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Geometric transformations can be distinguished into two types: active or alibi transformations which change the physical position of a set of points relative to a fixed frame of reference or coordinate system (alibi meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the ...
Affine transformation (Euclidean geometry) Bäcklund transform; Bilinear transform; Box–Muller transform; Burrows–Wheeler transform (data compression) Chirplet transform; Distance transform; Fractal transform; Gelfand transform; Hadamard transform; Hough transform (digital image processing) Inverse scattering transform; Legendre ...
In mathematics, transform theory is the study of transforms, which relate a function in one domain to another function in a second domain. The essence of transform theory is that by a suitable choice of basis for a vector space a problem may be simplified—or diagonalized as in spectral theory .
In mathematics, a transformation, transform, or self-map [1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. [ 2 ] [ 3 ] [ 4 ] Examples include linear transformations of vector spaces and geometric transformations , which include projective transformations , affine transformations , and ...
For example, if the affine transformation acts on the plane and if the determinant of is 1 or −1 then the transformation is an equiareal mapping. Such transformations form a subgroup called the equi-affine group. [13] A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance.
A Transformation Approach to Tenth Grade Geometry, The Mathematics Teacher, Vol. 65, No. 1 (January 1972), pp. 21-30. Zalman P. Usiskin. The Effects of Teaching Euclidean Geometry via Transformations on Student Achievement and Attitudes in Tenth-Grade Geometry, Journal for Research in Mathematics Education, Vol. 3, No. 4 (Nov., 1972), pp. 249-259.
For example, every integral transform is a linear operator, since the integral is a linear operator, and in fact if the kernel is allowed to be a generalized function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem).
A prime example – in mathematics and physics – would be the theory of spherical harmonics. Their role in the group theory of the rotation groups is that of being a representation space for the entire set of finite-dimensional irreducible representations of the rotation group SO(3). For this topic, see Rotation group SO(3) § Spherical harmonics