Introduction
Triangulations
Delaunay Triangulations
Delaunay Triangulations
Computational Geometry
Lecture 12: Delaunay Triangulations
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Motivation: Terrains
a terrain is the graph of a
function f : A ⊂ R2 → R
we know only height values for a
set of measurement points
how can we interpolate the
height at other points?
using a triangulation
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Motivation: Terrains
a terrain is the graph of a
function f : A ⊂ R2 → R
we know only height values for a
set of measurement points
how can we interpolate the
height at other points?
using a triangulation
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Motivation: Terrains
a terrain is the graph of a
function f : A ⊂ R2 → R
?
we know only height values for a
set of measurement points
how can we interpolate the
height at other points?
using a triangulation
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Motivation: Terrains
a terrain is the graph of a
function f : A ⊂ R2 → R
?
we know only height values for a
set of measurement points
how can we interpolate the
height at other points?
using a triangulation
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Motivation: Terrains
0
10
a terrain is the graph of a
function f : A ⊂ R2 → R
using a triangulation
– but which?
Computational Geometry
q 980
20
36
990
6
we know only height values for a
set of measurement points
how can we interpolate the
height at other points?
1000
1008
4
890
interpolated height = 985
28
23
1000
20
0
10
q 980
36
990
6
1008
4
890
interpolated height = 23
Lecture 12: Delaunay Triangulations
28
23
Introduction
Triangulations
Delaunay Triangulations
Triangulation
Let P = {p1 , . . . , pn } be a point set. A triangulation of P is a
maximal planar subdivision with vertex set P.
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Triangulation
Let P = {p1 , . . . , pn } be a point set. A triangulation of P is a
maximal planar subdivision with vertex set P.
Complexity:
2n − 2 − k triangles
3n − 3 − k edges
where k is the number of points in P
on the convex hull of P.
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Angle Vector of a Triangulation
Let T be a triangulation of P with m triangles and 3m
vertices. Its angle vector is A(T) = (α1 , . . . , α3m ) where
α1 , . . . , α3m are the angles of T sorted by increasing value.
Let T 0 be another triangulation of P.
We define A(T) > A(T 0 ) if A(T) is
lexicographically larger than A(T 0 ).
T is angle optimal if A(T) ≥ A(T 0 )
for all triangulations T 0 of P.
Computational Geometry
α5
α4
α2
α1
α6
α3
A(T) = (α1 , . . . , α6 )
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Angle Vector of a Triangulation
Let T be a triangulation of P with m triangles and 3m
vertices. Its angle vector is A(T) = (α1 , . . . , α3m ) where
α1 , . . . , α3m are the angles of T sorted by increasing value.
Let T 0 be another triangulation of P.
We define A(T) > A(T 0 ) if A(T) is
lexicographically larger than A(T 0 ).
T is angle optimal if A(T) ≥ A(T 0 )
for all triangulations T 0 of P.
Computational Geometry
α5
α4
α2
α1
α6
α3
A(T) = (α1 , . . . , α6 )
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Angle Vector of a Triangulation
Let T be a triangulation of P with m triangles and 3m
vertices. Its angle vector is A(T) = (α1 , . . . , α3m ) where
α1 , . . . , α3m are the angles of T sorted by increasing value.
Let T 0 be another triangulation of P.
We define A(T) > A(T 0 ) if A(T) is
lexicographically larger than A(T 0 ).
T is angle optimal if A(T) ≥ A(T 0 )
for all triangulations T 0 of P.
Computational Geometry
α5
α4
α2
α1
α6
α3
A(T) = (α1 , . . . , α6 )
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Edge Flipping
pl
α2
pi
α1
α4
pl
α3
α5
pj
edge flip
α20
α6
pi
pk
α10
α40
α30
α60
α50
pk
Change in angle vector:
α1 , . . . , α6 are replaced by α10 , . . . , α60 .
The edge e = pi pj is illegal if min1≤i≤6 αi < min1≤i≤6 αi0 .
Flipping an illegal edge increases the angle vector.
Computational Geometry
Lecture 12: Delaunay Triangulations
pj
Introduction
Triangulations
Delaunay Triangulations
Edge Flipping
pl
α2
pi
α1
α4
pl
α3
α5
pj
edge flip
α20
α6
pi
pk
α10
α40
α30
α60
α50
pk
Change in angle vector:
α1 , . . . , α6 are replaced by α10 , . . . , α60 .
