View source for Conformal linear transformation

A '''{{visible anchor|conformal linear transformation}}''' is a [[Linear map|linear transformation]] that is also a [[Conformal map|conformal map]]. Since it is a linear transformation, it transforms [[Vector (mathematics and physics)|vectors]] between [[Vector space|vector space]]s. In terms of [[Transformation matrix#Examples in 2 dimensions|high-level transform decomposition]], a conformal linear transformation may only be composed of [[Rotation (mathematics)|rotation]] and uniform [[Scaling (geometry)|scale]] (and [[Translation (geometry)|translation]] in the case of matrices with an origin vector), but not [[Shear matrix|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.<ref>{{cite web |last1=Amir-Moez |first1=Ali R. |title=Conformal Linear Transformations |url=https://www.jstor.org/stable/2688286 |website=JSTOR |publisher=Taylor & Francis, Ltd. |access-date=2023-07-26}}</ref><ref>{{cite web |title=Differential Geometry: Conformal Maps |url=https://www.cs.jhu.edu/~misha/Fall09/9-conformal.pdf |website=Johns Hopkins University |publisher=Johns Hopkins University |access-date=2023-07-26}}</ref>

== General properties ==
{{uncited section|date=October 2023}}
Across all [[Physics:Dimension|dimension]]s, a conformal linear [[Transformation matrix|transformation matrix]] has the following properties:

* The [[Basis (linear algebra)|basis]] matrix{{clarify|reason=These words do not make sense|date=October 2023}} must have the same number of [[Row and column vectors|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 [[Orthogonality (mathematics)|orthogonal]]).
* Each column vector must be the same magnitude as the other column vectors.
* 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 (geometry)|translation]].{{clarify|reason=Completely unclear what this is supposed to mean.  Linear maps cannot be translations.|date=October 2023}}
* The basis may be composed of [[Rotation (mathematics)|rotation]].{{clarify|reason=Completely unclear what this is supposed to mean.|date=October 2023}}
* The basis may be composed of a flip/[[Reflection (mathematics)|reflection]].{{clarify|reason=Completely unclear what this is supposed to mean.|date=October 2023}}
* The basis may be composed of uniform [[Scaling (geometry)|scale]].{{clarify|reason=Completely unclear what this is supposed to mean.|date=October 2023}}
* The basis must not be composed of non-uniform scale.{{clarify|reason=Completely unclear what this is supposed to mean.  What is "the basis"?  What is "non-uniform scale"?  What does it mean for the one to be composed of the other?|date=October 2023}}
* The basis must not be composed of [[Shear matrix|shear]]/skew.{{clarify|reason=Completely unclear what this is supposed to mean.|date=October 2023}}
* The basis must not be composed of [[Squeeze mapping|squeeze]].{{clarify|reason=Completely unclear what this is supposed to mean.|date=October 2023}}
* The basis must not be composed of [[Projection (linear algebra)|projection]].{{clarify|reason=Completely unclear what this is supposed to mean.|date=October 2023}}

== Two dimensions ==

In 2D, a conformal linear transformation has a special form. For a non-flipped conformal 2D basis, it looks like this:

:<math>\begin{bmatrix}a&-b\\b&a\end{bmatrix}</math>

Or, in the case of a flip/[[Reflection (mathematics)|reflection]], the form is similar but with the signs swapped in the second column:

:<math>\begin{bmatrix}a&b\\b&-a\end{bmatrix}</math>

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 matrix#Basic 3D rotations|rotation between two axes]].

== Practical applications ==

Ensuring a transformation is conformal has benefits in various domains.

When [[Transformation matrix#Composing and inverting transformations|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 [[Software:Physics engine|physics simulations]], a sphere (or circle, [[N-sphere|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 [[Symmetry group|symmetric]] and [[Invariant (mathematics)|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|ellipsoid]], therefore the distance check algorithm does not work correctly anymore.

==References==
{{Reflist}}

[[Category:Abstract algebra]]
[[Category:Functions and mappings]]
[[Category:Linear operators| ]]
[[Category:Transformation (function)]]
[[Category:Conformal mappings]]

{{Sourceattribution|Conformal linear transformation}}