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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).

curveradiant pointcatacaustic
Atzema spiralorigincircle
cardioid catacausticcardioid cuspnephroid
circle catacausticcircumferencecardioid
circle catacausticparallel raysnephroid
cissoid of Diocles catacausticfocuscardioid
cycloid catacaustic for 1 archparallel rays perpendicular to axis2 arches of a cycloid
deltoid catacausticparallel raysastroid
ellipse catacausticany pointunnamed curve
natural logarithm catacausticparallel rays parallel to axiscatenary
logarithmic spiral catacausticoriginequal logarithmic spiral
parabola catacausticparallel rays perpendicular to axisTschirnhausen cubic
Tschirnhausen cubic catacausticfocussemicubical parabola

See also

Atzema Spiral, Caustic, Diacaustic

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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/.

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Catacaustic

Cite this as:

Weisstein, Eric W. "Catacaustic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Catacaustic.html

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