresponding Lie algebra is s o, while in even dimension the Lie algebra is s o. In two dimensions, O is the group of all rotations about the origin and all reflections along a line through the origin, SO is the group of all rotations about the origin. These groups are related, SO is a subgroup of O of index 2. More generally, in any number of dimensions an even number of reflections gives a rotation, therefore, the rotations define a subgroup of O, but the reflections do not define a subgroup. A reflection through the origin may be generated as a combination of one reflection along each of the axes, the reflection through the origin is not a reflection in the usual sense in even dimensions, but rather a rotation
31.
Product topology
–
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. The product topology is called the Tychonoff topology. The open sets in the product topology are unions of sets of the form ∏ i ∈ I U i, in particular, for a finite product, the products of base elements of the Xi gives a basis for the product ∏ i ∈ I X i. The product topology on X is the topology generated by sets of the form pi−1, in other words, the sets form a subbase for the topology on X. A subset of X is open if and only if it is a union of intersections of many sets of the form pi−1. The pi−1 are sometimes called open cylinders, and their intersections are cylinder sets, in general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general, the box topology is finer than the product topology, if one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn. Several additional examples are given in the article on the initial topology and it follows from the above universal property that a map f, Y → X is continuous if and only if fi = pi o f is continuous for all i in I. In many cases it is easier to check that the component functions fi are continuous, checking whether a map f, Y → X is continuous is usually more difficult, one tries to use the fact that the pi are continuous in some way. In addition to being continuous, the projections pi, X → Xi are open maps. This means that any subset of the product space remains open when projected down to the Xi. The converse is not true, if W is a subspace of the space whose projections down to all the Xi are open, then W need not be open in X. The canonical projections are not generally closed maps, the product topology is also called the topology of pointwise convergence because of the following fact, a sequence in X converges if and only if all its projections to the spaces Xi converge. In particular, if one considers the space X = RI of all real valued functions on I, any product of closed subsets of Xi is a closed set in X. An important theorem about the topology is Tychonoffs theorem, any product of compact spaces is compact. This is easy to show for finite products, while the general statement is equivalent to the axiom of choice, however, an arbitrary product of locally compact spaces where all but finitely many are compact is locally compact. Connectedness Every product of connected spaces is connected Every product of hereditarily disconnected spaces is hereditarily disconnected, metric spaces Countable products of metric spaces are metrizable The axiom of choice is equivalent to the statement that the product of a collection of non-empty sets is non-empty. The proof is easy enough, one only to pick an element from each set to find a representative in the product
32.
Tangent space
–
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
33.
Magnetic declination
–
Magnetic declination or variation is the angle on the horizontal plane between magnetic north and true north. This angle varies depending on position on the Earths surface, and changes over time. The angle between magnetic and grid meridians is called grid magnetic angle, grid variation, or grivation. ”By convention, declination is positive when magnetic north is east of true north, and negative when it is to the west. Isogonic lines are lines on the Earths surface along which the declination has the constant value. The lowercase Greek letter δ is frequently used as the symbol for magnetic declination, Magnetic declination varies both from place to place and with the passage of time. As a traveller cruises the east coast of the United States, for example, the declination at London, UK is one degree 7 minutes west, and as the country is quite small that figure is fairly good for the whole of the country. It is reducing, and scientists predict that in about 2050 it will be zero, similarly, secular changes to these flows result in slow changes to the field strength and direction at the same point on the Earth. The magnetic declination in an area may change slowly over time, possibly as little as 2–2.5 degrees every hundred years or so. For a location closer to the pole like Ivujivik, the declination may change by 1 degree every three years and this may be insignificant to most travellers, but can be important if using magnetic bearings from old charts or metes in old deeds for locating places with any precision. As an example of how variation changes over time, see the two charts of the area, below, surveyed 124 years apart. The 1884 chart shows a variation of 8 degrees,20 minutes West, the 2008 chart shows 13 degrees,15 minutes West. The instrument used to perform this measurement is known as a declinometer, the approximate position of the north celestial pole is indicated by Polaris. In the northern hemisphere, declination can therefore be determined as the difference between the magnetic bearing and a visual bearing on Polaris. Polaris currently traces a circle 0. 75° in radius around the celestial pole. At high latitudes a plumb-bob is helpful to sight Polaris against an object close to the horizon. A rough estimate of the local declination can be determined from a general isogonic chart of the world or a continent, isogonic lines are also shown on aeronautical and nautical charts. Larger-scale local maps may indicate current local declination, often with the aid of a schematic diagram, unless the area depicted is very small, declination may vary measurably over the extent of the map, so the data may be referred to a specific location on the map. The current rate and direction of change may also be shown, the same diagram may show the angle of grid north, which may differ from true north
34.
