I have this bowl roughly the shape of a circular truncated cone, and the lamp shines on the bowl at an angle. The reflected lamp light hit the bottom of the bowl and created a light ring that looks just like a cardioid. Could someone help me, or give me a hint on how to really solve the equation for the light ring?
You're correct. If the angle at which the lamp shines is just right, the catacaustic (the envelope of rays reflected from a point) is a cardioid, as for example in this picture:
In particular, if the light reflects inside the bowl (which from above looks circular) from a point on the inner edge, you have the situation in the bottom left figure:
The general formula for a catacaustic generated by a source point and a reflecting curve is given on this wikipedia page. For a circle, the curve can be parametrized as $(\cos t, \sin t)$, and given a source point on the circle (e.g. $(1,0)$), the resulting catacaustic should be a cardioid (see equations (5) and (6) on this Wolfram page).
protected by Qmechanic♦ Sep 15 '15 at 15:17
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