Computational Geometry csci3250 � � � � � � Laura Toma � Bowdoin College
Today Triangulations • Delaunay triangulation • Properties • How to construct it • Applications •
Triangulation Input: P = {p 1 , p 2 ,…,p n } set of points in the plane � A triangulation T(P) is a maximal planar subdivision whose vertices are P. � maximal planar subdivision: no edge can be added without distorting planarity planar : edges can only intersect at endpoints � � � � � � � � � � � � � � � � � Every bounded face must be a triangle. The unbounded face : bounded by convex hull.
P = {p 1 , p 2 ,…,p n } set of points in the plane Many triangulations possible • In practice, want triangulations without small angles • small angles cause numerical instability with geometric predicates, etc •
P = {p 1 , p 2 ,…,p n } set of points in the plane Size of a triangulation Let k = number of points in P that lie on CH(P). • Any triangulation of P has 2n - 2 -k triangles and 3n - 3 - k edges. • � Proof: Use Euler’s formula: e - v + f = 2; •
P = {p 1 , p 2 ,…,p n } set of points in the plane Voronoi Diagram Vor(P) Vor(p i ): all points in the plane that are closer to p i than to any other site • p 2 v p 1 p 3
P = {p 1 , p 2 ,…,p n } set of points in the plane Voronoi Diagram Vor(P) The edges of Vor(P) are (segments of) perpendicular bisectors • p 2 v p 1 p 3
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Voronoi Diagram Vor(P) Every Voronoi vertex is the intersection of 3 Voronoi regions • (if no 4-points co-circular) p 2 v p 1 p 3
P = {p 1 , p 2 ,…,p n } set of points in the plane Voronoi Diagram Vor(P) Degeneracies • more than 3 points lie on same circle • collinear points •
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Voronoi Diagram Vor(P) Every Voronoi vertex is the intersection of 3 Voronoi regions • v is equidistant from p 1 , p 2 , p 3 • p 2 v p 1 p 3
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Voronoi Diagram Vor(P) Every Voronoi vertex is the intersection of 3 Voronoi regions • v is equidistant from p 1 , p 2 , p 3 • • C(v): circle of p 1 ,p 2 ,p 3 has center v p 2 v p 1 p 3
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Voronoi Diagram Vor(P) Every Voronoi vertex is the intersection of 3 Voronoi regions • v is equidistant from p1, p2, p3 • • C(v): circle of p1,p2,p3 has center at v • No other site can be inside C(v) <———— empty circle property p 2 v p 1 p 3 • Every Voronoi vertex is the center of a circle that has 3 sites on its boundary and no other sites inside
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular The dual of Vor(P) For every pair of Voronoi regions Vor(u) and Vor(v) that share and edge, draw • an edge between u and v ==> Each Voronoi vertex v defines a triangle p 2 v p 1 p 3
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Delaunay triangulation DT Theorem [Delaunay 1934]: The straight-line dual of Vor(P) is planar, and is a triangulation. Proof: not trivial (see p.197 in 4M book) p 2 v p 1 p 3
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Delaunay triangulation DT
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Delaunay triangulation DT By definition, DT(P) is the dual of Vor(P). Would be nice to have a direct way to characterize DT(P) (without Vor(P)).
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Delaunay triangulation DT What we know from Vor(P) p 1 p 2 p 3 is a triangle in DT(P) <==> circumcircle of p 1 p 2 p 3 is empty • � p 2 � p 2 � � v � p 1 p 3 p 1 p 3 � � � � Theorem: A triangulation T(P) is the DT(P) iff all triangles of T(P) have the empty-circle property. Proof: “==>” easy “<==“ requires a bit more work
Delaunay triangulation DT How to build a triangulation such that all triangles have empty-circle property ? • p 2 circumcircle of p 1 p 2 p 3 is empty p 1 p 3
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular How to build a triangulation? • Input: a set of points • Output: a partition of CH(P) into triangles • � � � � � � � �
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Many triangulations possible •
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Class work: Come up with an algorithm to compute a triangulation of P • � � � � � � � � � � � Possible ideas: incremental, plane sweep, divide-and-conquer, … •
Flipping edges edge flips
Flipping edges and empty circles edge flips How to get a DT? Compute an arbitrary triangulation T(P) • edge flips T ————> DT • c c c a b a b a b d d d flip ab to cd C(abc) contains d b outside C(acd) WHY?
