Delaunay Triangulations

1
Delaunay Triangulations

• Reading Chapter 9 of the Textbook
• Driving Applications
• Height Interpolation
• Constrained Triangulation

2
Height Interpolation

• A terrain is a 2D surface in 3D space with a
  special property every vertical line intersects
  it in a point. That is, it is the graph of a
  function f A ? R2? R that assigns a height f(p)
  to every point p in the domain.
• Problem From the height of the sample points we
  somehow have to approximate the height at the
  other points in the domain.
• ? Triangles with small angles are undesirable.
  So, rank the triangulations by their smallest
  angles.

3
Triangulations of Planar Point Sets

• Let P p1, p2, , pn be a set of points in
  the plane. A triangulation of P is the maximal
  planar subdivision S whose vertex set is P, s.t.
  no edge connecting 2 vertices can be added to S
  without destroying its planarity.
• Let P be a set of n points in the plane, not all
  collinear, and let k denote the number of points
  in P that lie on the boundary of the convex hull
  of P. Then any triangulation of P has 2n-2-k
  triangles and 3n-3-k edges.

4
Basic Properties

• Let C be a circle, l a line intersecting C in
  points a and b, and p, q, r, and s points lying
  on the same side of l. Suppose that p and q lie
  on C, that r lies inside C, and that s lies
  outside C, then
• ?arb
  gt ?apb ?aqb gt ?asb
• An edge is illegal if we can locally increase
  the smallest angle by flipping that edge.

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5
Basic Properties

• Let T be a triangulation with an illegal edge e,
  and T be a triangulation obtained from T by
  flipping e. Then A(T) gt A(T), where A(.) stands
  for the smallest angle.
• Let edge pipj be incident to triangles pipjpk and
  pipjpl, and let C be the circle through pi, pj,
  pk and pl. The edge is illegal if and only if
  the point pl lies in the interior of C.
  Furthermore, if the four points form a convex
  quadrilateral and do not lie on a common circle,
  then exactly one of pipj and pkpl is an illegal
  edge.

6
Legal Triangulation

• A legal triangulation is a triangulation that do
  not contain any illegal edge. Any angle-optimal
  triangulation is legal.
• We can compute legal triangulation by simply flip
  illegal edges till all edges are legal, given a
  triangulation. (see next)
• there are only finite number of triangulations
  and every iteration the smallest angle increases.
  So, it works, but too slow!

7
LegalTriangulation(T)

• Input Some triangulation T of a point set P.
• Output A legal triangulation T.
• 1. while T contains an illegal edge pi pj
• 2. do ( Flip pi pj )
• 3. Let pi pj pk and pi pj pl be the two
  triangles adjacent to pi pj
• 4. Remove pi pj from T, and add pk pl
  instead
• 5. return T

8
Voronoi Diagram Dual Graph

• The Voronoi diagram of P, Vor(P), is the
  subdivision of the plane into n regions, one for
  each site in P, s.t. the region of a site p? P
  contains all points in the plane for which p is
  the closest site. The region of a site p is
  called the Voronoi cell of p, denoted by V(p).
• The dual graph of Vor(P), G, has a node for every
  Voronoi cell (or every site). G has an arc
  between 2 nodes if the corresponding cells share
  an edge.

9
Delaunay Graph Triangulation

• Delaunay graph of P, DG(p) is the straight-line
  embedding of G, where the node corresponding to
  the Voronoi cell V(p) is the point p, and the arc
  connecting the nodes of V(p) is the segment pq.
• If P is in general position (i.e. no 4 points lie
  on a circle), then all vertices of the Voronoi
  diagram have degree three, and consequently all
  bounded faces of DG(p) are triangles. In this
  case, DG(p) is the Delaunay triangulation of P
  and unique.

10
Basic Properties of Delaunay Triangulations

• Let P be a set of points in the plane.
• 3 points are vertices of the same face of the
  DG(P) iff the circle thru them contains no point
  of P in interior.
• 2 points form an edge of DG(P) iff there is a
  closed disc C that contains them on its boundary
  and doesnt contain any other point.
• Let P be a set of points in the plane, and let T
  be a triangulation of P. Then T is a Delaunay
  triangulation of P, iff the circumcircle of any
  triangulation of T doesnt contain a point of P
  in its interior.

