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Reflecting a vector y over a second vector x is a common operation in computer graphics, and it is performed by inverting the perpendicular part of y while the parallel component of stays unaltered, as it can be seen in the image on the right. So, if y = pr(y) + ap(y) where pr and ap are the projection and antiprojection of y with respect to x, then the reflection R(y) = pr(y) – ap(y) = pr(y) – (y-pr(y)) = 2pr(y) – y. That’s the familiar (for anyone who ever wrote a raytracer or a shader)
When working in matrix form, we must invert the quadrature of y to get its reflection, which results again in a regular vector transformation by a matrix R = (2P-I):
If we were now to divide the reflected vector R(y) by x we would see that the complex part of the result gets negated when compared to the simple y/x division: if “Reflecting the numerator of a vector division results in the complex conjugate of the regular division” |
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as in previous posts, then 
or in other words