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Can we visualize from any book/report reference a spiky curvilinear triangle with each vertex angle zero and area (by Gauss-Bonnet theorem) $\pi a^2$ on any surface of constant Gaussian curvature $K=-1/a^2$ in $\mathbb R^3 ?$

Can portions of it exist across cuspidal edges?

Narasimham
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  • Look in Thurston's notes, one of the earlier chapters (I forget which one). The triangle cusps wrap around closed geodesics. A triangulation looks something like this. https://math.uchicago.edu/~dannyc/OUPbook/singular.png – Neal Aug 14 '17 at 12:59
  • Am afraid I cannot see the point.. Lines at center are visibly non- geodesic, more like level curves. – Narasimham Aug 14 '17 at 13:08
  • I think you cannot construct a real ideal triangle in euclidean 3d space , you can construct an triangle with one angle of 0 on a pseudo sphere ( the angle of 0 is far away from the cusp – Willemien Aug 15 '17 at 05:35
  • GB theorem cannot be visualized on the Beltrami pseudosphere for this case... Right? – Narasimham Aug 15 '17 at 06:19
  • I would not say that ,just that you cannot construct an hyperbolic ideal triangle in euclidean 3d space Gauss Bonnet with other angles (not more than one zero) is I think possible – Willemien Aug 15 '17 at 12:25

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