1.
Affine transformation
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In geometry, an affine transformation, affine map or an affinity is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation, an affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. If X and Y are affine spaces, then every affine transformation f, X → Y is of the form x ↦ M x + b, unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear, all Euclidean spaces are affine, but there are affine spaces that are non-Euclidean. In affine coordinates, which include Cartesian coordinates in Euclidean spaces, another way to deal with affine transformations systematically is to select a point as the origin, then, any affine transformation is equivalent to a linear transformation followed by a translation. An affine map f, A → B between two spaces is a map on the points that acts linearly on the vectors. In symbols, f determines a linear transformation φ such that and we can interpret this definition in a few other ways, as follows. If an origin O ∈ A is chosen, and B denotes its image f ∈ B, the conclusion is that, intuitively, f consists of a translation and a linear map. In other words, f preserves barycenters, as shown above, an affine map is the composition of two functions, a translation and a linear map. Ordinary vector algebra uses matrix multiplication to represent linear maps, using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. If A is a matrix, = is equivalent to the following y → = A x → + b →, the above-mentioned augmented matrix is called an affine transformation matrix, or projective transformation matrix. This representation exhibits the set of all affine transformations as the semidirect product of K n and G L. This is a group under the operation of composition of functions, ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending the additional coordinate 1 to every vector, one considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the coordinate is 1. Thus the origin of the space can be found at. A translation within the space by means of a linear transformation of the higher-dimensional space is then possible

2.
Rotation of axes
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A point P has coordinates with respect to the original system and coordinates with respect to the new system. In the new system, the point P will appear to have been rotated in the opposite direction. A rotation of axes in more than two dimensions is defined similarly, a rotation of axes is a linear map and a rigid transformation. Coordinate systems are essential for studying the equations of curves using the methods of analytic geometry, to use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of ellipses and hyperbolas, If the curve is not situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a transformation of coordinates, the solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin. The equations defining the transformation in two dimensions, which rotates the xy axes counterclockwise through an angle θ into the xy axes, are derived as follows, in the xy system, let the point P have polar coordinates. Then, in the xy system, P will have polar coordinates, the inverse transformation is or =. Find the coordinates of the point P1 = = after the axes have been rotated through the angle θ1 = π /6, or 30°. Solution, x ′ =3 cos +1 sin = + =2 y ′ =1 cos −3 sin = − =0. The axes have been rotated counterclockwise through an angle of θ1 = π /6, note that the point appears to have been rotated clockwise through π /6, that is, it now coincides with the x axis. Find the coordinates of the point P2 = = after the axes have been rotated clockwise 90°, the axes have been rotated through an angle of θ2 = − π /2, which is in the clockwise direction and the new coordinates are P2 = =. Again, note that the point appears to have been rotated counterclockwise through π /2, the most general equation of the second degree has the form Through a change of coordinates, equation can be put into a standard form, which is usually easier to work with. It is always possible to rotate the coordinates in such a way that in the new system there is no xy term. Substituting equations and into equation, we obtain where If θ is selected so that cot 2 θ = / B we will have B ′ =0 and the xy term in equation will vanish. When a problem arises with B, D and E all different from zero, they can be eliminated by performing in succession a rotation, a non-degenerate conic section given by equation can be identified by evaluating B2 −4 A C. The conic section is, { an ellipse or a circle, if B2 −4 A C <0, a parabola, if B2 −4 A C =0, a hyperbola, if B2 −4 A C >0. Suppose a rectangular xyz-coordinate system is rotated around its z axis counterclockwise through an angle θ, that is, the z coordinate of each point is unchanged and the x and y coordinates transform as above

3.
Rotation matrix
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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

4.
Transformation (function)
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In mathematics, particularly in semigroup theory, a transformation is a function f that maps a set X to itself, i. e. f, X → X. In other areas of mathematics, a transformation may simply be any function and this wider sense shall not be considered in this article, refer instead to the article on function for that sense. Examples include linear transformations and affine transformations, rotations, reflections and translations and these can be carried out in Euclidean space, particularly in R2 and R3. They are also operations that can be performed using linear algebra, a translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, in other words, if v is a fixed vector, then the translation Tv will work as Tv = p + v. For the purpose of visualization, consider a browser window and this window, if maximized to full dimensions of the screen, is the reference plane. Imagine one of the corners as the point or origin. Consider a point P in the corresponding plane, now the axes are shifted from the original axes to a distance and this is the corresponding reference axes. Replacing these values or using these equations in the equation we obtain the transformed equation or new reference axes, old reference axes. A reflection is a map that transforms an object into its mirror image with respect to a mirror, for example, a reflection of the small Latin letter p with respect to a vertical line would look like a q. In order to reflect a planar figure one needs the mirror to be a line, reflection may be considered as the limiting case of inversion as the radius of the reference circle increases without bound. Reflection is considered to be a motion since it changes the orientation of the figures it reflects. A glide reflection is a type of isometry of the Euclidean plane, the combination of a reflection in a line, reversing the order of combining gives the same result. Depending on context, we may consider a reflection as a special case where the translation vector is the zero vector. A rotation is a transformation that is performed by spinning the object around a point known as the center of rotation. You can rotate the object at any degree measure, but 90° and 180° are two of the most common, rotation by a positive angle rotates the object counterclockwise, whereas rotation by a negative angle rotates the object clockwise. Uniform scaling is a transformation that enlarges or diminishes objects. The result of scaling is similar to the original

