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assume billiard boundary Chebyshev closed curve coefficients computation cone confocal conics Consider construction continued fraction convex coordinates coplanar corresponding Crofton's formula cubic curve cubic equation curvature cuspidal edge cusps deformation Dehn invariant Dehn(P denote developable surface dihedral angles disc domain double points double tangent ellipse ellipsoid envelope equal example Exercise faces fold follows formula function Gauss-Bonnet Theorem geometry graph hence hexagonal horizontal hyperbola hyperboloid infinitely inflection points inside integer intersection point Lecture Lemma length loop mathematics Mobius band n-gon number of sign obtained octahedron orthogonal osculating circle oval pairs parabola parallel translation partition permutations planar polygon polyhedra polyhedron polynomial of degree problem projection Prove quadratic rectangle result roots rotation ruled surfaces Section segment self-intersecting sequence shown in Figure side sign changes solutions space sphere spherical straight lines symmetric tangent lines tiling total number triangle values vector vertex vertices winding number zero
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Francisco JP Lopes, Fernando MC Vieira, David M Holloway, Paulo M Bisch, Alexander V Spirov - 2008 - PLoS Computational Biology