Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015)

Pages 379–382 Short Papers

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.

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