The edge e = pi pj is illegal if min1≤i≤6 αi < min1≤i≤6 αi0 .
Flipping an illegal edge increases the angle vector.
Computational Geometry
Lecture 12: Delaunay Triangulations
pj
Introduction
Triangulations
Delaunay Triangulations
Edge Flipping
pl
α2
pi
α1
α4
pl
α3
α5
pj
edge flip
α20
α6
pi
pk
α10
α40
α30
α60
α50
pk
Change in angle vector:
α1 , . . . , α6 are replaced by α10 , . . . , α60 .
The edge e = pi pj is illegal if min1≤i≤6 αi < min1≤i≤6 αi0 .
Flipping an illegal edge increases the angle vector.
Computational Geometry
Lecture 12: Delaunay Triangulations
pj
Introduction
Triangulations
Delaunay Triangulations
Edge Flipping
pl
α2
pi
α1
α4
pl
α3
α5
pj
edge flip
α20
α6
pi
pk
α10
α40
α30
α60
α50
pk
Change in angle vector:
α1 , . . . , α6 are replaced by α10 , . . . , α60 .
The edge e = pi pj is illegal if min1≤i≤6 αi < min1≤i≤6 αi0 .
Flipping an illegal edge increases the angle vector.
Computational Geometry
Lecture 12: Delaunay Triangulations
pj
Introduction
Triangulations
Delaunay Triangulations
Characterisation of Illegal Edges
How do we determine if an edge is illegal?
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Characterisation of Illegal Edges
How do we determine if an edge is illegal?
Lemma: The edge pi pj is illegal
if and only if pl lies in the interior
of the circle C.
pl
pj
pi
pk
Computational Geometry
Lecture 12: Delaunay Triangulations
illegal
Introduction
Triangulations
Delaunay Triangulations
Thales Theorem
s
q
Theorem: Let C be a circle, ` a line
intersecting C in points a and b, and
p, q, r, s points lying on the same side
of `. Suppose that p, q lie on C, r lies
inside C, and s lies outside C. Then
p
r
b
`
a
C
]arb > ]apb = ]aqb > ]asb,
where ]abc denotes the smaller
angle defined by three points a, b, c.
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Thales Theorem
s
q
Theorem: Let C be a circle, ` a line
intersecting C in points a and b, and
p, q, r, s points lying on the same side
of `. Suppose that p, q lie on C, r lies
inside C, and s lies outside C. Then
p
r
b
`
a
C
]arb > ]apb = ]aqb > ]asb,
pl
where ]abc denotes the smaller
angle defined by three points a, b, c.
pj
pi
pk
Computational Geometry
Lecture 12: Delaunay Triangulations
illegal
Introduction
Triangulations
Delaunay Triangulations
Thales Theorem
s
q
Theorem: Let C be a circle, ` a line
intersecting C in points a and b, and
p, q, r, s points lying on the same side
of `. Suppose that p, q lie on C, r lies
inside C, and s lies outside C. Then
p
r
b
`
a
C
]arb > ]apb = ]aqb > ]asb,
pl
where ]abc denotes the smaller
angle defined by three points a, b, c.
α0 < α
β0 < β
αβ
pi
α0
β0
pk
Computational Geometry
Lecture 12: Delaunay Triangulations
pj
illegal
Introduction
Triangulations
Delaunay Triangulations
Legal Triangulations
A legal triangulation is a triangulation that does not contain
any illegal edge.
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Legal Triangulations
A legal triangulation is a triangulation that does not contain
any illegal edge.
Algorithm LegalTriangulation(T)
Input. A triangulation T of a point set P.
Output. A legal triangulation of P.
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
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Legal Triangulations
A legal triangulation is a triangulation that does not contain
any illegal edge.
Algorithm LegalTriangulation(T)
Input. A triangulation T of a point set P.
Output. A legal triangulation of P.
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
Question: Why does this algorithm terminate?
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Voronoi Diagram and Delaunay Graph
Let P be a set of n points in the
plane.
The Voronoi diagram Vor(P) is
the subdivision of the plane into
Voronoi cells V(p) for all p ∈ P.
Let G be the dual graph of
Vor(P).
