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A conformal linear transformation, also called a homogeneous similarity transformation or homogeneous similitude, is a similarity transformation of a Euclidean or pseudo-Euclidean vector space which fixes the origin. It can be written as the composition of an orthogonal transformation (an origin-preserving rigid transformation) with a uniform scaling (dilation). All similarity transformations (which globally preserve the shape but not necessarily the size of geometric figures) are also conformal (locally preserve shape). Similarity transformations which fix the origin also preserve scalar–vector multiplication and vector addition, making them linear transformations.
Every origin-fixing reflection, rotation, or dilation is a conformal linear transformation, as is any composition of these basic transformations, including improper rotations and most generally similarity transformations. However, shear transformations and non-uniform scaling are not.
As linear transformations, conformal linear transformations are representable by matrices once the vector space has been given a basis, composing with each-other and transforming vectors by matrix multiplication. The Lie group of these transformations has been called the conformal orthogonal group, the conformal linear transformation group or the homogeneous similtude group.
Alternatively any conformal linear transformation can be represented as a versor (geometric product of vectors); every versor and its negative represent the same transformation, so the versor group (also called the Lipschitz group) is a double cover of the conformal orthogonal group.
Conformal linear transformations are a special type of Möbius transformations (conformal transformations mapping circles to circles); the conformal ortogonal group is a subgroup of the conformal group.
Across all dimensions, a conformal linear transformation has the following properties:
In 2D, a conformal linear transformation has a special form. For a non-flipped conformal 2D basis, it looks like this:
Or, in the case of a flip/reflection, the form is similar but with the signs swapped in the second column:
This form occurs because in order for a transformation matrix to be conformal, the second column must be 90 degrees apart from the first column (orthogonal), and the same length (uniform scale). This gives only 2 possible locations for the second column, one flipped, and one non-flipped.
A similar form can occur in other dimensions when there is only rotation between two axes.[ citation needed ]
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When composing multiple linear transformations, it is possible to create a shear/skew by composing a parent transform with a non-uniform scale, and a child transform with a rotation. Therefore, in situations where shear/skew is not allowed, transformation matrices must also have uniform scale in order to prevent a shear/skew from appearing as the result of composition. This implies conformal linear transformations are required to prevent shear/skew when composing multiple transformations.
In physics simulations, a sphere (or circle, hypersphere, etc.) is often defined by a point and a radius. Checking if a point overlaps the sphere can therefore be performed by using a distance check to the center. With a rotation or flip/reflection, the sphere is symmetric and invariant, therefore the same check works. With a uniform scale, only the radius needs to be changed. However, with a non-uniform scale or shear/skew, the sphere becomes "distorted" into an ellipsoid, therefore the distance check algorithm does not work correctly anymore.
2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models and by techniques specific to them. It may refer to the branch of computer science that comprises such techniques or to the models themselves.
In Euclidean geometry, an affine transformation or affinity is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order . Any two square matrices of the same order can be added and multiplied.
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear 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). The orthogonal group is an algebraic group and a Lie group. It is compact.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
In mathematics, particularly in linear algebra, a skew-symmetricmatrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices.
Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space.
In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation. There are three degrees of freedom, so that the dimension of SO(3) is three. In numerous applications one or other coordinate system is used, and the question arises how to convert from a given system to another.
In linear algebra, a Householder transformation is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder.
In affine geometry, uniform scaling is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.
In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.
In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators.
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space Rp,q to null vectors in Rp+1,q+1. This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations.
In mathematics, a geometric transformation is any bijection of a set to itself with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both or both — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations.