Constrained DelaunayTriangulations

(CDT)Recall:“Definition 1” (Constrained triangulation). A triangulation with prespecified edges or breaklines between nodes.

Motivation: Geological faults in oil recovery Rivers, roads, lake boundaries in GIT Repr. non-convex boundaries and holes Linear features in CAD models Meshing for FEM (boundaries, interior & exterior)

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Algorithm III from PreliminariesGiven a predefined constraint Ei,j. Suppose that theendpoints pi and pj are in .

1. For all Ti in ,If IntTi Eij ,

remove Ti from We get one or more regions Ri on each side of Ei,jinside simple closed polygond. (More than one if Ei,jintersects nodes.)

2. Triangulate each region Ri with Algorithm I(Protruding point removal).

pipj

Ei,j

pipj

Ei,j

(a)

(b)

May include holes and arbitrary exterior boundary

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Delaunay Triangulation of a PSLG

We will generalize the theory of Delaunay triangulationfrom P to G where G GP,Ec is a PSLG (PlanarStraight Line Graph).

Ec are constrained edges. The endpoints of Ec are in P.

Recall fom conventional Delaunay triangulations: MaxMin angle criterion Lexicographical ordering of indicator vectors Definition of local- and global optimum The circle criterion (will be redefined) The dual (Voronoi diagram) will not be considered here!(involved and not so useful)

DEFINISJON (Visibility). pi and pj are visible to eachother if pipj does not intersect the interior of any edge inEc.

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DEFINISJON (CDT). A CDT G of a PSLG GP,Ec isa triangulation containing the edges Ec such that Ct ofany triangle t in G contains no point of P in its interiorwhich is visible from all the three nodes of t.

DEFINISJON: Modified circle criterion (relaxed)

DEFINISJON: Edges in G that are not in Ec arecalled Delaunay edges; triangles in G are calledDelaunay triangles.

(a) (b)

(c) (d)

a) GP,Ec b) DTP c) CDTG

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if Ec G P and G P, P;a conventional Delaunay triangulation.

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Generalization of the theory to CDTs(Brief, consult cited papers for details)

(a) (b)

1θ

2θ 2θ'e

e'e1p

2p

3p

4p

In (b), e is constrained

LEMMA. The modified circle criterion and the MaxMinangle criterion are equivalent for strictly convexquadrilaterals.

Recall: indicator vector, lexicographical measure, ...

LEMMA(*). The indicator vector (of a contrainedtriangulation) becomes lexicographically larger each timean edge of a strictly convex quadrilateral is swappedaccording to the Delaunay swapping criteria.

Basis for Lawson’s LOP Algorithm

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LOP applied to a G, where G is a PSLG:1. Start with an arbitrary G

2. Repeat swapping of edges according to modifiedcircle criterion

Does the LOP converge when applied to G? What does it converge to? Do we reach a global optimum?

Lemma (*) and the fact that the number of possibletriangulations of G is finite guarantees that the algorithmterminates after a finite number of edge swaps.

DEFINITION. Locally optimal edge:1. When the decision is not to swap it in the LOP.2. Edges in Ec and boundary edges are locally optimalby default.

DEFINITION. Locally optimal triangulation; accordingly.

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TEOREM. All interior edges of a triangulation G(G GP,Ec) are locally optimal

(if and only if)

the modified circle criterion holds for all triangles.

Proof. See Exercise in lecture notes.

So, LOP yields a CDT in accordance withDEFINITION (CDT).

THEOREM. A triangulation G is a CDT in accordancewith DEFINITION (CDT)

(if and only if)

its indicator vector is lexicographically maximum.

Proof. See Exercise with guidelines in lecture notes.(Difficult since a dual construction (Voronoi) cannot beused.)

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Uniqueness of a CDT can be deduced from the proof(under the usual assumption that no four points of P arecocircular).

