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?
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2 Answers
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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).
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$\begingroup$ Technically the curve is actually a nephroid rather than a cardiod. $\endgroup$– prideoutCommented May 13, 2018 at 14:15
