TTransform2::Inverse

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Description

Inverts a transform. So a transform from space A to space B results in a transform from space B to space A. Since this class applies the 2x2 transform followed by translation, our inversion logic needs to be able to recast the result as a M * T. It does it using the following identity: (M * T)^-1 == T^-1 * M^-1

In homogeneous form, we represent our affine transform like so: M * T [A B 0] [1 0 0] [A B 0] [C D 0] * [0 1 0] = [C D 0]. This class simply decomposes the 2x2 transform and translation. [0 0 1] [X Y 1] [X Y 1]

But if we were applying the transforms in reverse order (as we need to for the inverse identity above): T^-1 * M^-1 [1 0 0] [A B 0] [A B 0] where [X' Y'] = [X Y] * [A B] [0 1 0] * [C D 0] = [C D 0] [C D] [X Y 1] [0 0 1] [X' Y' 1]

This can be conceptualized by seeing that a translation effectively defines a new local origin for that frame of reference. Since there is a 2x2 transform AFTER that, the concatenated frame of reference has an origin that is the old origin transformed by the 2x2 transform.

In the last equation: We know that [X Y] is the translation induced by inverting T, or -Translate. We know that [[A B][C D]] == Inverse(M), so we can represent T^-1 * M^-1 as M'* T' where: M' == Inverse(M) T' == Inverse(Translate) * Inverse(M)

TTransform2 Inverse() const  
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TTransform2 Inverse() const