1. Delaunay Triangulation and Tetrahedrilization
Marc van Kreveld
Slides DDM

2. Triangulations and their quality
When going from points on a surface to triangles representing that surface, there are many ways to form the triangles
Some ways are better than others

3. Triangulations and their quality
This is not just true visually, but also when the triangulation is the interpolator for a terrain with elevation measurements

4. Triangulations and their quality

5. Triangulations and their quality
Triangulations are good if they
do not have long edges
do not have triangles with both short and long edges
do not have triangles with very small angles
do not have triangles with obtuse angles
do not have vertices with high degree …

6. Voronoi diagrams
For a set of points (sites), the Voronoi diagram is the subdivision of the plane (space) with faces where exactly one of the sites is closest, among all sites

7. Voronoi diagrams
Voronoi vertices “often” have degree 3, but it can be much larger
Voronoi edges can be bounded or unbounded
Voronoi edge
Voronoi vertex

8. Voronoi diagrams
With n point sites, there are < 2n Voronoi vertices and < 3n Voronoi edges
Voronoi edge
Voronoi vertex

9. Voronoi diagrams
The average number of edges bounding a Voronoi cell is six (or slightly less), but a single cell could be bounded by up to n – 1 edges
Voronoi edge
Voronoi vertex

10. Voronoi diagrams
Voronoi diagrams have the empty circle property for every Voronoi vertex and Voronoi edge

11. Voronoi diagrams
Efficient algorithms exist to construct 2D Voronoi diagrams of n point sites, namely O(n log n) time algorithms (as fast as sorting! [mergesort, quicksort])

12. Voronoi diagrams
3D Voronoi diagrams with n point sites can have up to (n2) Voronoi vertices (0D), Voronoi edges (1D), and Voronoi facets (2D) but of course just n cells (3D)
The descriptive complexity can be anything between linear and quadratic in n

13. Voronoi diagrams
3D Voronoi diagrams with n point sites often have linearly many Voronoi vertices (0D), Voronoi edges (1D), and Voronoi facets (2D)

14. Delaunay triangulation
For a set of points in the plane, suppose we compute its Voronoi diagram and use it to make an embedded planar graph G as follows:
all points are vertices of the graph G
for all points whose Voronoi cells share a Voronoi edge, we have an edge in G
faces of G are implied by the vertices and edges of G

15. Delaunay triangulation
all points are vertices of the graph G
edges in G correspond to edge-adjacent Voronoi cells
faces of G are implied by the vertices and edges of G

16. Delaunay triangulation
all points are vertices of the graph G
edges in G correspond to edge-adjacent Voronoi cells
faces of G are implied by the vertices and edges of G

17. Delaunay triangulation
This graph is called the Delaunay graph
Notice that an edge of the Delaunay graph does not necessarily intersect the corresponding Voronoi edge

18. Delaunay triangulation
The Delaunay graph is a triangulation of the original points if and only if all Voronoi vertices have degree 3  there is no circle through more than 3 sites and with no sites inside

19. Delaunay triangulation
Any (planar) completion of the Delaunay graph to a triangulation is a Delaunay triangulation of the original points

20. Delaunay triangulation
The empty-circle property for Voronoi diagrams transfers to Delaunay triangulations
for each triangle, its circumcircle contains no other points from the set inside
for each edge, a circle exists through its endpoints that has no other points of the set inside

21. Delaunay triangulation
A Delaunay triangulation of n points has at least n – 2 and at most 2n – 5 triangles (Delaunay triangles)
A Delaunay triangulation of n points has at least 2n – 3 and at most 3n – 6 edges (Delaunay edges)
The empty-circle property is “if and only if”: if three points have a circumcircle with no points inside or other points on the boundary, then they make a Delaunay triangle (a similar statement holds for edges)

22. Delaunay triangulation
When a Delaunay graph is not a triangulation, the non-triangular faces have all vertices on a circle

23. Delaunay triangulation

24. Delaunay triangulation

25. Delaunay triangulation

26. Delaunay triangulation

27. Delaunay triangulation

28. Delaunay triangulation
When we know four Delaunay edges that form an empty convex quadrilateral, we will have one of the two diagonals as a Delaunay edge
The circumcircles of the triangles of one choice will contain the fourth points, but not for the other choice

29. Delaunay triangulation
The Delaunay triangulation maximizes the smallest occurring angle, over all triangulations of the point set
In other words, any other triangulation will have a smaller (or the same) smallest angle

30. Delaunay triangulation
The Delaunay triangulation often connects points close together, e.g., every point is connected to its nearest neighbor
But it does not minimize total edge length

31. Delaunay triangulation
In a triangulation, when two triangles form a convex quadrilateral, their shared edge can be replaced by one other edge
This is called an edge flip
flip

32. Computing the Delaunay triangulation
Approach: insert points one-by-one, and restore the Delaunay triangulation after every insertion
Locate the triangle t con-taining the next point p
Connect p to t’s vertices
Restore the Delaunay triangulation by flipping non-Delaunay edges p t

