Conformal linear transformation

A  is a linear transformation that is also a conformal map. Since it is a linear transformation, it transforms vectors between vector spaces. In terms of high-level transform decomposition, a conformal linear transformation may only be composed of rotation and uniform scale (and translation in the case of matrices with an origin vector), but not shear/skew or non-uniform scale. This restriction ensures the transform preserves angles like a conformal map. Additionally, this union of linear transformations and conformal maps has a new property not generally present in either alone, that distance ratios are preserved by the transformation.

General properties 

Across all dimensions, a conformal linear transformation matrix has the following properties:

 The basis matrix must have the same number of rows and columns, such that its inputs and outputs are in the same dimensions.
 Each column vector must be 90 degrees apart from all other column vectors (they are orthogonal).
 Angles are preserved by applying the transformation (possibly with orientation reversed).
 Distance ratios are preserved by applying the transformation.
 Orthogonal inputs remain orthogonal after applying the transformation.
 The transformation may be composed of translation.
 The basis may be composed of rotation.
 The basis may be composed of a flip/reflection.
 The basis may be composed of uniform scale.
 The basis must not be composed of non-uniform scale.
 The basis must not be composed of shear/skew.
 The basis must not be composed of squeeze.
 The basis must not be composed of projection.

Two dimensions 

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.

Practical applications 

Ensuring a transformation is conformal has benefits in various domains.

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.

References

Abstract algebra
Functions and mappings
 
Transformation (function)
Conformal mappings