AUTOMATIC TRIANGULAR MESH
GENERATION BASED ON
LONGEST EDGE ALGORITHMS
Maria Cecilia Rivara Nancy Hitschfeld and Carlos Hasan
Department of Computer Science University of Chile
Casilla Santiago Chile
mcrivara dcc uchile cl nancy dcc uchile cl chasan dcc uchile cl
Abstract Several aspects of the LEPP algorithms based on the
use of the Longest Edge Propagation Path of the triangles for dealing
both with the triangulation improvement problem and with the auto
matic quality triangulation problem are reviewed and discussed Ap
plications producing quality nonobtuse triangulations and quality sim
pli cation for terrain modeling are also discussed
Key Words Mesh generation quality triangulatio n Delaunay algorithm
longest edge algorithms b oundary nonobtuse triangulatio n nite volume
metho ds terrain mo deling
Intro duction
In the adaptive nite element context several mathematical algorithms for
the re nement and or dere nement of unstructured triangulatio ns based on
the bisection of triangles by its longest edge have b een discussed and used
1{4
in the last years These algorithms guarantee the construction of re
ned nested and irregular triangulations of analogous quality as the input
Submitted to the International journal of numerical metho ds juli
triangulation However the use of a new and related mathematical con
5, 6
cept the longest edge propagation path of a triangle has allowed the
development of new longest edge algorithms for dealing with more general
asp ects of the mesh generation problem triangulation re nement prob
lem triangulation improvement problem automatic quality triangu
lation problem quality nonobtuse triangulation problem and terrain
mo deling triangulation problem In this pap er di erent asp ects of these
mesh generation problems and some of the algorithms prop osed to deal with
them are reviewed discussed and illustrated In particular the following
sp eci c applications are discussed quality triangulations as needed for nite
element metho ds quality nonobtuse b oundary triangulatio ns as needed for
mixed nite element and nite volume metho ds and quality surface terrain
simpli cation
Mesh generation related problems
The p olygon triangulatio n problem an imp ortant issue for nite element
applications can b e formulated as follows
De nition Polygon Triangulation Problem given N representative points
of a polygonal region join them by non intersecting straight line segments so
that every region internal to the polygon is a triangle The resulting triagu
lations is a conforming triangulation the intersection of adjacent triangles
is either a common vertex or a common side
Many criteria have b een prop osed as to what constitutes a go o d tri
angulation for numerical purp oses some of which involve maximizing the
smallest angle or minimizing the total edge length The Delaunay algorithm
which constructs triangulations satisfying the rst criteria has b een of com
mon use in engineering applications followed by a p ostpro cess step which
assures the b oundary resp ect of the p olygon
In the adaptive nite element context the triangulation re nement prob
lem is also critical To state this problem some requirements and criteria
ab out how to de ne the set of triangles to b e re ned and how to obtain
the desired resolution need to b e sp eci ed To simplify we shall intro duce a
subregion R to de ne the re nement area and a condition over the diameter
longest edge of the triangles given by a resolution parameter to x the
desired resolution
De nition Triangulation Re nement Problem given an acceptable trian
gulation of a polygonal region construct a local ly re ned triangulation such
that the diameters of the triangles that intersect the re nement area R are
less than and such that the smal lest or the largest angle is bounded
In the case we disp ose of a bad quality triangulation of the p olygonal
geometry having a non adequate distribution of vertices the triangulatio n
improvement problem has to b e considered To state this problem a triangle
quality indicator function q t a tolerance parameter and a lo cal triangle
improvement criterion need to b e sp eci ed
De nition Triangulation Improvement Problem given a non quality tri
angulation of a polygonal region having triangles such that its quality
0
indicator q t construct an improved triangulation such that each
triangle t satis es q t
Note that if an initial coarse triangulatio n of the b oundary p olygonal
vertices is considered the more general automatic quality p olygon triangu
lation problem can b e stated
De nition Quality Triangulation Problem Given an initial boundary
triangulation of the boundary vertices which de ne the polygonal geometry
0
construct a geometry adapted triangulation such that for each triangle t of
q t
For nite element nite volume applications the following Nonobtuse
Boundary Triangulati on Problem can b e stated
De nition Nonobtuse Boundary Triangulation Problem given a quality
Delaunay triangulation of a polygonal region construct a triangulation
0
