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Conformal linear transformation
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<p>A conformal linear transformation, also called a homogeneous similarity transformation or homogeneous similitude, is a <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similarity transformation</a> of a <a href="/facts/Euclidean_vector_space/R2UbzmzM">Euclidean</a> or <a href="/facts/Pseudo-Euclidean_vector_space/UW6sbGeK">pseudo-Euclidean vector space</a> which fixes the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a>. It can be written as the composition of an <a href="/facts/Orthogonal_transformation/s7LGBDFX">orthogonal transformation</a> (an origin-preserving <a href="/facts/Rigid_transformation/0KzWYQXl">rigid transformation</a>) with a <a href="/facts/Uniform_scaling/pFm6fvpu">uniform scaling</a> (dilation). All similarity transformations (which globally preserve the shape but not necessarily the size of geometric figures) are also <a href="/facts/Conformal_map/cEpQnMad">conformal</a> (locally preserve shape). Similarity transformations which fix the origin also preserve <a href="/facts/Scalar_multiplication/jroFDfbM">scalar–vector multiplication</a> and <a href="/facts/Vector_addition/bCLrdksy">vector addition</a>, making them <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a>. </p><p>Every origin-fixing <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflection</a> or dilation is a conformal linear transformation, as is any composition of these basic transformations, including <a href="/facts/Rotation/yChkz49x">rotations</a> and <a href="/facts/Improper_rotation/kcLdCD9r">improper rotations</a> and most generally similarity transformations. However, <a href="/facts/Shear_transformation/HVCYblfU">shear transformations</a> and non-uniform scaling are not. Conformal linear transformations come in two types, <i>proper</i> transformations preserve the <a href="/facts/Orientability/1SEQM2Pt">orientation</a> of the space whereas <i>improper</i> transformations reverse it. </p><p>As linear transformations, conformal linear transformations are representable by <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> once the vector space has been given a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a>, composing with each-other and transforming vectors by <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a>. The <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> of these transformations has been called the conformal orthogonal group, the conformal linear transformation group or the homogeneous similtude group. </p><p>Alternatively any conformal linear transformation can be represented as a <a href="/facts/Geometric_algebra/yykKMNYx">versor</a> (<a href="/facts/Geometric_algebra/yykKMNYx">geometric product</a> of vectors); every versor and its negative represent the same transformation, so the versor group (also called the Lipschitz group) is a <a href="/facts/Double_covering_group/IrRjtTL2">double cover</a> of the conformal orthogonal group. </p><p>Conformal linear transformations are a special type of <a href="/facts/M%25C3%25B6bius_transformation/YEkTTLfK">Möbius transformations</a> (conformal transformations mapping circles to circles); the conformal orthogonal group is a subgroup of the <a href="/facts/Conformal_group/mNqA5vLP">conformal group</a>. </p>