The Delaunay graph DG(P) is
the straight line embedding of G.
Question: How can we
compute the Delaunay graph?
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Voronoi Diagram and Delaunay Graph
Let P be a set of n points in the
plane.
The Voronoi diagram Vor(P) is
the subdivision of the plane into
Voronoi cells V(p) for all p ∈ P.
Let G be the dual graph of
Vor(P).
The Delaunay graph DG(P) is
the straight line embedding of G.
Question: How can we
compute the Delaunay graph?
Computational Geometry
Lecture 12: Delaunay Triangulations
Vor(P)
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Voronoi Diagram and Delaunay Graph
Let P be a set of n points in the
plane.
Vor(P)
The Voronoi diagram Vor(P) is
the subdivision of the plane into
Voronoi cells V(p) for all p ∈ P.
G
Let G be the dual graph of
Vor(P).
The Delaunay graph DG(P) is
the straight line embedding of G.
Question: How can we
compute the Delaunay graph?
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Voronoi Diagram and Delaunay Graph
Let P be a set of n points in the
plane.
Vor(P)
The Voronoi diagram Vor(P) is
the subdivision of the plane into
Voronoi cells V(p) for all p ∈ P.
DG(P)
Let G be the dual graph of
Vor(P).
The Delaunay graph DG(P) is
the straight line embedding of G.
Question: How can we
compute the Delaunay graph?
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Voronoi Diagram and Delaunay Graph
Let P be a set of n points in the
plane.
Vor(P)
The Voronoi diagram Vor(P) is
the subdivision of the plane into
Voronoi cells V(p) for all p ∈ P.
DG(P)
Let G be the dual graph of
Vor(P).
The Delaunay graph DG(P) is
the straight line embedding of G.
Question: How can we
compute the Delaunay graph?
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Planarity of the Delaunay Graph
Theorem: The Delaunay graph of a planar point set is a plane
graph.
contained in V(pi )
pi
Ci j
pj
contained in V(p j )
Computational Geometry
Lecture 12: Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Introduction
Triangulations
Delaunay Triangulations
Delaunay Triangulation
If the point set P is in general position then the Delaunay graph
is a triangulation.
f
Computational Geometry
v
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Empty Circle Property
Theorem: 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
if and only if the circumcircle of any triangle of T does not
contain a point of P in its interior.
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Delaunay Triangulations and Legal Triangulations
Theorem: 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.
pm
C(pi p j pm )
pi
pl
e
pj
pk
C(pi p j pk )
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Angle Optimality and Delaunay Triangulations
Theorem: Let P be a set of points in the plane. Any
angle-optimal triangulation of P is a Delaunay triangulation of
P. Furthermore, any Delaunay triangulation of P maximizes
the minimum angle over all triangulations of P.
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Randomized Incremental Construction
Algorithm DelaunayTriangulation(P)
Input. A set P of n + 1 points in the plane.
Output. A Delaunay triangulation of P.
1. Initialize T as the triangulation consisting of an outer
triangle p0 p−1 p−2 containing points of P, where p0 is the
lexicographically highest point of P.
2. Compute a random permutation p1 , p2 , . . . , pn of P \ {p0 }.
3. for r ← 1 to n
4.
do
5.
Locate(pr , T)
6.
Insert(pr , T)
7. Discard p−1 and p−2 with all their incident edges from T.
8. return T
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Randomized Incremental Construction
pr lies in the interior of a triangle
pr falls on an edge
pi
pk
pl
pr
pr
pi
pj
Computational Geometry
pj
Lecture 12: Delaunay Triangulations
pk
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Randomized Incremental Construction
Insert(pr , T)
1. if pr lies in the interior of the triangle pi pj pk
2.
then Add edges from pr to the three vertices of pi pj pk , thereby
splitting pi pj pk into three triangles.
3.
LegalizeEdge(pr , pi pj , T)
4.
LegalizeEdge(pr , pj pk , T)
5.
LegalizeEdge(pr , pk pi , T)
6.
else (∗ pr lies on an edge of pi pj pk , say the edge pi pj ∗)
7.
Add edges from pr to pk and to the third vertex pl of the
other triangle that is incident to pi pj , thereby splitting the
two triangles incident to pi pj into four triangles.
8.
LegalizeEdge(pr , pi pl , T)
9.
LegalizeEdge(pr , pl pj , T)
10.
