Catacaustic
A catacaustic is a curve that is the envelope of rays emanating from a specified point (or a point at infinite distance producing parallel rays) for a given mirror shape. The point from which the rays emanate is called the radiant point. The catacaustic is an evolute of the orthotomic (Lawrence 1972, p. 60).
The following table lists the catacaustics for some common curves, omitting the incorrect catacaustic listed for the quadrifolium (Lawrence 1972, p. 207).
curve radiant point catacaustic
circle catacaustic parallel
rays nephroid
cycloid catacaustic
for 1 arch parallel rays perpendicular to axis 2
arches of a cycloid
deltoid catacaustic parallel
rays astroid
ellipse catacaustic any
point unnamed curve
natural logarithm catacaustic parallel rays parallel to axis catenary
parabola catacaustic parallel rays perpendicular to axis Tschirnhausen
cubic
See also
Atzema Spiral, Caustic, DiacausticExplore with Wolfram|Alpha
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References
Borovikov, V. A. and Kinber, B. Y. Geometrical Theory of Diffraction. London, England: Institute of Electrical Engineering, 1994.Bruce, J. W.; Giblin, P. G.; and Gibson, C. G. "On Caustics of Plane Curves." Amer. Math. Monthly 88, 651-667, 1981. https://doi.org/10.1080/00029890.1981.11995337.Bruce, J. W.; Giblin, P. G.; and Gibson, C. G. "On Caustics by Reflexion." Topology 21, 179-199, 1982. https://doi.org/10.1016/0040-9383(82)90004-0.Bruce, J. W.; Giblin, P. G.; and Gibson, C. G. "Genericity of Caustics by Reflexion." Proc. Symposia Pure Math. 40/1, 179-193, 1983.Cornbleet, S. Microwave and Geometrical Optics. London, England: Academic Press, 1994.Ehlers, J. and Newman, E. T. "The Theory of Caustics and Wave Front Singularities with Physical Applications." J. Math. Phys. 41, 3344-3378, 2000. https://doi.org/10.1063/1.533316.Georgiou, C.; Hasanis, T.; and Koutroufiotis, D. "On the Caustic of a Convex Mirror." Geom. Dedicata 28, 153-169, 1988.Giblin, P. J. and Kingston, J. G. "Caustics by Reflexion in the Plane with Stable Triple Crossings." Quart. J. Math Oxford 37, 17-25, 1986. https://doi.org/10.1093/qmath/37.1.17.Hairer, E. and Wanner, G. Analysis by Its History. New York: Springer-Verlag, 1996.Hartman, P. and Valentine, F. A. "On Generalized Ellipses." Duke Math. J. 26, 373-385, 1959. https://doi.org/10.1215/S0012-7094-59-02635-3.Knill, O. "On Nonconvex Caustics of Convex Billiards." Elem. Math. 53, 89-106, 1998. https://doi.org/10.1007/S000170050038.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 60 and 207, 1972.Loe, B. L. and Beagley, N. "The Coffee Cup Caustic for Calculus Students." Coll. Math. J. 28, 277-284, 1997. https://doi.org/10.1080/07468342.1997.11973875.Porteous, I. R. Geometric Differentiation for the Intelligence of Curves and Surfaces. Cambridge, England: Cambridge University Press, 1994.Poston, T. and Stewart, I. Catastrophe Theory and Its Application. London, England: Pitman, 1978.Schupp, H. and Dabrock, H. Höhere Kurven. Mannheim, Germany: BI, 1995.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, pp. 9-11, 2004. https://www.mathematicaguidebooks.org/.Referenced on Wolfram|Alpha
CatacausticCite this as:
Weisstein, Eric W. "Catacaustic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Catacaustic.html