Computer Aided Geometric Design 7 (1990) 489-497

North-Holland

489

Minimal roughness property of the Delaunay triangulation

Samuel RIPPA

S&d Of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat Aviv 69978, Tel Aviv. Israel

Received November 1988

Revised October 1989

Abstract. A set of scattered data in the plane consists of function values measured on a set of data points in R2.

A surface model of this set may be obtained by triangulating the set of data points and constructing the

Piecewise Linear Interpolating Surface (PLIS) to the given function values. The PLIS is combined of planar

triangular facets with vertices at the data points. The roughness measure of a PLIS is the L2 norm squared of

the gradient of the piecewise linear surface, integrated over the triangulated region and obviously depends on

the specific triangulation. In this paper we prove that the Delaunay triangulation of the data points minimizes

the roughness measure of a PLIS, for any fixed set of function values. This Theorem connects for the first time,

as far as we know, the geometry of the Delaunay triangulation with the properties of the PLIS defined over it.

Keywords. Triangulations, Delaunay triangulation.

1. Introduction

Let D = {(xi, y,.. <)}y be a set of scattered data points V= { ui = (xi, y,)};’ consists of distinct and non-collinear points in R*. Suppose furthermore that H, is the convex hull of V. A common method for obtaining a surface model representation to D is to triangulate the set V

and to construct the Piecewise Linear Interpolating Surface (PLIS) defined over this triangula- tion. Before we proceed, let us define more precisely the terms triangulation and PLIS:

Definition 1.1. A set T = {q};’ of non-degenerate, open, triangles is a triangulation of H, if: - V is the set of all vertices of triangles in T.

- Every edge of a triangle in T contains only two points from V, namely its endpoints, l H,=UyL,T, TC7 7;=@, i#j.

Definition 1.2. Let D be a set of scattered data and T a triangulation of V. Define S:(T) to be the set of piecewise linear functions defined on a triangulation T of H,, i.e.

S:(T)= (gEC”(K): gI,=4)

where II, is the three-dimensional space of linear polynomials. The Piecewise Linear Interpolat-

ing Surface (PLIS) to a given (real) data vector F = (F,, . . . , F,) is the unique function fr E S,‘)(T) interpolating F, i.e.

fT(xjr y,) = Fj, i= l,..., N.

The roughness of the data vector F, relative to a triangulation T is defined by:

R(F, T> = I_& IT,, (1)

0167-8396/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

490 S. Rlppa / Delaunay triangulation

where 1 . IT,, is the Sobolev semi-norm defined as:

I& = I? I&J? i-l

l~r~,t=/,[(~)2+(~)2]dxdy.

An interesting question is to find, for a given a set of scattered data, a triangulation T * for which R( F, T) is minimal. The main result of this paper is: The Delaunay triangulation of H, is a minimal roughness triangulation for any set of scattered data.

The Delaunay triangulation is widely used, for various reason, within PLISes models (see

e.g. [Lee & Schachter ‘801, [Petrie & Kennie ‘871, [Watson & Philip ‘841) and is best described by using the Voroni diagram of the set V: The region V(i) which is the locus of points closer (in the Euclidian distance) to ui than to any other point of V is called the Voroni polygon

associated with the point ui. The Voroni polygons of all points in V define the Voroni tessellation of the entire plane. Two Voroni polygons V(i) and V(j) are called Voroni neighbors if they share the same edge. If no four points ui of Voroni neighbors are cocircular, then the dual graph of this tessellation, formed by connecting points ui and u, of each Voroni neighbors V(i) and V(j), is a triangulation of H, which is called the Delaunay triangulation. If four, or more, points ui of Voroni neighbors are cocircular, then the dual graph of the Voroni tessellation contains triangles and closed polygons having cocircular vertices. In that case any triangulation of these closed polygons result in a triangulation which is called a Delaunay triangulation.

The Delaunay triangulation was extensively studied in the last years. It has many nice geometric properties (see e.g. [Lawson ‘771, [Sibson ‘781) and efficient algorithms for its

construction exist (see e.g. [Cline & Renka ‘841, [Dwyer ‘871, [Lee & Schachter ‘SO]). In this paper the Delaunay triangulation is characterized, for the first time to our knowledge, in terms

of the PLIS which is defined over it, namely that the Delaunay triangulation is a minimal roughness triangulation.

2. The main theorem

Theorem 2.1. Let V be a set of N distinct and non-collinear points in Iw2 and let T * be a Delaunay triangulation of H,. Then

R(F, T*) Q R(F, T)

for euery triangulation T of H, and euety fixed data uector F = ( F,, . . . , FN).

The first step in the proof is to consider the local problem of triangulating a convex quadrilateral: Let T be a triangulation, e an internal edge of T and Q a strictly convex

Fig. 1. Switch diagonals according to the circle criterion.

S. Rippa / Delaunay triangulation 491

quadrilateral formed from the the two triangles in T having e as a common edge. Let also T’ be the triangulation created by replacing the edge e with the opposite diagonal e’ of Q.

Definition 2.1. The circle criterion replaces triangulation T by triangulation T’ if and only if the circle passing through the three vertices of any of the triangles (of T) forming Q contains the remaining point in its interior (see Fig. 1).

An edge e is called locally optimal if the application of the circle criterion to that edge leaves it unchanged. A triangulation is called locally optimal if all its edges are locally optimal.

Finally it is known (see [Lawson ‘771, [Sibson ‘781) that a triangulation T is a Delaunay triangulation if and only if it is locally optimal with respect to the circle criterion.

Lawson [Lawson ‘771 suggested the following Local Optimization Procedure (LOP) for constructing a locally optimal triangulation:

Algorithm 2.1. LOP 1. Construct an initial triangulation T(O) of H, and set T + T(O). 2. If T is locally optimal, end the procedure, else go to Step 3. 3. Let e be an internal edge of T which is not locally optimal and let Q be the strictly

convex quadrilateral formed from the two triangles in T having e as a common edge. (a) Swap diagonals of Q: Replace e by the other diagonal of Q, therefore transforming T

into a triangulation T’. (b) Set T + T’ and go to Step 2.

The above LOP produces, after finitely many edge swaps, a locally optimal, thus a Delaunay, triangulation T *. We intend to prove that T * is also a minimal roughness

triangulation. Suppose that II, lJ are two triangles in a triangulation T having e as a common edge and

that

Q=l&Uj, T,~ET

is a convex quadrilateral with vertices uk = (xi,, vi,), k = 1, 2, 3, 4, labeled in a counterclock- wise manner. Suppose, furthermore, that

I; = AV1V2VJ, 5 = AU,O,U,

and so T’ is obtained from T by replacing q, 3;. with T’, lJ’ where:

q.’ = AV,V2V4, Tj’ = A O,U,U, .

- Let u. = (x0, yo) be the intersection point of e = U,U~ and e’ = uzu4 (u. E Q since Q is strictly

convex). Finally define

rkl=d(uk, 0,) = ((xih-xi,)*+ (yi,-yi,)2)1’2, k, I=0 ,..., 4

and r, = ro, (see Fig. 2).

V2

w

Vl

V

V3

V4

Fig. 2. The quadrilateral Q.

492 S. Rippa / Delaunay triangulation

The following lemma is a reformulation of the circle criterion:

Lemma 2.1. The circle criterion replaces edge e by edge e’ if and only if r,r, c r,r,.

Proof. It is well known (see e.g [Jacobs ‘74, p. 4801) that u,, u2, uj, u, are cocircular if and only if r1r3 = r2r4. 0

In the next lemma the difference in the Sobolev semi-norm of the PLIS defined on triangulations T and T’ is computed.

Lemma 2.2. Let fr E S:(T) and frl E SF(T’) be the interpolators to the data vector F on triangulations T and T’ respectiuely. Then

I f?- $1 - I f?-• I:,., = A(r*r, - r2r4) (3)

where

A = (fr(xo* YO) -frP(xOv Yo)) 2 h+r3)(r2+r4) ,

( 2r,r,r,r, sin 13 i

B_Luuu

- 1 0 2. (4)

The proof of this lemma is quite involved and is thus left to Section 3.

Lemma 2.3. Let e be an edge in a triangulation T which is to be swapped by the circle criterion and let T’ denote the resulting triangulation. Then

R(F,T’)<R(F,T) foranyfixeddatauectorF=(F,,...,F,).

Proof. If the circle criterion decides to swap the edge e, then by Lemma 2.1 rlrs 7 r,r, and the

result follows from (3), the definition of R(F, T) and the fact that A > 0. •I

Corollary 2.1. Zf the points (xi*, yi,, 4,) E Iw3, k = 1, 2, 3, 4, are not coplanar, then the minimal

roughness criterion and the circle criterion are equiualent.

Proof. If the four points are not coplanar, then fr(xo, yo) # f,*(xo, yo). Thus

sign( I fr I& - l fr’ I&> = sign(rlr3 - r2r4)

and the result follows from Lemma 2.1. 0

Lemma 2.4. For any two Delaunay triangulations T’ and T2 and any fixed data vector F,

R(F, T’) = R(F, T2).

Proof. Any two Delaunay triangulations T’ and T2 differ only in the way closed polygons, with cocircular vertices, are triangulated. It is possible to transform triangulation T’ into triangulation T2 using only operations of swapping the diagonals of convex quadrilaterals within these polygons (see e.g. [Lawson ‘721). For any quadrilateral with cocircular vertices r,r, = r2r4 and thus, by Lemma 2.2, swapping the diagonals of such quadrilaterals can not

change the roughness measure. 0

Proof of Theorem 2.1. Let T’ be a minimal-roughness triangulation and suppose that there exists a Delaunay triangulation Td which is not a minimal-roughness triangulation, i.e.

R( F, Td) 7 R( F, T’). By applying LOP, with the circle criterion, it is possible to transform T’

S. Rippa / Delaunay rriangulation 493

into a Delaunay triangulation T *. From Lemma 2.3 R( F, T *) I R( F, T’), since application

of LOP with the circle criterion never increases the roughness measure. But from Lemma 2.4

R( F, Td) = R( F, T *) which contradicts the assumption that Td is not a minimal roughness triangulation. Thus any Delaunay triangulation is a minimal roughness triangulation for any fixed data vector F = (F,, . . . , FN). 0

Theorem 2.1 states that every Delaunay triangulation is a minimal roughness triangulation. The reverse is not true in general. For example the roughness of the constant data vector

F=(l, l,..., 1) is the same (zero) for every triangulation T. The following is a condition for a minimal roughness triangulation to be a Delaunay triangulation:

Corollary 2.2. Zf no four points from the data set D = { x,, y,, t;; };” are coplanar, then a triangulation T’ is a minimal-roughness triangulation if and only if it is a Delaunay triangulation.

Proof. Suppose that T’ is a minimal roughness triangulation. Then T’ is also a locally optimal triangulation according to the minimal roughness criterion. Since no four points in D are coplanar, then by Corollary 2.1, the circle criterion and the minimal roughness criterion are equivalent for any convex quadrilateral with vertices from V. Thus T’ is also locally optimal according to the circle criterion, i.e it is a Delaunay triangulation. On the other hand, from

Theorem 2.1, and Delaunay triangulation is a minimal roughness triangulation. Cl

Corollary 2.3. Zf no four points in V are cocircular and no four points from the data set

{Xi? Y;, I;r>: are coplanar, then the minimal-roughness triangulation is unique.

Proof. If no four points in V are cocircular, then Delaunay triangulation is unique and since no four points are coplanar, the minimal-roughness triangulation is that unique Delaunay triangu- lation.

3. Proof of Lemma 2.2

Consider, again, the quadrilateral Q of Fig. 2 and define the vectors g = (gl, g2, gs, g4, g,)T

and g’ = (gi, g2, gs, g4, 6)’ by

g, = g; = F. ‘k’ k = 1, 2, 3, 4

and

g, =frh Yo) = &g, + &3, (5)

d =f74% Y0) = $+2 + *g49 (6)

and let S be a triangulation of Q mto four triangles S, = AUaUjUj+i, j = 1, 2, 3, 4 (Fig. 2) where

us = ui. It is easy to verify that

I fT I& + I fT I& = I t-c, I& and I fTe I& + I fTp I&I = i&I

where g, and g; are the piecewise linear polynomials interpolating the data vectors g and g’ respectively. Thus Lemma 2.2 claims that:

I f, IA - I f** I:*.1 = ( I fr I& + I fr I&) - ( I f7-t I;,., + I fT* I$.,)

= I gs I& --l&l:., =A(r,r3-r2r,) with A given by (4). The proof of Lemma 2.2 consists of two additional lemmas.

494 S. Rippa / Delaunay triangularion

Lemma 3.1. Let h be a real, five-dimensional, vector and let hs be a piecewise linear function

defined on S such that

hs(vi) =hj, j= l,..., 4, h,(v,) =A,.

Then there exists a 5 X 5 matrix C = C(S) such that:

I hs I:., = (Ch, h)

where ( - , * ) is the usual scalar product in W 5.

Proof. Consider the triangle S, = Av,v,v, on which h, is a linear function (h,) 1 s, = Ho’ = (YX + p_y -t- y where the coefficients a, 8, y are determined from the interpolation conditions:

H”‘( vi) = h,, H’“( v2) = h,, H”‘( v,,) = h,.

It is easy to write the derivatives of H”) in cardinal form:

8H”’ - = a,h, + azh, + a,h,,

3X

3H’” - = b,h, + b,h, + b,h,

ay where

a, = (Y,, -yO)/21SI I, b, = (JG, - xi,)/2 Is* 0

al=(Yo_Yi,)/2l&I, b, = (xi, - x,))/2 I& I> (7) a,=(Yi,-yi2)/21S1 1, bS=(xi,-xi,)/2IS, I-

l S, l = fr,r2 sin 8, is the area of triangle S, and 8, = LV,V,V,. From (7) we can compute the Sobolev semi-norm of H”‘:

IH(1)l~,.,=i,[[~)‘+(~)2]dxdy=~~“)h, h) (8)

where E(‘) = ( Ei’j’))~,i,, is a matrix with elements:

E’?’ -_ ‘J

i

0 if i = 3,4 or j = 3,4,

I S, I ( aiaj -t- bibi) otherwise. (9)

An explicit expression for E$‘, E$j’, E,‘:’ is obtained by interpreting the elements E/j’, i #j, as scalar products and using the cosine law:

E$‘= -l&1(-a,a,-b5bl)= - rzr12 c4LvOv2vI)

4lSI I r/* + ri - rf r2* - r1r2 cos 8,

=- 8lS, I = - 2r,r2 sin 0, ’ (11)

E;;’ E -lS,l(-a,a,-bSb2)= - r1r12 cos(L wv2 1

41&I

rf2 + rf - r: r: - r,r, cos 0, =-

81 S, I = - 2rIr2 sin 8, ’ 02)

Similar expressions can be derived for the triangles S’, j = 2, 3, 4. Summing for all triangles Sj and using (8) we get:

I h, I:,, = i I H”’ If., = i (E”‘h, h)=(Ch, h), C= ; E”’ j-l j-1 j-l

and the proof of Lemma 3.1 is completed. 0

S. Rippo / Delaunay triangulation

From (5) and (6) it is easy to see that 4 4

(85 +gi) = C Gjgj, (85-g;) = C (-l)‘+‘+jgj*

j-1 j-l

495

(13)

'j+2 gBji- -

‘j+Q+2’ j= 1,...,4, r5 = r, , r6 = r2,

and using Lemma 3.1 we calculate:

= 2(g5 -IT;) t(g5 + gi>G5 + i Gjgj ( j-l

=2(g5-g;) i (Gj+ i+jc55)gj- (14) j-1

The next step is to represent the components of the sum in the right-hand side of (14) in terms of the distances r,, r2, rJ and r,.

Lemma 3.2

csj+ t4C~J=(-l)‘+‘4~(‘1~~-~2~4), j-1,2, 3,4

where

B ( r1 + 5c2 + r4)

= 2r,r2r3r4 sin 8 ’ 8 = Lu,u,u,.

Proof. Using (lo), (11) and (12) we compute:

C,, = i E;{) j-1

= &(iI +#..?I) =Bhr3+f2r4),

2 ‘i-1

2 ri+l

-+- 'I-l'i ‘i+1’i

and as usual the indices are taken in a cyclic manner, i.e. rs = r,, r, = r, and r,, = r,. Note that 8, = 8s = 8 and o2 = 8, = n - B (see Fig. 2) and thus X;_t cos ej = 0 and, also, sin 6’, = sin 8, j= 1, 2, 3, 4. Finally we get:

rlr3 + r2r4 $c,+fc,,=B-( 2 - 5+15+3 = $B(-l)‘+‘(r,r,-r2r4) J

and Lemma 3.2 is proved. 0

From (14), Lemma 3.2 and (13) we conclude that

1 fr I:., - i fr’ I:,., = I& I:,, - 1 d I:,, = B(g, - g;)‘(% - r2r4) = Ah3 - r2r4)

with A given by (4) and this completes the proof of Lemma 2.2.

496 S. Rippa / Delaunay triangulation

4. Conclusions and remarks

In this paper we have proved a property of the Delaunay triangulation for piecewise linear schemes defined over all triangulations of a fixed set of points, namely that the Delaunay triangulation is a minimal roughness triangulation. Several authors have proposed to use the

Delaunay triangulation for piecewise linear (and piecewise polynomial) schemes. The reason is the equiangular property of the Delaunay triangulation: it maximizes, over all possible triangulations T, the quantity

a(T) = p?=(smallest angle in T,).

This property is considered as ‘good’ for interpolation (see e.g. [Lee & Schachter ‘801, [Floriani et al. ‘851, [Watson & Philip ‘841) mainly because the interpolation error in early error bounds (see e.g. [Bramble & Zlamal ‘701) increases as the smallest angle in the triangle approaches zero. Improved estimates [Gregory ‘751 indicate that the interpolation error increases as the largest

angle in the triangle approaches 7, i.e. the smallest angle can be arbitrarily small as long as the largest angle is not too close to q. This refined error bound leads to the minmax triangulation criterion [Nielson & Franke ‘831: Choose a triangulation T * * minimizing the quantity

P(T) = Faxr(largest angle in T).

It was observed by numerical testing involving the generation of random data sets that the minmax triangulation T * * and the Delaunay triangulation of the same set of data points differ only slightly [Nielson & Franke ‘831. This observation and the fact that many efficient schemes for the construction of the Delaunay triangulation exist promote the use of the Delaunay triangulation within piecewise linear interpolation schemes.

In [Dyn et al. ‘901 the concept of data dependent triangulations, i.e. triangulations which depend on the data vector F, was introduced. It is shown, by various numerical testing, that the Delaunay triangulation is not always a good choice for use within linear interpolation schemes. In fact, it produces quite poor approximation to phenomenon (the function from which the data vector F was sampled), having a preferred direction, i.e. large second directional deriva- tives in a given direction, as compared to other directions. Based on Theorem 2.1, our conjecture is that the Delaunay triangulation is adequate for approximating functions which minimize the Sobolev semi-norm defined by (2), in an appropriate function space. Such are the harmonic functions satisfying the Laplace equation:

Ag=g,,+g,,=O

with appropriate boundary conditions.

Acknowledgments

I thank my advisors Nira Dyn and David Levin of the School of Mathematical Sciences, Tel-Aviv University for carefully reading and commenting upon the drafts of this paper. I would also like to thank the referees of this paper for their helpful suggestions.

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