Conformal linear transformation

Last updated February 09, 2024

A conformal linear transformation, also called a homogeneous similarity transformation or homogeneous similitude, is a similarity transformation of a Euclidean or pseudo-Euclidean vector space which fixes the origin. It can be written as the composition of an orthogonal transformation (an origin-preserving rigid transformation) with a uniform scaling (dilation). All similarity transformations (which globally preserve the shape but not necessarily the size of geometric figures) are also conformal (locally preserve shape). Similarity transformations which fix the origin also preserve scalar–vector multiplication and vector addition, making them linear transformations.

As linear transformations, conformal linear transformations are representable by matrices once the vector space has been given a basis, composing with each-other and transforming vectors by matrix multiplication. The Lie group of these transformations has been called the conformal orthogonal group, the conformal linear transformation group or the homogeneous similtude group.

Alternatively any conformal linear transformation can be represented as a versor (geometric product of vectors); 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.

Conformal linear transformations are a special type of Möbius transformations (conformal transformations mapping circles to circles); the conformal ortogonal group is a subgroup of the conformal group.

General properties

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

Distance ratios are preserved by the transformation.^[1]
Given an orthonormal basis, a matrix representing the transformation must have each column the same magnitude and each pair of columns must be orthogonal.
The transformation is conformal (angle preserving); in particular orthogonal vectors remain orthogonal after applying the transformation.
The transformation maps concentric $k$ -spheres to concentric $k$ -spheres for every $k$ (circles to circles, spheres to spheres, etc.). $k$ -spheres centered at the origin are mapped to $k$ -spheres centered at the origin.

Two dimensions

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

{\begin{bmatrix}a&-b\\b&a\end{bmatrix}}

Or, in the case of a flip/reflection, the form is similar but with the signs swapped in the second column:

{\begin{bmatrix}a&b\\b&-a\end{bmatrix}}

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.^{[ citation needed ]}

Practical applications

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.

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References

↑ Amir-Moez, Ali R. (1967). "Conformal Linear Transformations". Mathematics Magazine. Taylor & Francis, Ltd. 40 (5): 268–270. doi:10.2307/2688286. JSTOR 2688286 . Retrieved 2023-07-26.

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[1] Amir-Moez, Ali R. (1967). "Conformal Linear Transformations". Mathematics Magazine. Taylor & Francis, Ltd. 40 (5): 268–270. doi:10.2307/2688286. JSTOR 2688286 . Retrieved 2023-07-26.

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