5.
Vector projection
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The vector projection of a vector a on a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. It is a parallel to b, defined as a 1 = a 1 b ^ where a 1 is a scalar, called the scalar projection of a onto b. The scalar projection is equal to the length of the vector projection, the vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b, is the orthogonal projection of a onto the plane orthogonal to b. Both the projection a1 and rejection a2 of a vector a are vectors, and their sum is equal to a, typically, a vector projection is denoted in a bold font, and the corresponding scalar projection with normal font. In some cases, especially in handwriting, the projection is also denoted using a diacritic above or below the letter. The vector projection of a on b and the corresponding rejection are sometimes denoted by a∥b and a⊥b, the scalar projection of a on b is a scalar equal to a 1 = | a | cos θ where θ is the angle between a and b. A scalar projection can be used as a factor to compute the corresponding vector projection. The vector projection of a on b is a vector whose magnitude is the projection of a on b. The latter formula is more efficient than the former. By definition, a 2 = a − a 1 Hence, the scalar projection a on b is a scalar which has a negative sign if 90 < θ ≤180 degrees. It coincides with the length |c| of the vector projection if the angle is smaller than 90°, more exactly, a1 = |a1| if 0 ≤ θ ≤90 degrees, a1 = −|a1| if 90 < θ ≤180 degrees. The vector projection of a on b is a vector a1 which is null or parallel to b. More exactly, a1 =0 if θ = 90°, a1 and b have the direction if 0 ≤ θ <90 degrees, a1. The vector rejection of a on b is a vector a2 which is null or orthogonal to b. More exactly, a2 =0 if θ =0 degrees or θ =180 degrees, a2 is orthogonal to b if 0 < θ <180 degrees and it is also used in the Separating axis theorem to detect whether two convex shapes intersect. In some cases, the inner product coincides with the dot product, whenever they dont coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection. The projection of a vector on a plane is its orthogonal projection on that plane, the rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. The first is parallel to the plane, the second is orthogonal, for a given vector and plane, the sum of projection and rejection is equal to the original vector

6.
Glide reflection
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In 2-dimensional geometry, a glide reflection is a type of opposite isometry of the Euclidean plane, the composition of a reflection in a line and a translation along that line. A single glide is represented as frieze group p11g, a glide reflection can be seen as a limiting rotoreflection, where the rotation becomes a translation. It can also be given a Schoenflies notation as S2∞, Coxeter notation as, the combination of a reflection in a line and a translation in a perpendicular direction is a reflection in a parallel line. However, a reflection cannot be reduced like that. Thus the effect of a combined with any translation is a glide reflection. These are the two kinds of indirect isometries in 2D, for example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. It fixes a system of parallel lines, the isometry group generated by just a glide reflection is an infinite cyclic group. In the case of reflection symmetry, the symmetry group of an object contains a glide reflection. If that is all it contains, this type is frieze group p11g, example pattern with this symmetry group, Frieze group nr.6 is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a product of Z and C2. Example pattern with this group, A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach. For any symmetry group containing some glide reflection symmetry, the vector of any glide reflection is one half of an element of the translation group. This corresponds to wallpaper group pg, with additional symmetry it occurs also in pmg, pgg, if there are also true reflection lines in the same direction then they are evenly spaced between the glide reflection lines. A glide reflection line parallel to a true reflection line already implies this situation and this corresponds to wallpaper group cm. With additional symmetry it occurs also in cmm, p3m1, p31m, p4m, in 3D the glide reflection is called a glide plane. It is a reflection in a combined with a translation parallel to the plane. In the Euclidean plane 3 of 17 wallpaper groups require glide reflection generators, p2gg has orthogonal glide reflections and 2-fold rotations. Cm has parallel mirrors and glides, and pg has parallel glides, Glide symmetry can be observed in nature among certain fossils of the Ediacara biota, the machaeridians, and certain palaeoscolecid worms