Unique characterization of Delaunay edge and Delaunaytriangle in A CDT (Recall, edges in Ec arenot Delaunayedges):

THEOREM (Delaunay edge). Edge eij between points piand pj of P is Delaunay

(if and only if)

pi and pj are visible to each other and there exists acircle passing through pi and pj that does not contain anypoints of P in its interior visible from both pi and pj.

Proof. See Exercise in lecture notes.

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THEOREM (Delaunay triangle). Triangle t with nodes pi,pj and pk is Delaunay

(if and only if)

Ct contains no point of P in its interior which is visiblefrom both pi, pj and pk.

Proof. See Exercise in lecture notes.

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Algorithms for Constrained DelaunayTriangulation

Overview:Similar schemes as for conventional Delaunay, but1. predefined constrained edges, and

2. modified circle criterion.

Only incremental algorithms are considered here: basicoperations:

inserting a constrained edge into a CDT, and

inserting a node into a CDT.

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Given a PSLG GP,Ec where endpoints of Ec are in P.Algorithm (Compute G):1. Compute P, (conventional Delaunay)

2. for each e in Ecinsert e into P,Ec and update to a CDT.Ec Ec e

3. (add additional points into G)

Recall from conventional Delaunay: two schemes forinserting a point p:

remove triangles in (star shaped) influence region Rp andretriangulate

split t which contains the insertion point into three newtriangles and apply recSwap procedure three times.

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Inserting an Edge into a CDT

“Insert” constrained edge ec between pa and pb inP,Ec P,Ec ec.

(a)

(b)

LecQ ,

RecQ ,

ec

pa pb

pa pb

ec

a) Rec b) Qec,L and Qec,R

Influence region Rec of ec in G are trianglesintersected by ec

Note that no node is inserted or moved only Rec isaffected! (Recall Theorem)

Influence polygons, Qec,L and Qec,R on each side of ec.

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Obtain P,Ec ec by triangulating Qec,L and Qec,R oneach side of the constrained edge ec separately usingmodified circle criterion:

“Step-by-step” approach with base line and growingcircles:

Retriangulate one Q:1. Let ec be the first baseline

2. Start with a growing circle with r and grow it intoQ.

3. Make a triangle t with the first point p reached that isnot separated from ec with a constraint.

4. Chose a new base line ec (an edge of t) and use thesame circle.

5. GOTO 3

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peb

'p

t

Y

The growing

The point p is uniquely defined as the point in P that,i) makes the largest angle at p spanned by thebaseline eb, and

ii) p is visible from the endpoints of eb.

Note: No non-procedural approach as for point insertion.

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(a)

(b)

(c)

(d)

ec

LecQ ,

RecQ ,

Growing circles

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Edge Insertion and Swapping

Recall:T 2VI VB 2 IE 3VI 2VB 3 IIEI 3VI VB 3 IIIT E V 1 IV

,

that is, a new constraint edge between two existingnodes does not change the cardinality of edges ortriangles.

This suggests that a new constrained edge can be“swapped in place”.

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1. Swapping procedure for “inserting” a constraint ecinto P,Ec (operating inside Rec only).

2. LOP applied to edges inside each of Qec,L and Qec,Rto obtain P,Ec ec

The swapping procedure:Principle: Swap edges away from the constrained edgeec such that eventually ec is included as an edge in thetriangulation. Let pa,u1, ,un,pb define the closed influence polygonQec,L to the left of the directed line from pa to pb.

Note how the vertices of the polygon are enumeratedwhen it is multiply connected; see figure.

Consider a point um each time where the angle m and swap away edges radiating from um and intersectingec.

1. is it always possible to find a point um where m ?2. is there always a swappable edge at um?

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ec

u1 u3

u2

u5u4

pa pb

α1

α2

α3α4

α5

ec

u1pa pbα1

LecQ ,

ec

u1

u2 u4

u3

u6u5

pa pb

α1

α2

α3

α4α5

α6(a)

(b)

(c)

RecQ ,

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LEMMA . A closed and simply connected polygon Phas at least three interior angles smaller than .

