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I agree with your point about embedding. The game solves the geodesic equation using the instrinsic geometry of the torus in 2D space without reference to any
by musgravepeter 12y ago
I agree with your point about embedding.
The game solves the geodesic equation using the instrinsic geometry of the torus in 2D space without reference to any embedding. It is in this sense that one might call the physics "correct"
The motion is then mapped into either a 2D or 3D representation and I agree this has a strong effect on how things look.
I'll need to think about the four space example.
What is the metric for a mathematically abstract flat torus?
- waqf 12y agoIf your model for the torus is the unit square with edges identified together, then the metric is the usual Euclidean metric, dx² + dy². (Alternatively, the metric induced from the embedding into Euclidean 4-space, with a parameterization like φ=atan2(y,x), θ=atan2(z,w), should give you the same answer give or take some factors of 2π.) (Also, Mathoverflow relevant to the double torus question: http://mathoverflow.net/questions/194451/canonical-immersion-of-the-double-torus. http://mathoverflow.net/questions/194451/canonical-immersion...)