Indeed - thanks for the clarification! Even with this provision, the “wormholes” in the examples shown in the video will induce paths that will fail the triangle inequality using the shortest geodesics, right? I haven’t actually worked this out but it looks like it’s true.
Suppose you have points A, B, C and geodesics of minimal length connecting all the pairs: A->B, A->C, and B->C. Then the piecewise geodesic A->B->C either is an actual geodesic, or can be shortened. (Say, by rounding the corner at B.) In the first case A->B->C is either equal to A->C or is one of the geodesics which is longer than A->C. In the second, it is automatic that A->C is shorter. (Constructively, you could use a curve shortening flow to get a geodesic, which could still only be >= to A->C in length.)
Haha, maybe… I feel like some of difficulty in that era of gaming came just from dealing with the limitations of the hardware.
shakes tiny fist
It was on a dodecahedron
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