Delaunay Refinement
Grid Generation
Motivation:
 Meshing for the Finite Element Method (FEM)
 Good triangulations for surface modelling with
constraints
 A general means to compensate for “ill-conditioned”
behaviour of CDTs
Also called meshing or gridding
We still operate in R 2 (Delaunay)
1
FEM
Triangulations as a basis for solving partial differential
equations numerically.
Differential equations, for example representing heat
distribution or airflow, are approximated with piecewise
polynomial functions over a finite element mesh and
solved numerically to achieve an approximate solution
of the original problem.
FEM meshes have typically complicated exterior and
interior boundaries (holes) and linear features:
That is: CDTs (Constrained Delaunay Triangulations)!
2
The Refinement Process
(very brief)
Input:
 A PSLG G (Planar straight line graph)
1. Make the initial CDT of G
Loop (refinement):
2. Split long edges (constraints)
3.
Split skinny triangles
The triangulation is always maintained as a CDT
The basic operation is insertNode (into the CDT)
3
Preview of algorithm
Algorithm (Delaunay refinement)
1. Make the initial CDT of the PSLG.
Remove triangles outside the triangulation domain.
2. while skinny triangles remain
3.
while any segment s is encroached upon
4.
SplitSegment(s)
5.
Let t be a skinny triangle and v its circumcenter.
6.
If v encroaches upon any segments s 1 , s 2 ,  , s k
7.
for i  1,  , k
8.
SplitSegment(s k )
9.
goto 3
10.
else
11.
KillTriangle(t)
12.
goto 5
4
Illustration from Shewchuk.
5
Illustration from Shewchuk.
6
Requirements for FEM Meshes
Triangle size:
 Not too small: size of equation system (and CPU-time)
is linear in the number of triangles
 Sufficiently dense: Accuracy of the solution increases
with increasing density of triangles
Solution: Adaptive mesh with good grading
Triangle shape:
 Skinny triangles gives numerical unstability,
 ill-conditioned equation systems in FEM problems
 numerical unstability in interpolation with
polynomials (splines) defined over triangles

Recall the Delaunay property

Delaunay Refinement
7
Node Insertion
Node insertions should “kill” triangles with
poor quality.
Fundamental questions:
1. What exactly is a well-shaped triangle or a triangle
with poor quality?
2. Where should the next node be inserted during the
refinement process?
3. Will the refinement procedure terminate?
4. Can we control/guarantee good spatial grading?
8
Circumradius-to-shortest-edge ratio r/l
10
9
8
αmin
αmin
r
7
6
5
r/l
4
3
2
1
0
5
10
15
20
l
25
30
αmin
35
40
45
50
55
60
Relationship between  min and r/l
DEFINITION (Circumradius-to-shortest-edge ratio r/l).
Ratio between radius r of the circumcircle of a triangle,
and the length of its shortest edge l.
The smallest angle is always opposite to the shortest
edge; see Figure.
Equivalent with smallest angle measure:
1
r 
.
2 sin  min
l
Maximizing  min corresponds to minimizing r/l.
9
Example:
The equilateral triangle with angles 60  obtains the
largest possible  min and the smallest r/l.
r 
1
1  0. 58.

l
2 sin 60 
3
Also used in the literature is aspect ratio of a triangle:
Ratio between length of the longest edge and the
shortest altitude.
Bounds the minimum angle  min of a the triangle as,
1
2
 Aspect ratio 
,
sin  min
sin  min
10
Upper bound B on the
circumradius-to-shortest-edge ratio r/l.
Delaunay refinement algorithms operate with upper
bound B for the circumradius-to-shortest-edge ratio r/l.
 min  arcsin 1
2B
 max  180   2 arcsin 1/2B
Any triangle with r/l ratio greater than a required upper
bound B will be called skinny.
Example with B 
2 and B  1:
B max r/l  min  arcsin
1
2B
 max  180   2 min
1
30 
120 
2
 20. 7 
 138. 6 
Note: Not possible to obtain if the input PSLG has
angles smaller than  min between incident segments.
11
Where should the next node p be inserted in
a CDT ?
Assumption (temporary): No constraints in .
Let t be a skinny triangle (with r/l  B) and Ct its
circumcircle.
Insertion of a new node inside Ct kills t !
If p is inserted too close to any existing nodes we get
dissatisfactory r/l ratio.

Insert p “as far from existing nodes as possible” !

At the center of Ct
(equidistant from the three nodes of t.)
DEFINITION Circumcenter  center of Ct.
12
Recall the definition of Voronoi diagram, and Delaunay
triangulation as its dual construction:
The circumcenter of t is at the Voronoi point where the
Voronoi edges corresponding to t intersect.