33. Computing the Delaunay triangulation
Locate: Recall that we store triangulations in some convenient structure (triangle-neighbor structure, half-edge structure)
From any access point (triangle, half-edge), walk along a straight line to the location of the p, finding the triangle t containing p
The details of the walk depend on the mesh structure used p

34. Computing the Delaunay triangulation
Connect: Make edges from p to t’s vertices; this replaces t by three new triangles
Adapt the mesh representation structure accordingly Claim: These three new edges are Delaunay p

35. Computing the Delaunay triangulation
Proof of claim:
Triangle t was Delaunay before the insertion of p, so there was an empty circle C through its vertices C v
The growing circle starts as a single point equal to v, and then the growing circle has its center move in a straight line to the center of C, while keeping passing through v. This keeps the growing circle tangent to C at v. No other point than p can be hit in the growing process, because we know that p is the only point inside C. p t

36. Computing the Delaunay triangulation
Proof of claim:
Triangle t was Delaunay before the insertion of p, so there was an empty circle C through its vertices
Consider the edge between p and any vertex v of t, there is always a circle through p and v completely inside C (grow a small circle from v tangent to C at v, until it hits p) C v
The growing circle starts as a single point equal to v, and then the growing circle has its center move in a straight line to the center of C, while keeping passing through v. This keeps the growing circle tangent to C at v. No other point than p can be hit in the growing process, because we know that p is the only point inside C. p t

37. Computing the Delaunay triangulation
Proof of claim:
Triangle t was Delaunay before the insertion of p, so there was an empty circle C through its vertices
Consider the edge between p and any vertex v of t, there is always a circle through p and v completely inside C (grow a small circle from v tangent to C at v, until it hits p)
Since v and p have an empty circle, they define a Delaunay edge C v
The growing circle starts as a single point equal to v, and then the growing circle has its center move in a straight line to the center of C, while keeping passing through v. This keeps the growing circle tangent to C at v. No other point than p can be hit in the growing process, because we know that p is the only point inside C. p t

38. Computing the Delaunay triangulation
Restore: The three new edges are always Delaunay, but the three new triangles need not be …
Delaunay p p
maybe not Delaunay

39. Computing the Delaunay triangulation
In the picture: triangle qrs had an empty circle C(q,r,s) before the insertion of p, but maybe p lies inside now  test p C(q,r,s), and if so, edge-flip qr to ps
Delaunay q q
C(q,r,s) p p s s r r
maybe not Delaunay

40. Computing the Delaunay triangulation
If flipped, the edges pq, pr, and also ps must be Delaunay, but maybe pqs and prs are not Delaunay triangles …  recurse on them, using the triangles opposite qs and rs
Delaunay
Delaunay q q p p s s r r
maybe not Delaunay
maybe not Delaunay

41. Computing the Delaunay triangulation
Code for recursive flipping (restore algorithm):
TestFlipEdge (p, qr)
Let s  p be the third point of the triangle incident to qr
if p is inside C(q,r,s)
{ flip: delete qr and insert ps in the triangulation
TestFlipEdge(p, qs)
TestFlipEdge(p, sr) }

42. Computing the Delaunay triangulation
Comments:
The edge flip must be done in our triangle-mesh representation structure (triangle-neighbor, half-edge, …)
With suitable triangle-mesh structure, a flip takes O(1) time (with an indexed mesh, a flip takes O(n) time)
Every edge flip connects p to an extra vertex  on the average, about 3 flips are needed
The main geometric test is the so-called in-circle test: does a point p lie inside the circle defined by three points q,r,s?

43. The in-circle test
The in-circle test: does a point p lie inside the circle defined by three points q,r,s? (1) The bad way:
Compute the bisectors of q,r and r,s
Intersect them to get the center c of C(q,r,s)
Compute dist(c,p) and dist(c,q) (the radius of C(q,r,s)); if the former is smaller, then p lies inside C(q,r,s)

44. The in-circle test
The in-circle test: does a point p lie inside the circle defined by three points q,r,s? (2) The good way:
Compute the plane through (qx, qy, qx2 + qy2), (rx, ry, rx2 + ry2), and (sx, sy, sx2 + sy2)
Test whether the point (px, py, px2 + py2) lies above or below this plane: below  p is inside; above  p is outside
Geometrically, we are lifting the points q,r,s from the plane onto the unit paraboloid in 3D (z=x^2+y^2). The plane through these three 3D points intersects the unit paraboloid exactly in a shape that projects back to a circle on the xy-plane.

