Conformal linear transformation

A conformal linear transformation 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.[1][2]

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
  1. Amir-Moez, Ali R. "Conformal Linear Transformations". JSTOR. Taylor & Francis, Ltd. Retrieved 2023-07-26.
  2. "Differential Geometry: Conformal Maps" (PDF). Johns Hopkins University. Johns Hopkins University. Retrieved 2023-07-26.
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