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February 5th, 2013, 05:57 AM   #1
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Calculate reflection point in 3D

Hi,

I've been trying to solve the following for days now, and I just can't seem to get to a solution:

We have a cone of which the apex-angle is . The tip of the cone, with length , has been cut off, and the cone-height (length of its optical axis *before* the tip was cut off) is .
The top of the cone (which is not the apex anymore, since this has been cut off) is at position , and the optical axis is the z-axis, so that the bottom of the cone is on . So, if you imagine looking at the cone on your display, the X-axis goes to the right, the Y-axis points towards you, and the Z-axis goes to the bottom of your screen. The equation of this cone then is .

The normal vector on the cone is thus .

On coordinates , so on the same height as the top of the cone, there is a gun that can shoot a tennisball at any angle we want it to. The ball always flies in a straight line at an angle we choose.

On coordinates (and obviously not hiding inside the cone), there is the target we would like to hit.

The question now is this; Where on the cone do we shoot the tennisball, such that the ball, after reflection off the cone, hits the target?

If the question would have been 'If we shoot at location L on the cone, where will the ball end up after reflection?', then I would be able to solve it. But regrettably that is not the question and I'm really stuck here.

Any ideas?

Thanks a lot!
Arno
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