11
Basic Properties of Delaunay Triangulations

• Let P be a set of points in the plane. A
  triangulation T of P is legal if and only if T is
  a Delaunay triangulation of P.
• Let P be a set of points in the plane. Any
  angle-optimal triangulation of P is a Delaunay
  triangulation of P. Any Delaunay triangulation
  of P maximizes the minimum angle over all
  triangulations of P.

12
Computing Delaunay Triangulation

• Use Voronoi diagram to get Delaunay
  Triangulation.
• Use Randomized Incremental Construction
• Start with a big triangle that contains all the
  points. The vertices of this big triangle should
  not lie in any circle defined by 3 points in P.
• Add one point pr at a time, then add edges from
  pr to the vertices of the existing triangle.
• There are 2 cases to consider
• pr lies in the interior of a triangle
• pr falls on an edge (need to make sure new edges
  are legal by flipping edges if necessary).

13
DelaunayTriangulation(P)

• Input A set P of n points in the plane
• Output A Delaunay triangulation of P
• 1. Let p-1 , p-2 p-3 be 3 point s.t. P is
  contained in triangle p-1 p-2 p-3
• 2. Initialize T as triangulation consisting of a
  single triangle p-1 p-2 p-3
• 3. Compute a random permutation p1 , , pn of P
• 4. for r ? 1 to n
• 5. do ( Insert pr into T )
• 6. Find a triangle pi pj pk ? T
  containing pr
• 7. if pr lies in the interior of the
  triangle pi pj pk
• 8. then Add edges from pr to 3
  vertices of pi pj pk and split it
• 9. LegalizeEdge(pr , pi pj
  , T)
• 10. LegalizeEdge(pr , pj pk ,
  T)
• 11. LegalizeEdge(pr , pk pi ,
  T)

14
DelaunayTriangulation(P)

• 12. else ( pr lies on an edge of pi
  pj pk, say the edge pi pj )
• 13. Add edges from pr to pk
  and to the third vertex of
• other triangle that is
  incident to pi pj , thereby
• splitting 2 triangles
  incident to pi pj into 4 tris
• 14. LegalizeEdge(pr , pi pl ,
  T)
• 15. LegalizeEdge(pr , pl pj ,
  T)
• 16. LegalizeEdge(pr , pj pk ,
  T)
• 17. LegalizeEdge(pr , pk pi ,
  T)
• 18. Discard p-1 , p-2 and p-3 with all their
  incident edges from T
• 19. return T

15
LegalizeEdge(pr , pi pj , T)

• 1. ( The point being inserted is pr , and pi pj
  is the edge of T
• that may need to be flipped )
• 2. if pi pj is illegal
• 3. then Let pi pj pk be triangle adjacent to
  pr pi pj along pi pj
• 4. ( Flip pi pj ) Replace pi pj
  with pr pk
• 5. LegalizeEdge(pr , pi pk , T)
• 6. LegalizeEdge(pr , pk pj , T)

16
Find Triangle Containing Pr

• While we build the Delaunay triangulation, we
  also build a point location structure D. The
  leaves of D correspond to the triangles of
  current triangulation T. We maintain cross
  pointers between the leaves and the
  triangulation. The internal nodes of D
  correspond to triangles that were in
  triangulation at some earlier stage, but have
  been destroyed.
• We initialize D as a DAG with a single leaf node
  that corresponds to the big triangle.
• For the rest, see the pictorial example.

17
How to Pick a Big Triangle

• Let M be the maximum value of any coordinate of a
  point in P. Then, the triangle has the vertices
  at (3M, 0), (0, 3M), and (- 3M, - 3M).
• Then, handle the cases where the edge that Pr
  lies on has one vertex with a negative index.

18
Algorithm Analysis

• The Delaunay triangulation of a set of n points
  in the plane can be computed in O(n log n)
  expected time, using O(n) expected storage
• see the proof in 9.4 (similar to previous
  analysis for randomized algorithms)