The circumcenter of t is more distant from other nodes
in  than it is from the three nodes of t.
Assumption (temporary): Circumcenters of skinny
triangles are inside .
13
Node insertion at circumcenters of skinny
triangles:
(b)
(a)
p
t
t
(d)
(c)
p
t
p
Kill skinny triangle t with r/l  2. 14  B
14
Recall node insertion of p: Connect p to all boundary
nodes of the influence region R p :
 Three of the new edges have length equal to the
circumradius r of t, and the other new edges are longer
than r.
 t was chosen because,
r/l  B  new edges has length at least Bl.
 If B  1 ( min  30  ), no new edges are shorter than the
edges of t.
(This will be used later when examining termination)
 Also obtains nice spatial grading from small to large
triangles.
15
Suppose (temporary) that a skinny triangle is one with:
“circumradius greater than the shortest edge in the
entire mesh”.

An upper bound r/l  1 would be achieved
corresponding to  min  30  ; and
The mesh will be uniform with all triangles
approximately the same size (and more triangles); see
figure.
16
Splitting Encroached Segments
Assumptions above:
1. No constraints
2. circumcenters inside 
DEFINITION. Segment  constrained edge.
Edges of exterior and interior boundaries are segments.
The input PSLG and  are segment-bounded.
DEFINITION. Subsegments  a segment that has
been split.
Let s be a segment or subsegment of a CDT.
DEFINITION. Diametral circle of s is the circle with s
“as diameter”.
DEFINITION. s is encroached upon if a node different
from the endpoints of s lies on or inside the diametral
circle of s and the node is visible from the interior of s.
17
(a)
(b)
(c)
Splitting of encroached segment.
Rule for splitting:
Eliminate all encroached segments in the CDT .
1. Insert a new node at the midpoint of an encroached
segment (or subsegment) s.
2. s is replaced by two new segments s 1 and s 2 .
3. Update  to be Delaunay
(see figure with recursive splitting).
18
Termination is not guaranteed:
p
s2'
s1'
×
α
s1
×
q
s2
  45 
Assumption (temporary):   45  between incident
segments in the input PSLG; and no other configuration
gives endless loop.
Observation:If a CDT  has no encroached segments
and the only boundary of  is the convex hull of its
nodes, then  is a conventional Delaunay triangulation.
Why:
Theorem: e is a Delaunay edge   a point-free circle
interpolating the endpoints of e.
19
C (t )
C (s' )
t
v
s'
s
20
C (t )
C (s' )
t
v
s'
s
LEMMA. A triangle t in a segment-bounded Delaunay
triangulation is never separated from its circumcenter v
by a segment s if the segment is not encroached upon.
Proof.
Contradiction: Suppose that v is separated from t by a
segment s that is not encroached upon; see figure.
 Let s be the ’closest’ segment to t that is not encroached
upon.
 Let s  be the part of s inside Ct
 Because t is (constrained) Delaunay, both endpoints of s
are outside Ct.
 Since v and t lie on opposite sides of s  , the portion of
Ct on the same side of s  as t lies inside Cs  .
 At least one node of t must be strictly inside Cs  .
 But Cs   is inside Cs, so then at least one of the nodes
of t encroaches upon s
 Contradiction
21
Consequences.
When all encroached segments have been split:
1. The visibility between v and t is never obscured by
a segment.
 t will always disappear when a node is inserted
at v.
2. Recall: edges of the exterior boundary and interior
boundaries enclosing holes are also segments
 v never falls outside the triangulatation or inside
holes where there are no triangles.
22
The Delaunay Refinement
Algorithm
Input: a segment-bounded PSLG.
Two basic operations:
 SplitSegment(s):
Insert a node at the midpoint of a segment s.
 KillTriangle(t):
Insert a node at the circumcenter of a (skinny) triangle t.
KillTriangle: Only applied when there are no
encroached segments present in .
Both: Updates  to be constrained Delaunay.
23
Algorithm (Delaunay refinement)
1. Make the initial CDT of the PSLG.
Remove triangles outside the triangulation domain.
2. while skinny triangles remain
3.
while any segment s is encroached upon
4.
SplitSegment(s)
5.
Let t be a skinny triangle and v its circumcenter.
6.
If v encroaches upon any segments s 1 , s 2 ,  , s k
7.
for i  1,  , k
8.
SplitSegment(s k )
9.
goto 3
10.
else
11.
KillTriangle(t)
12.
goto 5
 The skinny triangle t may disappear during segment
splitting in Step 8 if one of its edges is swapped.
 New nodes used for segment splitting may encroach
upon other segments  Step 9 goes to segment splitting
again.
 The result depends on the specific order in which
encroached segments and skinny triangles are chosen.
24
Termination
The algorithm terminates when there are no
encroached segments and no skinny triangles left.
Is termination guaranteed?
Why shouldn’t it terminate?
 Because both KillTriangle and SplitSegment
may create new skinny triangles.

