![]() ![]() 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. Edges of the Voronoi diagram going to infinity are not defined by this relation in case of a finite set P.Four or more points on a perfect circle, where the triangulation is ambiguous and all circumcenters are trivially identical.Three or more collinear points, where the circumcircles are of infinite radii.Special cases where this relationship does not hold, or is ambiguous, include cases like: In the 2D case, the Voronoi vertices are connected via edges, that can be derived from adjacency-relationships of the Delaunay triangles: If two triangles share an edge in the Delaunay triangulation, their circumcenters are to be connected with an edge in the Voronoi tesselation. The circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi diagram for P. ![]() Relationship with the Voronoi diagram Ĭonnecting the centers of the circumcircles produces the Voronoi diagram (in red). However, in these cases a Delaunay triangulation is not guaranteed to exist or be unique. Generalizations are possible to metrics other than Euclidean distance. For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors.īy considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions. įor a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). The triangulation is named after Boris Delaunay for his work on this topic from 1934. Delaunay triangulations maximize the minimum of all the angles of the triangles in the triangulation they tend to avoid sliver triangles. In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT( P) such that no point in P is inside the circumcircle of any triangle in DT( P). A Delaunay triangulation in the plane with circumcircles shown
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