ZenoRogue | Three-Point Equidistant Projection @ZenoRogue | Uploaded March 2021 | Updated October 2024, 4 hours ago.
We try to map every point of three-dimensional hyperbolic space to three-dimensional Euclidean space, in the following way:
For a given point X, we compute the distances from three points (A,B,C), and then find X' the point in Euclidean space which has the same distance from points A', B', C'.
In this animation, we rotate the triangle ABC and also rotate the triangle A'B'C' (which is shown). The dodecahedra are from the {5,3,4} honeycomb.
The plane ABC splits the hyperbolic space to two half-spaces, and so does A'B'C'; we map each hyperbolic half-space to its respective Euclidean half-space. In some cases, for points in the ABC plane, there will be two (symmetric) Euclidean points, making the projection appear "broken" as we rotate the triangles. In other cases, there will be no correct Euclidean points (in this case, we just map to a point on the A'B'C' plane).
#nonEuclidean
We try to map every point of three-dimensional hyperbolic space to three-dimensional Euclidean space, in the following way:
For a given point X, we compute the distances from three points (A,B,C), and then find X' the point in Euclidean space which has the same distance from points A', B', C'.
In this animation, we rotate the triangle ABC and also rotate the triangle A'B'C' (which is shown). The dodecahedra are from the {5,3,4} honeycomb.
The plane ABC splits the hyperbolic space to two half-spaces, and so does A'B'C'; we map each hyperbolic half-space to its respective Euclidean half-space. In some cases, for points in the ABC plane, there will be two (symmetric) Euclidean points, making the projection appear "broken" as we rotate the triangles. In other cases, there will be no correct Euclidean points (in this case, we just map to a point on the A'B'C' plane).
#nonEuclidean