such that the boundary triangles having at least one boundary or interface
edge do not have an obtuse angle opposite to any boundary or interface edge
At this p oint the following remarks are in order
The triangulation problems stated in De nitions to are essentially
di erent than the classical triangulation problem in the following sense in
stead of having a xed set of p oints to b e triangulated one has the freedom
to cho ose the p oints to b e added in order to construct a mesh either with a
desired resolution or with a given mesh quality The construction of the mesh
is dynamically p erformed Furthermore it is p ossible to exploit the existence
of the reference triangulation constructed for instance by means of the De
launay algorithm in order to reduce the computational cost to construct the
output mesh
To cop e with the triangulatio n Re nement Problem the longest edge
re nement algorithms guarantee the construction of go o d quality irregular
triangulations section This is due in part to their natural re nement
propagation strategy farther than the re nement area of interest R Fur
thermore asymptotically the numb er N of p oints inserted in R to obtain
triangles of prescrib ed size is optimal and in spite of the unavoidable prop
agation outside the re nement region R the time cost of the algorithm is
7
linear in N indep endent of the size of the triangulatio n
In the remaining of this pap er longest edge based solutions b oth for the
improvement and quality triangulation problems of De nition and will
b e discussed in the context of automatic quality triangulations section
quality nonobtuse b oundary triangulation s section and triangulation s
for terrain mo deling section Note that in all these applications the
algorithms take advantage of an LEPP p oint insertion technique based on
following the longest edge propagation path of the target triangles over De
launay triangulations
The longest edge propagation path of a tri
angle
The longest edge propagation path of a triangle concept is de ned as follows
De nition For any triangle t of any conforming triangulation the
0
Longest Edge Propagation Path of t wil l be the ordered list of al l the triangles
0
t t t t such that t is the neighbor triangle of t by the longest edge
0 1 2 n i i 1
of t for i n In addition we shal l denote it as the LEPP t
i 1 0
Prop osition For any triangle t of any conforming triangulation of any
0
bounded dimensional geometry the fol lowing properties hold a for any
t the LEPP t is always nite b The triangles t t t have strictly
0 1 n 1
increasing longest edge if n c For the triangle t of the Longest Edge
n
Propagation Path of any triangle t it holds that either i t has its longest
0 n
edge along the boundary and this is greater than the longest edge of t or
n 1
ii t and t share the same common longest edge
n n 1
De nition Two adjacent triangles t t wil l be cal led a pair of terminal
triangles if they share their respective common longest edge In addition t
wil l be a terminal boundary triangle if its longest edge lies along a boundary
edge
It should b e p ointed out here that the Longest Edge Propagation Path of
any triangle t corresp onds to an asso ciated p olygon which in certain sense
measures the lo cal quality of the current p oint distribution induced by t To
illustrate these ideas see Figure a where the Longest Edge Propagation
Path of t corresp onds to the ordered list of triangles t t t t Moreover
0 0 1 2 3
the pair t t is a pair of terminal triangles
2 3
LEPP Bisection Algorithm for the re ne
ment of quality triangulations
By using the LEPP t concept an improved Longest Edge re nement algo
5, 6
rithm for non Delaunay triangulation s can b e formulated where the pure
longest edge re nement of a target triangle t see Figure essentially means
0
the rep etitive longest edge partition of pairs of terminal triangles asso ciated
with the current LEPP t until the triangle t itself is partitioned
0 0
The Figure illustrates the re nement of the triangle t over the initial
0
triangulation of Figure a with asso ciated LEPP t ft t t t g The
0 0 1 2 3
triangulations b and c illustrate the rst steps of the LEPP Bisection
pro cedure and their resp ective current LEPP t while that triangulation d
0
is the nal mesh obtained Note that the new vertices have b een enumerated
in the order they were created
The LEPP Bisection pro cedure schematically describ ed in Figure is a
non recursive algorithm essentially based on re ning pairs of terminal trian
gles where the concept of the Longest Edge Propagation Path of the triangle
t is rep eatedly used over the current triangulation in order to nd the last
terminal triangles of the path until the initial triangle i is bisected
Since the LEPP Bisection algorithm is an improved version of the orig
inal longest edge bisection algorithm the following Theorem based on the
prop erties of the longest edge bisection also holds t 0 t 0
t 1 t 1 t t 2 t 3 2 1
a) b)
t 0 6
t 1 t 4 5 4 t 2 t 3 3 2 2
1 1
c) d)
Figure LEPP Bisection of triangle t a Initial Triangulatio n b First
0
