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For a given set of points in the 3D space or in a 2D space, can two different triangulation that conform to the Delanuay rules of empty circumcirle be created? If yes what would it depend on, the starting point?

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A Delaunay triangulation is unique iff the circumcircle of every triangle contains exactly three points on its circumference: the vertices of the triangle.

For instance, the Delaunay diagram of the four vertices of a square is a square, and can be converted into a triangulation in two different ways.

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  • $\begingroup$ in the constrained Delaunay triangulation it says "Delaunay triangulation is almost always unique" but there is no reference. Can you refer to that too? many thanks $\endgroup$
    – Rott
    Commented Sep 30, 2017 at 9:42
  • $\begingroup$ @Rott, "almost always" means that the probability of four points to be cocircular is zero. $\endgroup$
    – lhf
    Commented Sep 30, 2017 at 12:04

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