 The algorithm may run forever creating more and more
(and smaller and smaller) triangles and never get rid of
skinny triangles.
Under what conditions will the algorithm terminate?
25
DEFINITION (Local feature size). Given a PSLG G.
The local feature size at an arbitrary point p  R 2
relative to G, denoted lfs G p (or only lfsp) is the radius
of the smallest disk centered at p that intersects two
non-incident nodes or segments of G.
Local feature size lfs at 
Local feature size lfs:
 lfs is relative to G and never changed during
refinement
 lfs continuous
 lfsp  0 for all points p
 Steepness in the range 1, 1  in any direction:
26
LEMMA. Given a PSLG and two points p and q in the
plane. Then
lfsq  lfsp  dp, q,
where dp, q is the Euclidean distance between p and
q.
DEFINITION (Insertion radius). The insertion radius r v
of an insertion node v is the length of the shortest
Delaunay edge connected to v immediately after
KillTriangle or SplitSegment has executed.
Other cases:
1. If v is an input node of the PSLG:
r v  length of shortest edge connected to v after the
initial CDT has been constructed and before the
refinement process starts.
2. If v is a rejected node (encroaches upon a
segment):
r v  shortest Delaunay edge connected to v as if v
had been inserted.
27
Recall the Delaunay property:
1. New edges created when inserting v have v as a
common endpoint.
2. The closest node visible from v defines the shortest
edge connected to v.
28
(b)
(a)
p
t
t
(d)
(c)
p
t
p
Example: Insertion radius is the shortest edge of t.
29
Termination, continued...
First goal:
Find lower bound on insertion radius r v in all possible
cases of KillTriangle and SplitSegment.
(That is such that r v  “something” )
DEFINITION (Parent node). The parent node of an
insertion node v is the node “responsible” for the
insertion of v.
(Explained under “cases” below.)
30
(a)
(b)
Input PSLG
t
p
p
rv
l ≥ rg
g
rp
v
rv
v
s
s
(c)
(d)
s’
s’
p
p
l
a α
rv
l
β
a
v
α
rv
β
s
v
s
Encroachment cases
31
Cases for KillTriangle and SplitSegment:
Different roles of the parent node p.
Case 1. (KillTriangle(t)): p is the last inserted
node of the two endpoints of the shortest edge of t.
Case 2. (SplitSegment(s)): p is the nearest node
that encroaches upon s.
Facts:
 p lies inside Cs.
 The shortest edge connected to v, which define
the insertion radius r v , has p as the other
endpoint if p is in  (and not rejected in Step 6).
Three possible roles of p:
2a. p is an input node, or p is a node inserted in a
segment not incident with s. (Figure (a))
2b. p is at the circumcenter of a skinny triangle and
thus rejected since it encroaches upon s.
(Figure (b), and Step 6–8 of Algorithm .)
2c. p is a node on a segment s  incident with s that
makes an angle 45     90  with s.
(Figure (c)).
2d. p is a node on a segment s  incident with s that
makes an angle   45  with s. (Figure (d))
32
(a)
(b)
Input PSLG
t
p
p
rv
l ≥ rg
g
rp
v
rv
v
s
(c)
s
(d)
s’
s’
p
p
l
a α
rv
l
β
a
v
α
rv
β
v
s
s
Encroachment cases
LEMMA. The insertion radius r v has the following lower
bounds.
Case 1. r v  Br p ,
Case 2a. r v  lfsv,
Case 2b. r v  r p / 2 ,
Case 2c. r v  r p /2 cos ,
Case 2d. r v  r p sin .
33
Proof.
rp
p
t
l
Case 1. KillTriangle(t) - case.
Recall: t was chosen because r/l  B.
 new edges has length at least Bl.
Facts:
- shortest new edge has length r v (by definition)
- l  rp
 r v  Br p . 
34
(a)
Input PSLG
p
rv
v
s
Case 2a. Consult Figure (a).
Follows directly from definition of local feature size:
r v  lfsv.
35
(b)
(e)
t
p
l ≥ rg
g
g
p
rp
rp
rv
l ≥ rg
t
rv
v
s
v
s
Case 2b. Consult Figure (b).
 p is rejected, so r v  |s |/2.
 r p  circumradius of t.
 Lower bound on r v is achieved when r v /r p is
minimized under the constraint that p lies on or
inside Cs.
 Delaunay property ensures that Ct cannot
contain endpoints s.
Figure (e) illustrates the position of p and Ct for
min r v /r p  1/ 2 .
 r v  r p / 2 .
36
s’
(c)
p
l
a α
rv
β
v
s
Case 2c. Consult Figure (c).
 l  rp.
 r v equals length of vp.
Law of sines gives,
r v  sin  .
l
sin 
Let  be fixed.
 Since l  r p , a lower bound on r v is achieved by
minimizing r v /l under the constraint that p lies on or
inside Cs.
  45     90   sin   1.
 min r v /l when  is maximum under the given
constraint.
 When p lies on Cs, which gives r v  l/2 cos .
 r v  r p /2 cos .
37
(d)
s’
p
l
a
α
rv
β
v
s
Case 2d. Consult Figure (d).
Use the same analysis as in Case 2c; law of sines,
r v  sin  .
l
sin 
But now  is not bounded by 90  .
 r v /l is minimized when sin   1, i.e.,   90  .
 r v  l sin , and since l  r p
r v  r p sin .
38
Goal:
Each node insertion generates edges that are either
 longer than an existing edge, or
 longer than a fixed feature in the PSLG.
(avoid chains of shorter and shorter edges and thus
endless loops)
Recall LEMMA:
Case 1. r v  Br p ,
Case 2a. r v  lfsv,
Case 2b. r v  r p / 2 ,
Case 2c. r v  r p /2 cos ,
Case 2d. r v  r p sin .
(where r v is the shortest edge created when inserting v)
Case 2d: r v  r p sin  for   45
Recall   45  generates endless loop with
SplitSegment in Step 3 and 4 of the Algorithm.
Impossible to get r v  r p .
Case 2c: r v  r p /2 cos  for 45     90 
r v  r p if   60  .
39
(e)
l ≥ rg
t
g
p
rp
rv
v
s
Case 2b: r v  r p / 2 .
Note that node p was not inserted (rejected). Case 1
defines a lower bound for p as r p  Br g ; see figure.
Combines to r v  r p / 2  Br g / 2 .
r v  r g when B 
2.
... and this also holds for
Case 1: r v  Br p .
B
2 from Case 2b sufficient for r v  r p .
40
p
(a)
Input PSLG
rv
v
s
Case 2a: r v  lfsv. . . that is, the input PSLG bounds
(determines) the the shortest edge connected to v!
But, since v and p are non-incident, there is a stronger
bound on r v :
Let d min be the shortest distance between two
non-incident entities of the input PSLG.
Then for any node v inserted by Algorithm,
and d min
r v  d min ,
 min lfs (and therefore it is a better bound)!
The same bound is (of course) valid for edges of the
input PSLG.