Proof. Let 1, ,N, N 3, be the interior angles of Pand suppose that k of them are smaller than . Fromelementary geometry we know that the sum of theinterior angles is N 2 and thus,

N 2 i1

N

i i

i i

i.

We have i

i N k, which inserted above gives,

k 2 i

i.

The right hand side is non-negative, so we have k 2.But since k 0 the right hand side is in fact positive andthis gives k 3.

Simply connected influence polygon:Lemma; at least one vertex different from pa and pbwhere m .

multiply connected: see exercise.

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ec

w1 w2 wr

um10 −= muw

11 ++ = mr uw

...

αm

Illustration for Lemma . Suppose that there are r edges radiating from um andintersecting ec; see figure.

The endpoints of the r edges on the opposite side of ecfrom um are denoted w1, ,wr, numberedcounterclockwise. Conventions w0 um1 andwr1 um1.

LEMMA . There is at least one edge radiating fromum and intersecting ec, where m , that is a diagonal ina convex quadrilateral (and thus swappable).

Proof. The closed polygon defined by the sequencew0,w1, ,wr,wr1 (not including um) has at least threeinterior angles smaller than by Lemma. Thus, there isat least one point ws, 1 s r such that the anglews1wsws1 is smaller than . Then the quadrilateralwith um,ws as a diagonal must be convex sincews1,um,ws1 is also smaller than .

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Thus, all edges um,ws, 1 s r, where m , canbe swapped away from um such that there are no edgesleft radiating from um and intersecting ec.

Result: um is eliminated from the influence polygon.

Algorithm (eliminate um, with m ):1. while r 12. Let um,ws be a diagonal in a convex

quadrilateral um,ws1,ws,ws1.3. Swap um,ws to ws1,ws14. r r 1

Eventually, r 1 and and only one edge um,w1 radiatesfrom um and intersects ec.

um,w1 is a diagonal in the quadrilateralum,um1,w1,um1 that is convex by Lemma.

When um,w1 is swapped to um1,um1 in the r’th cycleof the algorithm, um is isolated from ec and eliminatedfrom Q; see u1 in figure.

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Repeat Algorithm on

Q um pa,u1 ,um1,um1,un,pbetc. and eventually on Q pa,u1,pb.

Algorithm (Insert constrained edge ec1. while n 12. Find a point um, 1 m n where m 3. Apply Algorithm to um4. n n 15. Qec,L pa,u1, ,um1,um1, ,un,pb

When n 1, Q pa,u1,pb and the interior angle 1 atu1 is smaller than by Lemma; see figure.

When u1,w1 is swapped to pa,pb it takes the role asthe constrained edge ec that has pa and pb as endpoints.

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Finally, apply LOP to edges inside influence polygonsQec,L and Qec,R.

LOP terminates as CDT with ec as a constraint!

Existence follows from Lemma , extended to multiplyconnected polygons, and Lemma .

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Inserting a Point into a CDT

ce

(a) (b)

p

ce

Influence polygon Qp in a).

Let P p,Ec be the CDT obtained by inserting a pointp into a CDT P,Ec.

Exact limitation of Rp:

LEMMA. A triangle t in P,Ec will be modified wheninserting a point p to obtain P p,Ec if and only if thecircumcircle of t contains p in its interior and p is visiblefrom all the three nodes of t.

Proof. The proof follows directly fromTheorem (Delaunay triangle).

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How can P p,Ec be obtained?

Alt. 1:

THEOREM. All new triangles of P p,Ec have p as acommon node.

Proof. See Exercise in lecture notes.

P p,Ec can be obtained from P,Ec by removing alltriangles of Rp and connecting p to all points of Qp; seeFigure (b).

Alt 2, swapping procedure:As for conventional Delaunay with this modification only:recSwapDelaunay must not swap constrained edges.

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