One can get an idea of what the caustic looks like in the focal plane, the x,y-plane with z=1, by plotting curves of constant x for the reflection point on the mirror. The x values were stepped by 0.2 from -1.0 to 1.0 and on each curve the y values step by 0.1 from -1.0 to 1.0. The small red marker indicates the point on a curve where y=-1.0.
The curves for -x and +x appear to be mirror images of each other. The bounds of the caustic would be the outer edge or envelope of these curves so one gets a symmetrical shape for the caustic.
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