step of the pro cess c Second step in the pro cess d Final triangulation
Theorem a The repetitive use of the pure longest edge bisection algo
rithms in order to re ne t and its descendants triangles nested in t
0 0
tends to produce local quasi equilateral triangulations b The smal lest angle
of any triangle t obtained throughout this process satis es that
t t 0
where is the smal lest angle of t c For any conforming triangulation
0 0
the global iterative application of the algorithm covers in a monotonical ly
increasing form the area of t with quasi equilateral triangles
Theorem guarantees the construction of go o d quality irregular and
nested triangulatio ns Theorem assures in exchange that the LEPP Bisection
algorithm solves the triangulation re nement problem with linear time com
plexity provided that an initial go o d quality triangulation is used To this
end a suitable data structure that explicitly manage the neighb or triangle
relation should b e used In addition since at each iteration within the while
lo op the LEPP t may or not b e shortened and may include new triangles
not previously included in the LEPP t see Figure the current LEPP t
should b e up dated rather than computed from scratch in order to get the
linear running time Furthermore the new LEPP Re nement Algorithm
LEPP Bisection t T
while t remains without b eing bisected do
Find the LEPP t
if t the last triangle of the LEPP t is a terminal b oundary triangle
then
bisect t
else
bisect the last pair of terminal triangles of the LEPP t
end if
end while
Figure LEPP Bisection pro cedure
pro duces the same triangulation as the previous recursive algorithm in a
simpler cleaner easy to implement and more direct way
Theorem Let be any conforming triangulation of any bounded polygonal
region Then for any circular re nement subregion C of radious r the
use of the LEPP Bisection algorithm to produce triangles of size inside C
asymptotical ly introduces N points inside C and N points outside C where
i o
r
2
N O n N O n log n and n
i o
LEPP Delaunay algorithm for the improve
ment of triangulations
The LEPP Delaunay improvement algorithm uses the Longest Edge Prop
agation Path of the target triangles to b e improved in the mesh over the
Delaunay triangulation of the vertices in order to decide which is the b est
5, 6
p oint to b e inserted to pro duce a go o d quality distribution of p oints This
8, 9
basic algorithm generalizes the ideas intro duced in Rivara et al
LEPP Delaunay Improvement t T f
Input T Delaunay triangulatio n
while t remains without b eing mo di ed do
Find the Longest Edge Propagation Path of t
Perform a Delaunay insertion of the p oint p midp oint of the longest
edge of the last triangle in the Lepp t
end whileg
Figure LEPP Delaunay improvement pro cedure
For an illustration of the algorithm see Figure where the triangula
tion a is the initial Delaunay triangulation with LEPP t ft t t t g
0 0 1 2 3
and the triangulations b c and d illustrate the complete sequence of
p oint insertions needed to improve t Note that in this example the im
0
provement mo di cation of t implies the automatic Delaunay insertion of
0
three additional Steiner p oints Each one of these p oints is the midp oint
of the longest edge of the last triangle of the current LEPP t It should
0
b e p ointed out here that each Delaunay p oint insertion lo cally improves the
triangulation in the current LEPP t and in this sense this algorithm im
0
proves the triangulation s obtained with the pure LEPP Bisection pro cedure
of section
Note that we have used the word improvement instead of bisection or
re nement This is to make explicit the fact that one step of the pro cedure
do es not necessarily pro duce a smaller triangle More imp ortant however is
the fact that the pro cedure improves the triangle t in the sense of Theorem
of section t0 t0
t1 t1 t2 2 t3
t2 1
t3
a) b)
t0 3 t1 3 2 t2
1
c) d)
Figure LEPP Delaunay improvement of triangle t
0
Automatic algorithm for pro ducing quality
triangulations
By combining the basic LEPP pro cedure over constrained Delaunay trian
gulations with adequate b oundary considerations a simple dimensional
automatic quality triangulati on algorithm can b e formulated see Figure
where is a threshold parameter less than or equal to that can b e easily
adjusted Furthermore the following theorem holds
Theorem For any Delaunay triangulation T the repetitive use of the
LEPP Delaunay Improvement algorithm over the worst triangles of the mesh
with smal lest angle produces a quality triangulation of smal lest an
gles greater than or equal to
At this p oint the following remarks are in order
Even when Theorem guarantees the construction of quality trian
gulations it says nothing ab out the size of these triangulations More math
ematical results in this sense are certainly needed However in practice the
dimensional triangulatio ns obtained are size optimal In fact they are of
analogous quality as those obtained with the circumcenter p oint insertion
10
strategy
Quality Polygon Tria ngul ati on P f