Flight dynamics
–
Flight dynamics is the study of the performance, stability, and control of vehicles flying through the air or in outer space. It is concerned with how forces acting on the vehicle influence its speed, in fixed-wing aircraft, the changing orientation of the vehicle with respect to the local air flow is represented by two critical parameters, angle of attack and angle of sideslip. These angles describe the direction of airspeed, important because they are the principal source of modulations in the aerodynamic forces. Spacecraft flight dynamics involve three forces, propulsive, gravitational, and lift and drag, because aerodynamic forces involved with spacecraft flight are very small, this leaves gravity as the dominant force. These angles are the product of the equations of motion. For all flight vehicles, these two sets of dynamics, rotational and translational, operate simultaneously and in a fashion to evolve the vehicles state trajectory. This section focuses on fixed-wing aircraft, flight dynamics is the science of air-vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the center of mass, known as roll, pitch and yaw. Aircraft engineers develop control systems for a vehicles orientation about its center of mass, for example, a pitching moment is a vertical force applied at a distance forward or aft from the center of gravity of the aircraft, causing the aircraft to pitch up or down. Roll, pitch and yaw refer, in context, to rotations about the respective axes starting from a defined equilibrium state. The equilibrium roll angle is known as wings level or zero bank angle, a fixed-wing aircraft increases or decreases the lift generated by the wings when it pitches nose up or down by increasing or decreasing the angle of attack. The roll angle is known as bank angle on a fixed-wing aircraft. The forces acting on spacecraft are of three types, propulsive force, gravitational force exerted by the Earth and other celestial bodies, the vehicles attitude must be taken into account because of its effect on the aerodynamic and propulsive forces. There are other reasons, unrelated to flight dynamics, for controlling the attitude in non-powered flight. Also, most of a flight time is usually unpowered
35.
Attitude control
–
Attitude control is controlling the orientation of an object with respect to an inertial frame of reference or another entity. The integrated field that studies the combination of sensors, actuators and algorithms is called Guidance, Navigation, a spacecrafts attitude must typically be stabilized and controlled for a variety of reasons. Propulsion system thrusters are fired only occasionally to make desired changes in spin rate, if desired, the spinning may be stopped through the use of thrusters or by yo-yo de-spin. The Pioneer 10 and Pioneer 11 probes in the solar system are examples of spin-stabilized spacecraft. Three-axis stabilization is a method of spacecraft attitude control in which the spacecraft is held fixed in the desired orientation without any rotation. One method is to use thrusters to continually nudge the spacecraft back. Thrusters may also be referred to as control systems, or reaction control systems. The space probes Voyager 1 and Voyager 2 employ this method, another method for achieving three-axis stabilization is to use electrically powered reaction wheels, also called momentum wheels, which are mounted on three orthogonal axes aboard the spacecraft. They provide a means to trade angular momentum back and forth between spacecraft and wheels, to rotate the vehicle on a given axis, the reaction wheel on that axis is accelerated in the opposite direction. To rotate the back, the wheel is slowed. This is done during maneuvers called momentum desaturation or momentum unload maneuvers, most spacecraft use a system of thrusters to apply the torque for desaturation maneuvers. A different approach was used by the Hubble Space Telescope, which had sensitive optics that could be contaminated by thruster exhaust, there are advantages and disadvantages to both spin stabilization and three-axis stabilization. Many spacecraft have components that require articulation, Voyager and Galileo, for example, were designed with scan platforms for pointing optical instruments at their targets largely independently of spacecraft orientation. Many spacecraft, such as Mars orbiters, have solar panels that must track the Sun so they can provide power to the spacecraft. Cassinis main engine nozzles are steerable, knowing where to point a solar panel, or scan platform, or a nozzle — that is, how to articulate it—requires knowledge of the spacecrafts attitude. Because AACS keeps track of the attitude, the Suns location. It logically falls to one subsystem, then, to manage both attitude and articulation, the name AACS may even be carried over to a spacecraft even if it has no appendages to articulate. Many sensors generate outputs that reflect the rate of change in attitude and these require a known initial attitude, or external information to use them to determine attitude
36.
Plane of rotation
–
In geometry, a plane of rotation is an abstract object used to describe or visualise rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, mathematically such planes can be described in a number of ways. They can be described in terms of planes and angles of rotation and they can be associated with bivectors from geometric algebra. They are related to the eigenvalues and eigenvectors of a rotation matrix, and in particular dimensions they are related to other algebraic and geometric properties, which can then be generalised to other dimensions. The main use for them is in describing more complex rotations in higher dimensions and this can be done using geometric algebra, with the planes of rotations associated with simple bivectors in the algebra. For this article, all planes are planes through the origin, such a plane in n-dimensional space is a two-dimensional linear subspace of the space. If the bivector a ∧ b is written B, then the condition that a point lies on the associated with B is simply x ∧ B =0. This is true in all dimensions, and can be taken as the definition on the plane. In particular, from the properties of the product it is satisfied by both a and b, and so by any vector of the form c = λ a + μ b. As λ and μ range over all numbers, c ranges over the whole plane. A plane of rotation for a rotation is a plane that is mapped to itself by the rotation. The plane is not fixed, but all vectors in the plane are mapped to other vectors in the plane by the rotation. This transformation of the plane to itself is always a rotation about the origin, every rotation except for the identity rotation has at least one plane of rotation, and up to ⌊ n 2 ⌋ planes of rotation, where n is the dimension. The maximum number of planes up to eight dimensions is shown in table, When a rotation has multiple planes of rotation they are always orthogonal to each other. This can only happen in four or more dimensions, in two dimensions there is only one plane, while in three dimensions all planes have at least one nonzero vector in common, along their line of intersection. In more than three planes of rotation are not always unique. In two-dimensional space there is one plane of rotation, the plane of the space itself. In a Cartesian coordinate system it is the Cartesian plane, in numbers it is the complex plane
37.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
