computational geometry csci3250 - bowdoin.edu · triangulation input: p = {p1, p2,…,p n} set of...

Computational Geometry csci3250

!!!!!!

Laura Toma !

Bowdoin College

Today

• Triangulations • Delaunay triangulation

• Properties • How to construct it • Applications

Triangulation

Input: P = {p1, p2,…,pn} 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.

• Many triangulations possibleP = {p1, p2,…,pn} set of points in the plane

• In practice, want triangulations without small angles • small angles cause numerical instability with geometric predicates, etc

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 = {p1, p2,…,pn} set of points in the plane

Voronoi Diagram Vor(P)

• Vor(pi): all points in the plane that are closer to pi than to any other site


v

p1

p2

p3


• The edges of Vor(P) are (segments of) perpendicular bisectors


v

p1

p2

p3


• Every Voronoi vertex is the intersection of 3 Voronoi regions

(if no 4-points co-circular)

P = {p1, p2,…,pn} set of points in the plane no 4 points co-circular

v

p1

p2

p3


• Degeneracies • more than 3 points lie on same circle • collinear points



• Every Voronoi vertex is the intersection of 3 Voronoi regions • v is equidistant from p1, p2, p3

v

p1

p2

p3



• 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 v

v

p1

p2

p3



• 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

v

p1

p2

p3

• Every Voronoi vertex is the center of a circle that has 3 sites on its boundary and no other sites inside


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

v

p1

p2

p3


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)

v

p1

p2

p3


Delaunay triangulation DTP = {p1, p2,…,pn} set of points in the plane no 4 points co-circular


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)).


v

p1

p2

p3 p1

p2

p3


What we know from Vor(P) • p1p2p3 is a triangle in DT(P) <==> circumcircle of p1p2p3 is empty

!!!!!!!!!

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



• How to build a triangulation such that all triangles have empty-circle property?

p1

p2

p3

circumcircle of p1p2p3 is empty

• How to build a triangulation? • Input: a set of points • Output: a partition of CH(P) into triangles

!!!!!!!!


• Many triangulations possible


• Come up with an algorithm to compute a triangulation of P

!!!!!!!!!!!

• Possible ideas: incremental, plane sweep, divide-and-conquer, …

Class work: P = {p1, p2,…,pn} set of points in the plane no 4 points co-circular

Flipping edgesedge flips

How to get a DT? • Compute an arbitrary triangulation T(P) • T ————> DTedge flips

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edge flipsFlipping edges and empty circles

C(abc) contains d b outside C(acd)flip ab to cd

WHY?

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Lemma: d inside C(abc) <==> b outside C(abd)

• Why? !• basic geometry of angles (Thales theorem)

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a > b > c

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d

Note condition is symmetrical: d inside C(abc) <==> c inside C(abs)

• Why? !• basic geometry of angles (Thales theorem)

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a > b > c

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))

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edge flipsFlipping edges and empty circles

ab illegal cd legalflip

Theorem, revisited:

A triangulation where all edges are locally Delaunay (legal) is the DT.

(locally Delaunay ==> globally Delaunay)

a

c

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da

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Flipping edges and empty circles

ab illegal cd legalflip ab to cd

DT via edge flipping!• construct an arbitrary triangulation T • mark all edges in T and put them in S • while S not empty

• pop edge ab from S and un-mark it • 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

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c

b

d

Delaunay triangulation

Does this ever terminate?.. If it terminates, T is DT (by theorem)

a1

a2

a3a4 a5a6

d

a b

cEdge flipping and angles

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))

b1

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b4b5b6a

Edge flipping and angles

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.

a1

a2

a3a4 a5a6

d

a b

c

b1

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b2 b3

b4b5b6a

More generally..

A triangulation T.

consider all angles in each of its m triangles => 3m angles

sort angles increasingly: a1 ≦ a2 ≦ a3 ≦…… ≦ a3m

A(T) = [a1, a2, a3, ……,a3m]

!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.

a1

a2

a3a4 a5a6

d

a b

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b1

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b4b5b6a

DT Applications

• DT used because it is angle-optimal. • Small angles

• Numerical instability, Interpolation errors • Issues with geometric predicates (leftOf, tangents, derivatives)

Computing DT: RIC

RIC (Randomized Incremental Construction)!• Start with a large triangle T that contains P. • Compute a random permutation of the points in P • For each point p in P do:

//insert p in T • find triangle abc of T that contains p • 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


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p

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b

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) • LegalizeEdge(p, qv, T)


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computational geometry csci3250 - bowdoin.edu · triangulation input: p = {p1, p2,…,p n} set of...

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