Conformal affine transformation in 2D

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A conformal affine transformation (CAT) in $RR^n$ is a function $f : RR^n -> RR^n : f(x) = sQx + t$ , where $s in RR$ , $Q$ is an orthogonal matrix, and $t in RR^n$ .

Theory

We shall identify $f = [s, Q, t]$ , because of an obvious bijection. The CATs are closed under composition. For if $g(x) = s'Q'x + t$ ’, then:

$(f o g)(x) = f(g(x))$

$= sQ(s'Q'x + t') + t$

$= ss'QQ'x + sQt' + t$

$= ss'QQ'x + f(t')$

$=> f o g = [ss', QQ', f(t')]$

A CAT is invertible if and only if $s != 0$ :

$f(x) = sQx + t$

$=> f(x) - t = sQx$

$=> x = (1 / s) Q^T (f(x) - t)$

$=> x = (1 / s) Q^T f(x) - (1 / s) Q^T t$

$=> f^-1(x) = (1 / s) Q^T x - (1 / s) Q^T t$

$=> f^-1 = [1 / s, Q^T, -(1 / s) Q^T t]$

From above we see that the inverse of a CAT is also a CAT. Thus the invertible CATs form a group under composition.

Packed representation in 2D

In 2D, CATs can be identified with the 4-tuples $[s', alpha, t']$ by the mapping

$Q = [[cos(alpha), -sin(alpha)], [sin(alpha), cos(alpha)]]$

$t' = t$

$s' = s$

Therefore, in 2D, if $f(x) = sQx + t$ , we shall identify $f = [s, alpha, t]$ . This compact representation is specific to 2D and can’t be generalized to higher dimensions. It follows that:

$f o g = [ss', alpha + alpha', f(t')]$

$f^-1 = [1 / s, -alpha, -(1 / s) Q^T t]$

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Conformal affine tranformation in 2D

Conformal affine transformation in 2D

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Packed representation in 2D

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