Lemma: d inside C(abc) <==> b outside C(abd) c a b d c a b • Why? � • basic geometry of angles (Thales theorem) a > b > c
Note condition is symmetrical: d inside C(abc) <==> c inside C(abs) c a b d c a b • Why? � • basic geometry of angles (Thales theorem) a > b > c
Flipping edges and empty circles edge flips Notation: edge ab is illegal iff d is inside C(abc) • (a legal edge also called a locally Delaunay edge) edge ab is legal iff not illegal (d on or outside C(abc)) • c c c a b a b a b d d d flip ab illegal cd legal
Flipping edges and empty circles Theorem, revisited: A triangulation where all edges are locally Delaunay (legal) is the DT. (locally Delaunay ==> globally Delaunay) c c a b a b d d flip ab to cd ab illegal cd legal
Delaunay triangulation DT via edge flipping � c • construct an arbitrary triangulation T • mark all edges in T and put them in S • while S not empty a b • pop edge ab from S and un-mark it d • if ab not legal: • flip edge and update T • for each new edge ac, ad, bc, bc: if not marked, push it on S and mark it Does this ever terminate?.. If it terminates, T is DT (by theorem)
Edge flipping and angles c c a2 b2 b3 a1 a3 b4 b1 a b a4 a5 b a a6 b5 b6 d d angle vector before A(T) = [a1, a2, a3, a4, a5, a6] after A(T’) = [b1, b2, b3, b4, b5, b6] � Claim: For each new angle, there is an old angle that’s <= to it. Proof: b1 = a1+a4 > a1 b2 > a5 (b outside C(acd)) b3 > … similar b4 = a3 + a5 > a3 b5 > … similar b6 > a3 (b outside C(acd))
Edge flipping and angles c c a2 b2 b3 a1 a3 b4 b1 a b a4 a5 b a a6 b5 b6 d d Flipping an illegal edge to a legal edge increases minimum angle. Proof: Follows because each angle in T’ is larger than an angle in T. � DT maximizes minimum angle. Proof: DT has only legal edges. Any other flip would cause the min angel to decrease.
c c More generally.. a2 b2 b3 a1 a3 b4 b1 a4 a5 b a b a a6 b5 b6 d d A triangulation T. consider all angles in each of its m triangles => 3m angles sort angles increasingly: a 1 ≦ a 2 ≦ a 3 ≦ …… ≦ a 3m A(T) = [a 1, a 2, a 3, ……,a 3m ] � We say A(T) > A(T’) if A(T) > A(T’) lexicographically. [2,5,6] > [1,2,3] [1,2,3] > [1,1,9] � Of all possible triangulations, DT maximizes the angle vector. That is, for all T’, A(DT) > A(T’). Proof: An edge flip increases the angle vector.
DT Applications DT used because it is angle-optimal. • Small angles • Numerical instability, Interpolation errors • Issues with geometric predicates (leftOf, tangents, derivatives) •
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Computing DT: RIC b RIC (Randomized Incremental Construction) � Start with a large triangle T that contains P. • Compute a random permutation of the points in P • p For each point p in P do: • a //insert p in T find triangle abc of T that contains p • c splitting into 3 triangles: abp, bcp, cap and update T • LegalizeEdge(p, ab, T) • LegalizeEdge(p, bc, T) • LegalizeEdge(p, ca, T) • discard the initial triangle T and its edges • return T •
P = {p 1 , p 2 ,…,p n } set of points in the plane no 4 points co-circular Computing DT: RIC LegalizeEdge(p, uv, T) � //the point being inserted is p, and edge uv may need to be flipped if uv is illegal • let uvq be the triangle adjacent to uv, on the other side of p • flip uv and update T • LegalizeEdge(p, uq, T) • b LegalizeEdge(p, qv, T) • p a c
b p a c
b p a c
b p a c q
b p a c q
b p a c q
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