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Delaunay Triangulation and Tetrahedrilization

Marc van KreveldSlides DDM

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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

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• This is not just true visually, but also when the triangulation is the interpolator for a terrain with elevation measurements

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• 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– …

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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

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Voronoi diagrams

• Voronoi vertices “often” have degree 3, but it can be much larger

• Voronoi edges can be bounded or unbounded

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Voronoi edge

Voronoi vertex

Voronoi diagrams

• With n point sites, there are < 2n Voronoi vertices and < 3n Voronoi edges

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Voronoi edge

Voronoi vertex

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

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Voronoi edge

Voronoi vertex

Voronoi diagrams

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

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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])

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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

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Voronoi diagrams

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

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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

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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

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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

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• This graph is called the Delaunay graph• Notice that an edge of the Delaunay graph does not

necessarily intersect the corresponding Voronoi edge

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• 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


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

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• The empty-circle property for Voronoi diagrams transfers to Delaunay triangulations– for each triangle, its circumcircle

contains no other pointsfrom the set inside

– for each edge, a circleexists through itsendpoints that hasno other points of the set inside


• 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)

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

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• 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

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• 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

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• 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

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• 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

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flip

Computing the Delaunay triangulation

• Approach: insert points one-by-one, and restore the Delaunay triangulation after every insertion

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1. Locate the triangle t con-taining the next point p

2. Connect p to t’s vertices3. Restore the Delaunay

triangulation by flipping non-Delaunay edges

pt


• Locate: Recall that we store triangulations in some convenient structure (triangle-neighbor structure, half-edge structure)

• From any access point (triangle, half-edge), walkalong a straight line to thelocation of the p, finding the triangle t containing p

• The details of the walk depend on the mesh structure used

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p


• Connect: Make edges from p to t’s vertices; this replaces t by three new triangles

• Adapt the mesh representationstructure accordingly

Claim: These three new edges are Delaunay

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p

C


• Proof of claim:– Triangle t was Delaunay before the insertion of p,

so there was an empty circle C through its vertices

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pt

v

C



so there was an empty circle C through its vertices– Consider the edge between p and

any vertex v of t, there is always acircle through p and v completely inside C (grow a small circle fromv tangent to C at v, until it hits p)

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pt

v

C



so there was an empty circle C through its vertices– Consider the edge between p and

any vertex v of t, there is always acircle through p and v completely inside C (grow a small circle fromv tangent to C at v, until it hits p)

– Since v and p have an empty circle,they define a Delaunay edge

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pt

v


• Restore: The three new edges are always Delaunay, but the three new triangles need not be …

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p

p

Delaunay

maybe not Delaunay


• 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

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p

Delaunay

maybe not Delaunay

q

rs

pq

rs

C(q,r,s)


• 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

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p

Delaunay

maybe not Delaunay

q

rs

pq

rs

Delaunay

maybe not Delaunay


• Code for recursive flipping (restore algorithm):

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TestFlipEdge (p, qr)

Let s p be the third point of the triangle incident to qrif p is inside C(q,r,s){ flip: delete qr and insert ps in the triangulation

TestFlipEdge(p, qs)TestFlipEdge(p, sr)

}


• 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?

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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)

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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, qx

2 + qy2), (rx, ry, rx

2 + ry2),

and (sx, sy, sx2 + sy

2)

– Test whether the point (px, py, px2 + py

2) lies above or below this plane: below p is inside; above p is outside

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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, qx

2 + qy2), (rx, ry, rx

2 + ry2),

and (sx, sy, sx2 + sy

2)

– Test whether the point (px, py, px2 + py

2) lies above or below this plane

• This is equivalent to computing the sign of the 4x4 determinant:

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negative inside

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

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(triangulated)

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)

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The whole algorithm

1. Initialize a triangulation T with a bounding box2. For each point pi

i. Locate the triangle t of T containing pi by traversing the triangle-mesh structure from some access point

ii. Connect pi to the vertices of t

iii. (Restore) For each edge e of t, TestFlipEdge(p, e)

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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) time49some fixed starting point


• 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

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• 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(nn) time algorithm

Note: worst-case O(n log n) time algorithms exist; these are more complicated

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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

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Delaunay tetrahedrilization

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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)

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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

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• Useful when known linear features must be included in the triangulation– some polygon boundary– lake boundaries in a terrain with elevation values known

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• Input: set of points and set of edges (constraints)

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constrained Delaunay triangulation



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normal 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

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• 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

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Build Delaunay triangulation


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Insert constraining edge by removing intersected edges and triangles


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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


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Triangulate the cavity by shrinking a circle through the constraint endpoints until it reaches a vertex


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Make two new edges to this vertex from the other points defining the circle, and recurse


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• 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

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Constrained Delaunay in 3D

• Constrained Delaunay tetrahedrilizations do not always exist

• We would need to subdivide constraining facets using extra vertices and edges

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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

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• 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

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• A common example is triangulating a simple polygon with Steiner points on the edges and inside

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Questions1. Prove that edge ps (slide 38/39) is always Delaunay, using the

empty circles known from the situation before p was inserted2. Why is the running time of inserting all constraints

incrementally into a constrained Delaunay triangulation cubic in the number of points and constraints?

3. Why do we expect much better performance than cubic when inserting all constraints?

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