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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 or dilation is a conformal linear transformation, as is any composition of these basic transformations, including rotations and improper rotations and most generally similarity transformations. However, shear transformations and non-uniform scaling are not. Conformal linear transformations come in two types, proper transformations preserve the orientation of the space whereas improper transformations reverse it.
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);[1] 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 orthogonal group is a subgroup of the conformal group.
^Staples, G.S.; Wylie, D. (2015). "Clifford algebra decompositions of conformal orthogonal group elements". Clifford Analysis, Clifford Algebras and Their Applications. 4: 223–240.
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