No edge in the final mesh can be shorter than d min !
(why? see next slide)
41
Because:
 Case 2a, with r v  d min , is the only case that bounds r v
(the shortest edge) in terms of the input PSLG,
and:
 all the other cases has bounds at least r v  r p ,
where p is either
 a previous inserted node, or.
 a node of the input PSLG.
“Track the history back in time” from an insertion node v
to its parent, grandparent, etc. and you will find that
r v  d min represents a “last barricade” that bounds the
shortest edge in the final mesh.

No edge in the final mesh can be shorter than d min !
42
Summary so far:
1. Ever-decreasing edges cannot occur if:
   60  between incident segments of the input
PSLG, and
 B  2.
2. Lower bound d min on the shortest edge in the final
mesh.
Is this sufficient for termination of Algorithm?
YES, because:
Every edge is Delaunay (in a CDT):
 The shortest edge connected to v has the closest node
visible from v as its other endpoint.
 A new node can never be inserted closer to an existing
node than d min , unless the visibility is obscured by a
segment.
Draw circles with radius d min around each new node.
 Each such circle is node-free.
 Sooner or later there is no space left to draw circles.
 TERMINATION.
43
THEOREM. Let d min be the shortest distance between
two nonincident entities (nodes or segments) in the
input PSLG. Let a triangle be considered skinny if its
circum-radius-to-shortest-edge ratio r/l  B, where
B  2 ; and suppose that the angle between two
incidents segments in the input PSLG is at least 60  .
Then the Algorithm terminates with no edge shorter
than d min .
arcsin 1    180   2 arcsin 1
2B
2B
 20. 7       138. 6  
But with the constraint that the angle between incident
segments is at least 60  !
We can handle this restriction with “corner-lopping”:
44
Corner-Lopping for Handling
Small Input Angles
Illustration from Shewchuk.
1. Lop off corners with sharp angles in the input PSLG
G and obtain G  with angles  60  . Lopping-circles
with radius, e.g., r  lfs G a/3.
2. Run Algorithm on G 
3. Fill in lopped off corners again.
(see lecture notes for details)
45
Examples
Illustration from Ruppert
46
Illustration from Ruppert
47