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Costa Minimal Surface


A complete embedded Minimal Surface of finite topology. It has no Boundary and does not intersect itself. It can be represented parametrically by

x&=& {\textstyle{1\over 2}}\Re\left\{{-\zeta(u+iv)+\pi u +{\p...
...\pi} \ln\left\vert{\wp(u+iv)-e_1\over \wp(u+iv)+e_1}\right\vert,

where $\zeta(z)$ is the Weierstraß Zeta Function, $\wp(g_2,g_3;z)$ is the Weierstraß Elliptic Function, $c=189.07272$, $e_1=6.87519$, and the invariants are given by $g_2=c$ and $g_3=0$.


Costa, A. ``Examples of a Complete Minimal Immersion in $R^3$ of Genus One and Three Embedded Ends.'' Bil. Soc. Bras. Mat. 15, 47-54, 1984.

do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986.

Gray, A. Modern Differential Geometry of Curves and Surfaces.Boca Raton, FL: CRC Press, 1993.

Gray, A. Images of the Costa surface.

Nordstrand, T. ``Costa-Hoffman-Meeks Minimal Surface.''

Peterson, I. ``The Song in the Stone: Developing the Art of Telecarving a Minimal Surface.'' Sci. News 149, 110-111, Feb. 17, 1996.

© 1996-9 Eric W. Weisstein