Circle Catacaustic
Consider a unit circle and a radiant point located at (mu,0). There are four different regimes of caustics, illustrated above.
For radiant point at mu=infty, the catacaustic is the nephroid
(Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.)
For radiant point a finite distance mu>1, the catacaustic is the curve
which is apparently incorrectly described as a limaçon by Lawrence (1972, p. 207).
For radiant point on the circumference of the circle (mu=1), the catacaustic is the cardioid
with Cartesian equation
| -1-8x-18x^2+27x^4-18y^2+54x^2y^2+27y^4=0. |
(7)
|
For radiant point inside the circle, the catacaustic is a discontinuous two-part curve.
If the radiant point is the origin, then the catacaustic degenerates to a single point at the origin since all rays reflect upon themselves back through the origin.
See also
Catacaustic, Caustic, CircleExplore with Wolfram|Alpha
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References
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 207, 1972.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. https://www.mathematicaguidebooks.org/.Referenced on Wolfram|Alpha
Circle CatacausticCite this as:
Weisstein, Eric W. "Circle Catacaustic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CircleCatacaustic.html