Input A general p olygon P de ned by a set of vertices and edges and
a tolerance parameter
Construct T a constrained b oundary Delaunay triangulation of P
Find S the set of the worst triangles t of T of smallest angle
for each t in S do
Backward LE Delaunay Improvement T t
Up date the set S by adding the new small angled triangles and elimi
nating those destroyed throughout the pro cess
end forg
LEPP Delaunay Improvement T t f
while t remains without b eing mo di ed do
if t has a b oundary edge l and l is not the smallest edge of t then
select p the midp oint of l
else
Find the Lepp t and t the last triangle in the Lepp t
select p midp oint of the longest edge of t
end if
Perform the Delaunay insertion of p
end whileg
Figure LEPP Delaunay pro cedure with b oundary considerations
The triangulatio n of Figure illustrates the practical b ehavior of the
algorithm Note that the input data was the p olygon with the minimum
numb er of vertices to describ e the geometry
Figure Automatic triangulation obtained smallest angles greater than
Nonobtuse b oundary triangulations
This section discusses an algorithm to solve the nonobtuse b oundary prob
lem of De nition such as needed for mixed control volume discretization
11
and nite element metho d This extends the LEPP Delaunay algorithm
of section Nonobtuse b oundary triangulatio ns are also very useful when
problems are solved combining b oth metho ds This requires the combination
of go o d quality meshes and well shap ed Voronoi b oxes which implies b oth
that the minimum angle should b e b ounded and that b oundary triangles
should not have obtuse angles opp osite to any b oundary edge or interface
edge
The generation of quality nonobtuse b oundary Delaunay triangulation s
consists of two steps The construction of a go o d quality constrained De
launay triangulation CDT of the p olygon having interior angles comprised
b etween and A p ostpro cess step which eliminates b oundary
obtuse triangles by combining a longest edge pro cedure for selecting p oints
the Delaunay algorithm for inserting the p oints and a nite numb er of some
sp eci c Delaunay p oint insertions for b oundary obtuse triangles with edge
b oundary constrained acute angles angles de ned by two b oundary edges
The construction of the go o d quality constrained Delaunay triangu
lation consists of a The generation of an initial constrained Delaunay
triangulation essentially using the p olygon vertices and b the use of the
LEPP Delaunay algorithm describ ed in Figure section with
which pro duces a mesh with angles b etween and
The step designed to eliminate b oundary obtuse triangles of p olygonal
regions considers two cases a triangles with only one b oundary edge opp o
site to the obtuse angle edge b oundary obtuse triangles and b triangles
with two b oundary edges and one of them opp osite to the obtuse angle
edge b oundary obtuse triangles
A detailed description of the algorithm and the pro of of the theorems of
12
this section can b e found in Hitschfeld et al
Elimination of edge b oundary obtuse triangles
Each edge b oundary obtuse triangle is simply eliminated by Delaunay in
sertion of the midp oint of the b oundary edge Since the obtuse angle is
smaller than or equal to the insertion of only one p oint is required see
Theorem even when some some diagonal swappings might b e necessary
Theorem Let be any improved Delaunay triangulation of any polygonal
geometry with smal lest angle greater than or equal to Let consider
either an edge boundary obtuse triangle or two edge interface triangles
being at least one of them an obtuse triangle as shown in Figures b and c
one or two edge interface obtuse triangles In any of both cases let con
sider e the unique boundary or interface edge involved Then a the obtuse
triangle is eliminated by Delaunay insertion of the midpoint of e and b the
new generated boundary triangles are nonobtuse triangles
The three p ossible cases considered by Theorem are illustrated by Fig
ure Figure a shows a edge b oundary obtuse triangle Figure b two
edge interface triangles where one of the angles opp osite to the interface
edge is obtuse and Figure c shows two edge interface triangles where
b oth angles opp osite to the interface edge are obtuse The Delaunay inser
tion of the interface edge midp oint in case c destroys the two obtuse angles
not generating new obtuse angles opp osite to interface edges
(a) (b) (c)
Figure Cases of edge b oundary and interface obtuse triangles
Corollary For any improved Delaunay triangulation of any polygonal geom
etry the number of point insertions V to eliminate N edge boundary ob
1 1b
N
1i
tuse triangle and N interface obtuse triangles is N V N N
1i 1b 1 1b 1i
2
N
1i
The insertion of N is p ossible when the improved triangulatio n
1b
2
has an o dd numb er of interface obtuse triangles and all of them corresp ond
to the case c of Figure
Elimination of edge b oundary obtuse triangles
The elimination of edge b oundary obtuse triangles requires a sp ecial treat
ment when the smallest edge is an interior edge Note that in such a case the
