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.
Across all dimensions, a conformal linear transformation has the following properties: In the Euclidean vector plane, an improper conformal linear transformation is a reflection across a line through the origin composed with a positive dilation.
Given an orthonormal basis, it can be represented by a matrix of the form A proper conformal linear transformation is a rotation about the origin composed with a positive dilation.
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.
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.