Tchirnhausen's Cubic

We are looking at light rays reflected in a parabola. You can set the angle the light rays make with the axis of the parabola.
If you set the angle to 0, the rays all go through the focus.
If you set it to 90, the reflected rays make a curve called Tchirnhausen's Cubic.
What happens if the light comes in at some other angle?

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Light angle

See if you can dilate and rotate Tchirnhausen's cubic to mathch the light curve (called a caustic).
The center of the dilation and rotation is D which you can drag.
Can you guess a formula for the scale factor and rotation?

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