45. The in-circle test
The in-circle test: does a point p lie inside the circle defined by three points q,r,s? (2) The good way:
Compute the plane through (qx, qy, qx2 + qy2), (rx, ry, rx2 + ry2), and (sx, sy, sx2 + sy2)
Test whether the point (px, py, px2 + py2) lies above or below this plane
This is equivalent to computing the sign of the 4x4 determinant:
qx qy rx ry qx2 + qy2 1 rx2 + ry2 1 sx sy px py sx2 + sy2 1 px2 + py2 1
negative  inside

46. The whole algorithm
Initialization: we want to make sure that the next point p is always in some triangle of the current triangulation  start with a suitable bounding box
(triangulated)

47. The whole algorithm
The four extra vertices need special treatment when they are involved in an in-circle test (because they do not count for the empty-circle property)

48. The whole algorithm Initialize a triangulation T with a bounding box
For each point pi
Locate the triangle t of T containing pi by traversing the triangle-mesh structure from some access point
Connect pi to the vertices of t
(Restore) For each edge e of t, TestFlipEdge(p, e)

49. Efficiency of the algorithm
Locate: if the points are distributed “reasonably”, a line typically intersects O(n) triangles (certainly true if the points lie in a regular square grid pattern)
Worst-case O(n) time
Recall that a triangulation with n vertices has at most 2n triangles and at most 3n edges
some fixed starting point

50. Efficiency of the algorithm
Connect: obviously O(1) time in the triangle-neighbor structure and half-edge structure
Restore: typically 3 flips are needed: the average degree of a vertex in a triangulation is  6, connect leads to a starting degree of 3 for pi, and every flip increases the degree of pi by 1  typically O(1) time, but worst-case O(n) time

51. Efficiency of the algorithm
Initialization O(n) time
Insertion of the i-th point takes worst-case O(i) time but typically O(i) time
Worst-case O(n2) time algorithm
Typically O(nn) time algorithm Note: worst-case O(n log n) time algorithms exist; these are more complicated

52. Delaunay tetrahedrilization
Global approach the same
The in-circle test becomes an in-sphere test (solved with a 5x5 determinant)
The edge-flip becomes a bi-stellar flip: 2  3 tetrahedra

53. Delaunay tetrahedrilization

54. Terrains and 2D/3D Delaunay
Note the difference:
A polyhedral terrain (triangular mesh representing elevation) obtained by computing the Delaunay triangulation of the (x,y) points and using the triangles in 3D
A 3D Delaunay tetrahedrilization of the (x,y,z) points
A triangle in the first case does not necessarily occur as a triangular facet in the second case (the 2D circle can be small when the 3D ball is huge)

55. Constrained Delaunay triangulation
Variation on the Delaunay triangulation where certain edges must be included (even if they are not Delaunay)
Endpoints of such constraining edges are also vertices of the constrained Delaunay triangulation

56. Constrained Delaunay triangulation
Useful when known linear features must be included in the triangulation
some polygon boundary
lake boundaries in a terrain with elevation values known

57. Constrained Delaunay triangulation
Input: set of points and set of edges (constraints)

58. Constrained Delaunay triangulation
Input: set of points and set of edges (constraints)
constrained Delaunay triangulation

59. Constrained Delaunay triangulation
Input: set of points and set of edges (constraints)
normal Delaunay triangulation

60. Constrained Delaunay triangulation
Relaxed empty-circle property: three points define a circle in the constrained Delaunay graph if and only if their circumcircle has no other points (including constraint endpoints) inside or on the boundary, with the possible exception of points on the other side of constraints that cut the circle fully

61. Constrained Delaunay triangulation
To compute the constrained Delaunay triangulation, first compute the normal Delaunay triangulation of the points and constraint endpoints
Then insert the constraints one by one, removing the intersected edges/triangles, and re-triangulating the holes

62. Constrained Delaunay triangulation
Build Delaunay triangulation

63. Constrained Delaunay triangulation
Insert constraining edge by removing intersected edges and triangles

64. Constrained Delaunay triangulation
Note: all other edges and triangles remain in the CDT, because the empty circle of the DT will also be an empty circle with the constraint added

65. Constrained Delaunay triangulation
Triangulate the cavity by shrinking a circle through the constraint endpoints until it reaches a vertex

66. Constrained Delaunay triangulation
Make two new edges to this vertex from the other points defining the circle, and recurse

67. Constrained Delaunay triangulation

68. Constrained Delaunay triangulation
A constrained Delaunay triangulation can be constructed in O(n log n) time in the worst case (for n points and constraints)
The given algorithm takes O(n3) time in the worst case but one expects much better performance

69. Constrained Delaunay in 3D
Constrained Delaunay tetrahedrilizations do not always exist
We would need to subdivide constraining facets using extra vertices and edges

70. Steiner points in triangulations
The Delaunay triangulation of the given points may not give sufficiently well-shaped triangles  add extra points (Steiner points)
Useful for point sets with or without constraints

71. Steiner points in triangulations
Steiner points can lower the maximum vertex degree and improve the smallest occurring angle
Steiner points placed on a constraint can break it into several shorter constraints

72. Steiner points in triangulations
A common example is triangulating a simple polygon with Steiner points on the edges and inside

73. Questions
Prove that edge ps (slide 38/39) is always Delaunay, using the empty circles known from the situation before p was inserted
Why is the running time of inserting all constraints incrementally into a constrained Delaunay triangulation cubic in the number of points and constraints?
Why do we expect much better performance than cubic when inserting all constraints?