b oundary constrained angle is the smallest angle of the triangle is the
angle de ned by the two b oundary edges The strategy used for the edge
b oundary obtuse triangles can not b e applied in this particular case since
after two applications of the strategy a new triangle similar to the original
one will b e obtained t is similar to t in Figure One additional problem
o 4
is that due to the b oundary restrictions the minimum angle of this triangle
can b e less than and consequently the obtuse angle can b e greater than
C C C
M2 t t t t3 t β o α 2 1 t 4 1
BA BM1 A BM1 A
Figure t is similar to t
o 4
The essential ideas of the algorithm to handle case illustrated in Figure
are the followings An edge b oundary isosceles triangle of largest edges
equal to half the smallest b oundary edge of the target triangle is constructed
triangle BMN in Figure b Since the p oints M and N are inserted using
the Delaunay criterion this construction maintains the Delaunay prop erty
of the mesh but can pro duce an edge b oundary obtuse triangle t which is
1
in turn eliminated by the Delaunay insertion of the midp oint of the largest
edge of t Figure c This pro cedure can again pro duce a new b oundary
1
obtuse triangle t with largest angle smaller than the previous one and so on
1
The b oundary obtuse triangles are eliminated after the Delaunay insertion
of a nite numb er of p oints
Theorem Let t be a edge boundary obtuse triangle with interior smal lest
edge Then If the constrained angle is greater than with
0 0
the obtuse angle is eliminated by Delaunay insertion of exactly two points ob
tained by creating a edge boundary isosceles triangle If the constrained
angle is less than or equal to a new edge boundary obtuse triangle can
0
be generated which is eliminated by Delaunay insertion of a nite number of
C M
0
i points bounded by i see Figure
M N
0 C C
Mo β α t 1 BA B N A (a) (b) C C
t 1 B A BA
(c) (d)
Figure Elimination of edge b oundary obtuse triangles
Note that the elimination of edge b oundary obtuse triangles with small
est interior edge might intro duce triangles with angles less than The
numb er of involved triangles dep ends on the numb er of p oint insertions and
on the numb er of diagonal swapping made to eliminate the edge b oundary
obtuse triangles
Figure illustrates the more complex case arising when interfaces with
several interface edges converge to a common vertex A In this case we
eliminate the obtuse angles by Delaunay insertion of the midp oint M of the
smallest interface edge and a p oint N on each edge j so that the distance
j
b etween N and A is equal to the distance b etween M and A Thus isosceles
j
triangles are generated around A
B4
B N 4 3 t 3 N 3 t A 2 t N 1 2 B2
N 1
B1
Figure Adjacent b oundary obtuse triangles
Prop osition The number of point insertions to eliminate N boundary
obtuse angles is O N
Corollary The nal triangulation obtained having nonobtuse boundary
and interfaces triangles is a Delaunay triangulation
Examples
Figure a shows a strip geometry with an interior interface edge As
exp ected the numb er of inserted p oints to destroy the b oundary and interface
obtuse angles is less than or equal to the numb er of b oundary obtuse triangles
as shown in Table
a b
c d
Figure Example
Example
Delaunay LEPP Del Final mesh
vertices
triangles
min angle
aver min angle
max angle
aver max angle
b obtuse triangles
Table Statistical information for the example Figure
Figure a shows a two circles p olygon with interior interface edges
The numb er of inserted p oints to eliminate b oundary and interface obtuse
triangles shown in Table is in complete agreement with the theory After
the elimination of edge b oundary obtuse triangles a small numb er of in
terior triangles with minimum angle less than might app ear This is the
case of this example where triangles with minimum angle less than
still remain in the nal mesh obtained after the elimination of the b oundary
or interface obtuse triangles These bad quality triangles could b e easily
eliminated by using the LEPP Delaunay algorithm once more It should b e
p ointed out here however that some small geometry constrained angles that
can not b e improved due to the geometry constraints of the problem also
app ear in the nal mesh
a b
c d
Figure Example
Example
Delaunay LEPP Del Final mesh
vertices
triangles
min angle
aver min angle
max angle
aver max angle
b obtuse triangles
Table Statistical information for the example Figure
Terrain mo deling application
In this section we consider the application of a D surface LEPP Delaunay
algorithm for improving a D surface triangulation combined with a sim
pli cation algorithm for terrain mo dels in order to obtain go o d quality tri
angular meshes from digital elevation mo dels
A terrain is the graph of an scalar function of two variables The function
gives the elevation of each p oint in the domain Terrain mo dels are widely
used in visualization and computer graphics applications such as geographic
information systems ight simulators and video games
