Surfaces with Natural Ridges

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David Brander and Steen Markvorsen: Surfaces with Natural Ridges. In: Bridges 2015.



We discuss surfaces with singularities, both in mathematics and in the real world. For many types of mathematical surface, singularities are natural and can be regarded as part of the surface. The most emblematic example is that of surfaces of constant negative Gauss curvature, all of which necessarily have singularities. We describe a method for producing constant negative curvature surfaces with prescribed cusp lines. In particular, given a generic space curve, there is a unique surface of constant curvature K = -1 that contains this curve as a cuspidal edge. This is an effective means to easily generate many new and beautiful examples of surfaces with constant negative curvature.

Extended Abstract


 author      = {David Brander and Steen Markvorsen},
 title       = {Surfaces with Natural Ridges},
 pages       = {379--382},
 booktitle   = {Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture},
 year        = {2015},
 editor      = {Kelly Delp, Craig S. Kaplan, Douglas McKenna and Reza Sarhangi},
 isbn        = {978-1-938664-15-1},
 issn        = {1099-6702},
 publisher   = {Tessellations Publishing},
 address     = {Phoenix, Arizona},
 note        = {Available online at \url{ }},
 url         = { },

Used References

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[3] N.V. Efimov. Generation of singularities on surfaces of negative curvature. Mat. Sb., 64:286-320, 1964.

[4] D. Hilbert. U¨ ber Fla¨chen von konstanter Gaußscher Kru¨mmung. Trans. Amer. Math. Soc. 2:87-99, 1901.

[5] G. Ishikawa and Y. Machida. Singularities of improper affine spheres and surfaces of constant Gaussian curvature. Internat. J. Math., 17:269-293, 2006.

[6] A. Popov. Lobachevsky geometry and modern nonlinear problems. Birkh¨auser, 2014.


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Pages 379–382