![]() Unfortunately, this can take Ω( n 2) edge flips. This leads to a straightforward algorithm: construct any triangulation of the points, and then flip edges until no triangle is non-Delaunay. When A, B, C are sorted in a counterclockwise order, this determinant is positive only if D lies inside the circumcircle.Īs mentioned above, if a triangle is non-Delaunay, we can flip one of its edges. The Delaunay triangulation contains O ( n ⌈ d / 2 ⌉ ).The union of all simplices in the triangulation is the convex hull of the points.Let n be the number of points and d the number of dimensions. Halfway through, the triangulating edge flips showing that the Delaunay triangulation maximizes the minimum angle, not the edge-length of the triangles. Properties Įach frame of the animation shows a Delaunay triangulation of the four points. Nonsimplicial facets only occur when d + 2 of the original points lie on the same d- hypersphere, i.e., the points are not in general position. As the convex hull is unique, so is the triangulation, assuming all facets of the convex hull are simplices. This may be done by giving each point p an extra coordinate equal to | p| 2, thus turning it into a hyper-paraboloid (this is termed "lifting") taking the bottom side of the convex hull (as the top end-cap faces upwards away from the origin, and must be discarded) and mapping back to d-dimensional space by deleting the last coordinate. The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in ( d + 1)-dimensional space. It is known that there exists a unique Delaunay triangulation for P if P is a set of points in general position that is, the affine hull of P is d-dimensional and no set of d + 2 points in P lie on the boundary of a ball whose interior does not intersect P. ![]() Edges going to infinity start from a circumcenter and they are perpendicular to the common edge between the kept and ignored triangle.įor a set P of points in the ( d-dimensional) Euclidean space, a Delaunay triangulation is a triangulation DT( P) such that no point in P is inside the circum-hypersphere of any d- simplex in DT( P). If the Delaunay triangulation is calculated using the Bowyer–Watson algorithm then the circumcenters of triangles having a common vertex with the "super" triangle should be ignored.
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