The most common source of terrain elevation data is the digital eleva
tion mo del DEM which is basically a two dimensional oating p oint height
array Several alternative representations have b een prop osed including con
tour lines quad trees and triangular irregular networks TIN
Triangulatio ns stand out as b eing one of the most convenient formats
for rendering and other geometric manipulation op erations The automatic
generation of TINs from DEM mo dels is an imp ortant research area and is
the main topic of this section
The D surface LEPP Delaunay algorithm
In this context the sphere criterion is used in the Delaunay algorithm the
sphere criterion for data dep endent Delaunay triangulatio ns states that a
triangulation T of a set of p oints P is Delaunay if and only if no p oint of P
is interior to the circumsphere of any triangle of T It means that most of
13
the three dimensional triangles are nearly equiangular
If the diagonal of any quadrilateral formed by two adjacent triangles of a
Delaunay triangulation T is replaced by the opp osite diagonal then p oints
of T will b e interior to the circumsphere of these adjacent triangles If the
application of this criterion to each edge of an arbitrary triangulatio n T do es
not cause any edge swapping ie every edge is lo cally optimal then T is a
data dep endent Delaunay triangulation
The LEPP concept of the De nition is easily extended to D surface
3
triangulations by using the Euclidean distance in R
However since the D surface LEPP Delaunay algorithm do es not work
prop erly for triangles lo cated in areas with high lo cal curvatures a new
geometric strategy has b een develop ed to handle this problem n 1 n 2 θ t1
t2
Figure Lo cal curvature angle used for three dimensional triangulations
The lo cal curvature angle of a pair of adjacent triangles t and t Fig
1 2
ure is de ned as arccos n n where n is the normal to the triangle
1 2 i
t We de ne constrained edges e as all those edges lo cated at the b oundary
i c
of the mesh or whose lo cal curvature angle is greater than or equal to
c
a threshold These edges will not b e swapp ed by the data dep endent
max
Delaunay algorithm
Triangles with minimum angle formed by two constrained edges will
not b e improved the data dep endent Delaunay triangulation algorithm will
never swap constrained edges and consequently these minimum angles will
not b e removed from the triangulation
Finally by combining all the previous techniques of this section we can
formulate a simple and e ective algorithm to improve the geometric quality
of terrain triangulatio ns which guarantees the construction of go o d quality
triangulations The algorithm dep ends b oth on a threshold angle used
max
to decide which edges are constrained having value smaller than or equal to
0
and on a threshold angle used to sp ecify the minimum angle of
min
all the triangles in the nal triangulation
Terrain simpli cation
Simpli cation of terrain surfaces is an imp ortant research area in computer
graphics and geometric mo deling The terrain simpli cation algorithm is
14
based on the combination of a greedy p oint insertion and the extended
LEPP Delaunay algorithm for terrain meshes The algorithm starts with a
simple triangulation of the domain and on each step applies the re nement
algorithm over the worst triangles in the triangulatio n to reduce the inter
p olation error in the current approximation of the digital elevation mo del
Whenever a new p oint is inserted in the triangulation by the LEPP Delaunay
algorithm its elevation is adjusted to t the underlying digital elevation
mo del
Empirical results
The simpli cation LEPP Delaunay algorithm for terrain surfaces pro duces
go o d quality approximations including go o d quality triangles from DEM
mo dels
Figure illustrates di erent triangulations generated by the LEPP sim
pli cation algorithm from a digital elevation mo del of x p oints The
triangulations obtained include and vertices resp ectively with
maximal vertical error of and meters resp ectively with
the error de ned as follows
E r r or max H x y H x y
(x y )
where H x y is the height value corresp onding to DEM mo del and H x y
is the height value corresp onding to the triangular approximation
Figure Triangulations generated by the simpli cation algorithm from a
digital elevation mo del
Figure Triangular irregular networks generated by the LEPP simpli ca
tion algorithm from digital elevation mo dels of terrain surfaces
Acknowledgements
Some of the results presented in this pap er were supp orted by Fondap AN
and Fondecyt
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Device Modeling PhD thesis ETH Zuric h published by Hartung
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Chile June
M S Shepard H L Cougny Surface meshing using vertex insertion
Scienti c Computation Research Center Rensselaer Polytechnic Insti
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P Heckb ert M Garland Fast p olygonal approximation of terrains and
height elds School of Computer Science Carnegie Mel lon University
Pittsburg PA