LegalizeEdge(pr , pj pk , T)
11.
LegalizeEdge(pr , pk pi , T)
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Randomized Incremental Construction
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 the 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)
Computational Geometry
pj
pk
pr
Lecture 12: Delaunay Triangulations
pi
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Randomized Incremental Construction
=⇒
pr
All edges created are incident to pr .
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Randomized Incremental Construction
=⇒
pr
All edges created are incident to pr .
Correctness: Are new edges legal?
Computational Geometry
Lecture 12: Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Introduction
Triangulations
Delaunay Triangulations
Randomized Incremental Construction
pl
pj
pr
C
C0
pi
Correctness:
For any new edge there is an empty circle through endpoints.
New edges are legal.
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Randomized Incremental Construction
Initializing triangulation: treat p−1 and p−2 symbolically.
No actual coordinates.
Modify tests for point location and illegal edges to work as if
far away.
Point location: search data structure.
Point visits triangles of previous triangulations that contain it.
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Randomized Incremental Construction
flip pi p j
∆1
∆1 ∆2 ∆3
∆1
pi
∆2
∆5
∆3
∆3
pj
∆1 ∆2
∆3
∆4
∆5
pk
flip pi pk
split ∆1
pr
∆2
∆4
∆1
∆3
∆2
∆3
∆4
∆5
∆4
∆2
∆7
pi
∆6
∆3
∆6
Computational Geometry
Lecture 12: Delaunay Triangulations
∆7
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Analysis
1
Expected total number of triangles created in O(n)
2
Expected total number of triangles visited while search
point location data structure: O(n log n)
We will only consider the first (see book for second)
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Analysis
Lemma: Total number of triangles created is at most 9n + 1.
How many triangles are created when inserting pr ?
Backwards analysis: Any point of p1 , . . . , pr has the same
probability 1/r to be pr .
Expected degree of pr ≤ 6.
Number of triangles created ≤ 2degree(pr ) − 3
(Why? Count flips.)
2·6−3 = 9
+ outer triangle
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Analysis
Lemma: Total number of triangles created is at most 9n + 1.
How many triangles are created when inserting pr ?
Backwards analysis: Any point of p1 , . . . , pr has the same
probability 1/r to be pr .
Expected degree of pr ≤ 6.
Number of triangles created ≤ 2degree(pr ) − 3
(Why? Count flips.)
2·6−3 = 9
+ outer triangle
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Analysis
Lemma: Total number of triangles created is at most 9n + 1.
How many triangles are created when inserting pr ?
Backwards analysis: Any point of p1 , . . . , pr has the same
probability 1/r to be pr .
Expected degree of pr ≤ 6.
Number of triangles created ≤ 2degree(pr ) − 3
(Why? Count flips.)
2·6−3 = 9
+ outer triangle
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Analysis
Lemma: Total number of triangles created is at most 9n + 1.
How many triangles are created when inserting pr ?
Backwards analysis: Any point of p1 , . . . , pr has the same
probability 1/r to be pr .
Expected degree of pr ≤ 6.
Number of triangles created ≤ 2degree(pr ) − 3
(Why? Count flips.)
2·6−3 = 9
+ outer triangle
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Analysis
Lemma: Total number of triangles created is at most 9n + 1.
How many triangles are created when inserting pr ?
Backwards analysis: Any point of p1 , . . . , pr has the same
probability 1/r to be pr .
Expected degree of pr ≤ 6.
Number of triangles created ≤ 2degree(pr ) − 3
(Why? Count flips.)
2·6−3 = 9
+ outer triangle
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Analysis
Lemma: Total number of triangles created is at most 9n + 1.
How many triangles are created when inserting pr ?
Backwards analysis: Any point of p1 , . . . , pr has the same
probability 1/r to be pr .
Expected degree of pr ≤ 6.
Number of triangles created ≤ 2degree(pr ) − 3
(Why? Count flips.)
2·6−3 = 9
+ outer triangle
Computational Geometry
Lecture 12: Delaunay Triangulations
Introduction
Triangulations
Delaunay Triangulations
Properties
Randomized Incremental Construction
Analysis
Analysis
Theorem: The Delaunay triangulation of n points can be
computed in O(n log n) expected time.
Computational Geometry
Lecture 12: Delaunay Triangulations