7.
Homography
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In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation, in general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation, at the end of 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. The term projective transformation originated in these abstract constructions and these constructions divide into two classes that have been shown to be equivalent. For sake of simplicity, unless stated, the projective spaces considered in this article are supposed to be defined over a field. Equivalently Pappuss hexagon theorem and Desargues theorem are supposed to be true, a large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold. The projection is not defined if the point A belongs to the passing through O. The notion of space was originally introduced by extending the Euclidean space. Given another plane Q, which does not contain O, the restriction to Q of the projection is called a perspectivity. With these definitions, a perspectivity is only a partial function, therefore, this notion is normally defined for projective spaces. Originally, a homography was defined as the composition of a number of perspectivities. It is a part of the theorem of projective geometry that this definition coincides with the more algebraic definition sketched in the introduction. A projective space P of dimension n over a field K may be defined as the set of the lines through the origin in a K-vector space V of dimension n +1. If a basis of V has been fixed, a point of V may be represented by a point of Kn+1. A point of P, being a line in V, may thus be represented by the coordinates of any point of this line. Given two projective spaces P and P of the dimension, an homography is a mapping from P to P. Such an isomorphism induces a bijection from P to P, because of the linearity of f, two such isomorphisms, f and g, define the same homography if and only if there is a nonzero element a of K such that g = af. This may be written in terms of coordinates in the following way, A homography φ may be defined by a nonsingular n+1 × n+1 matrix

8.
Reflection (mathematics)
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In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points, this set is called the axis or plane of reflection. The image of a figure by a reflection is its image in the axis or plane of reflection. For example the image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b, a reflection is an involution, when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term reflection is used for a larger class of mappings from a Euclidean space to itself. Such isometries have a set of fixed points that is an affine subspace, for instance a reflection through a point is an involutive isometry with just one fixed point, the image of the letter p under it would look like a d. This operation is known as a central inversion, and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation, other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term reflection means reflection in a hyperplane, a figure that does not change upon undergoing a reflection is said to have reflectional symmetry. Some mathematicians use flip as a synonym for reflection, in a plane geometry, to find the reflection of a point drop a perpendicular from the point to the line used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure, step 2, construct circles centered at A′ and B′ having radius r. P and Q will be the points of intersection of two circles. Point Q is then the reflection of point P through line AB, the matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1,1,1. The product of two matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an number of reflections in hyperplanes through the origin. Thus reflections generate the group, and this result is known as the Cartan–Dieudonné theorem. Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes, in general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups, note that the second term in the above equation is just twice the vector projection of v onto a

9.
Transformation matrix
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In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping Rn to Rm and x → is a vector with n entries, then T = A x → for some m×n matrix A. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors, matrices allow arbitrary linear transformations to be represented in a consistent format, suitable for computation. This also allows transformations to be concatenated easily, linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on an n-dimensional Euclidean space Rn can be represented as linear transformations on the n+1-dimensional space Rn+1 and these include both affine transformations and projective transformations. For this reason, 4×4 transformation matrices are used in 3D computer graphics. With respect to a matrix, an n+1-dimensional matrix can be described as an augmented matrix. The distinction between active and passive transformations is important, by default, by transformation, mathematicians usually mean active transformations, while physicists could mean either. Put differently, a passive transformation refers to description of the object as viewed from two different coordinate frames. In other words, A = For example, the function T =5 x is a linear transformation, nevertheless, the method to find the components remains the same. Being diagonal means that all coefficients a i, j but a i, i are zeros leaving only one term in the sum ∑ a i, j e → i above. The surviving diagonal elements, a i, i, are known as eigenvalues and designated with λ i in the defining equation, the resulting equation is known as eigenvalue equation. The eigenvectors and eigenvalues are derived from it via the characteristic polynomial, with diagonalization, it is often possible to translate to and from eigenbases. In two dimensions, linear transformations can be represented using a 2×2 transformation matrix, a stretch in the xy-plane is a linear transformation which enlarges all distances in a particular direction by a constant factor but does not affect distances in the perpendicular direction. We only consider stretches along the x-axis and y-axis, a stretch along the x-axis has the form x = kx, y = y for some positive constant k. In formats such as SVG where the y axis points down, for shear mapping, there are two possibilities. A shear parallel to the x axis has x ′ = x + k y and y ′ = y. Then use the matrix, A =1 ∥ u → ∥2 As with reflections

10.
Translation (geometry)
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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.
Homothetic transformation
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In Euclidean geometry homotheties are the similarities that fix a point and either preserve or reverse the direction of all vectors. Together with the translations, all homotheties of a space form a group. These are precisely the transformations with the property that the image of every line L is a line parallel to L. In projective geometry, a transformation is a similarity transformation that leaves the line at infinity pointwise invariant. In Euclidean geometry, a homothety of ratio λ multiplies distances between points by |λ| and all areas by λ2, the first number is called the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1, the image of a point after a homothety with center and scale factor λ is given by. Hadamard, J. Lessons in Plane Geometry, meserve, Bruce E. Homothetic transformations, Fundamental Concepts of Geometry, Addison-Wesley, pp. 166–169. Tuller, Annita, A Modern Introduction to Geometries

12.
Linear map
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In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE