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1 Automatic Grid Generation with Almost regular Delaunay Tetrahedra Alexander Fuchs Department of Mathematics A University of Stuttgart Pfaenwaldring 57, D Stuttgart, Germany fuchs@mathematik.uni-stuttgart.de Abstract. An algorithm for constructing almost regular triangulations (ARTs) for three-dimensional polygonal domains is described. Ideally such triangulations consist entirely of congruent tetrahedra with nearly equal edges. The new feature of this method is that the position of the vertices is adjusted, before any connecting edges are assigned. This leads to grids with very few irregular vertices, i.e. most interior vertices are shared by 24 tetrahedra as for regular tessellations of R 3 with congruent tetrahedra. keywords. delaunay tetrahedra, tessellation of R 3, almost regularity Introduction Grid generation algorithms are essential tools in many areas of modern applied mathematics. Typical examples are nite element methods (FEM, [2]) and geometric modelling (CAGD, [4]). In both applications the quality of the grids strongly inuences the accuracy of numerical computations. For example, irregularly shaped tetrahedra result in large condition numbers of the stiness matrices for Galerkin methods. Similarly, spline approximation on irregular partitions requires non standard smoothness conditions [8, 5, 4, 5]. Ideally, a triangulation should consist entirely of equilateral tetrahedra. Unfortunately, this is not possible. However, as will be shown in section 2, there exist congruent tetrahedra having nearly equal edges. These tessellations have a regular lattice structure, i.e. every vertex has 4 neighbours and is shared by 24 tetrahedra. While such regularity can seldom be achieved it is natural to insist on \almost regularity". An almost regular triangulation (ART) should have only few irregular vertices, i.e. most of the interior vertices are shared by exactly 24 tetrahedra. Such combinatorial optimization also leads to good geometric properties. Clearly for an ART smoothing of the vertices usually results in very small deviations of the angles from the optimal value. Figure shows an example. The triangulation consists of 860 vertices and has just 83 irregular vertices. The percentage of angles less than 2:0 is lower than 0:2% and the maximum angle is 68:2. Ohand the construction of ARTs seems rather dicult since the combinatorial requirements may result in global constraints on the tetrahedra. We therefore employ an indirect approach, and extend our 2D-algorithm (see [0]) to 3D. We rst construct a set of inner vertices according to a density function %. This function is dened over the entire domain and describes the edge-length of the triangulation. In the second step we optimize the position of the vertices without assigning any connecting edges. Then, we form a Delaunay triangulation. More precisely, our algorithm for constructing an ART of a polygonal domain consists of three steps: (ART) construct an initial conguration and scale the density function, (ART2) adjust the position of the vertices by minimizing a penalty functional, (ART3) form a Delaunay triangulation. As usual, the density function controls the size of the tetrahedra which may vary throughout the domain. We do not discuss its construction here, which may depend on the geometry as well as on the local error of nite element calculations. The functional in the second step is minimal for regular triangulations. The optimization will therefore yield an almost regular conguration of the vertices. After these preprocessing steps the Delaunay triangulation gives excellent results, although the geometric advantages of two-dimensional Delaunay ([9],[7]) do not hold for three dimensions. supported by SFB 404, University of Stuttgart

2 Figure : Example of an ART In sections 3{5 the steps of the ART-Algorithm and the underlying theory will be described in detail. In section 6 we discuss a number of examples illustrating the performance of our method. We conclude with some remarks on possible extensions. 2 Regular Tessellations with Tetrahedra A regular triangulation is a partition of R 3 into congruent tetrahedra T i, so that their intersection is either empty or a vertex, edge or face. Unlike in two dimensions, there exists no canonical regular triangulation since one cannot partition R 3 with equilateral tetrahedra. Of course for applications in nite element methods and geometric modelling, the tetrahedra should be nearly equilateral. Another natural requirement is invariance under subdivision. If a tetrahedron T is split into 8 tetrahedra by halving the edges as is indicated in Figure 2, the small tetrahedra should by congruent to T scaled by the factor. Subdivision invariance is particularly important for grid renement. 2 Moreover, it is easy to show that a subdivision invariant tetrahedron generates a regular triangulation of R 3. One should note, however, that neither an equilateral tetrahedron nor the one produced by the tessellation of the standard cube (corners (0,0,0),(,0,0), (,,0) and (,,)) are subdivision invariant. p 4 q 6 q 4 q 5 p 3 T : p ; q ; q 3; q 4 T 2 : p 2; q ; q 2; q 5 T 3 : p 3; q 2; q 3; q 6 T 4 : p 4; q 4; q 5; q 6 T 5 : q ; q 2; q 4; q 5 T 6 : q ; q 2; q 4; q 3 T 7 : q 2; q 4; q 6; q 5 T 8 : q 2; q 4; q 6; q 3 q 3 q 2 p q p 2 Figure 2: Subdivision of a tetrahedron T.

3 With the following theorem we characterize all subdivision invariant tetrahedra. Theorem A tetrahedron in standard coordinates with vertices p = 0; p 2 = v = 0 0 A ; p3 = v 2 = x y 0 0 A and p4 = v 3 x2 y 2 z having x ; x 2; y 2 0 and y ; z > 0 (cf. Figure 2), is subdivision invariant, i one of the following 4 conditions is fullled: A ; Condition : 0 x < 2 x 2 = 2x y 2 =? x 2 y 2 ( + x)(? 2x)2 2 =? x Condition 2: y 2 > 0 z 2 ( + x)(? 2x) =? x x = 3 y = 2y 2 x 2 = 2 3 z = p 3y 2 Condition 3: 0 x < 2 x 2 =? x y 2 =? x 2 y 2 2 = x2 ( + x )? x Condition 4: 0 < x < 3 z 2 ( + x)(? 2x) =? x x 2 = 2 (x + ) y2 = x (3? x ) y2 2 = x(3? x) 4 z2 = (3? x) 4 Proof. It is easy to show that tetrahedra T, T 2, T 3 and T 4 are congruent to T. In the interior of T there are 2 2 dierent tetrahedra which have to be mapped to the tetrahedron T by suitable Euclidean movements. This results in the conditions to 4. The explicit computation of the congruence mappings is straightforward but rather tedious. Since the arguments are not crucial for the subsequent development of the grid generation algorithm we refer to [9] v v 2 v 3 Figure 3: Tetrahedra forming a parallelepiped. Figure 4: Tetrahedron T. Any of the tetrahedra dened by Theorem gives rise to a regular triangulation. Analyzing the structure of such triangulations in more detail, we rst notice that 6 tetrahedra form a parallelepiped as shown in Figure 3. The labels at the vertices in Figure 3 refer to the number of tetrahedra sharing this vertex. Moreover, we see that 24 tetrahedra meet in a regular vertex of a partition with parallelepipeds and each regular vertex has 4 neighbours. As a prototype for our grid generation algorithm we choose a subdivision invariant tetrahedron which diers from an equilateral tetrahedron as little as possible. Figure 5 shows the standard deviation of the dihedral angles of the tetrahedra satisfying the conditions of Theorem as a function of the free parameters. We select a tetrahedron T satisfying condition with x = =3 (see Figure 5).

4 Condition x x Condition Condition x Figure 5: Standard deviation of dihedral angles T has 3 congruent faces with angles 54:74 ; 54:74 ; 70:52 and the standard deviation to 60 is 7:44. The lengths of the edges of the tetrahedron T are ; ; ; ; 2= p 3 :5; 2= p 3 with a standard deviation of (2? p 3)=3 0:089 from the equilateral case, and the dihedral angles are 60 ; 60 ; 60 ; 60 ; 90 ; 90, having a standard deviation of 4:5 from 70:53, the dihedral angle for the equilateral tetrahedron. 3 Construction of an initial Conguration We can obtain an initial conguration with the following simple algorithm. We choose an approximate centre C 0 of the domain and construct the polyhedron formed by the 24 tetrahedra sharing C 0 (see Figure 6). Now we scale the polyhedron, until it covers the entire domain. Then we divide the tetrahedra which, according to the density function, are too large and repeat this, until all tetrahedra are small enough. For a constant density function, we get a fully regular triangulation. Finally we project points outside the domain, which have connection to an inner point, on the boundary of. All points within and on the boundary of the domain are then used as an initial set of vertices. At this point the exact position of the points on the boundary is not crucial, since no connecting edges are assigned in the subsequent optimization procedure. The number of vertices obtained in this way will, in general, not exactly match the density function. Since the density function plays an important role in the following optimization process (cf. Section 4), we have to rescale the density

5 Figure 6: Polyhedron, formed by 24 tetrahedra (solid and upper half transparent) function %! s % (s ) with an appropriate scaling factor s, in order to match the number of the constructed vertices best. Therefore, we have to analyse the relation between the density function and the number of vertices. The polygonal domain to be triangulated can be multiply connected and is specied by the edges and vertices making up the boundary polygons. For example the domain in Figure 7 has 2 boundary polyhedrons with a total of 24 edges and 6 vertices. Figure 7: Polygonal Domain To estimate the number of interior vertices X j, j = ; : : : ; n i in terms of the density function %, we assume that the triangulation is regular, i.e. that 24 tetrahedra share an interior and 2 tetrahedra share a at boundary vertex. The number of tetrahedra sharing a vertex on a boundary edge varies as can be seen in Figure 3. But in general all \types" of edge vertices occur with the same frequency, so we assume that 6 tetrahedra share an edge vertex. Denoting by n i, n b and n e the number of interior, boundary and edge vertices of the triangulation, it follows that the total number of tetrahedra is n t = (24n i + 2n b + 6n e)=4: () From Figure 3 it can be seen that 6 triangles share a boundary vertex and an edge vertex. It follows that the number of triangles on the boundary of the domain is n f = (6n b + 6n e)=3: (2) Hence we obtain for the number of vertices n = n i + n b + n e the formula n = 6 nt + 4 n f + ne: (3) 4

6 Now we relate n t and n f to the density function %, dropping our assumption that the triangulation is regular. The length of the edges should be approximately equal to the value of % at their midpoints. If we denote by V 0 the volume of the tetrahedron T (cf. Section 2), a regular triangulation with constant density function yields n t V 0 % 3 = vol(); and we can compute n t in terms of the volume of the domain. For non constant % we use the approximation Z % 3 X vol() % 3 X V 0 = V 0n t; (4) where the sums are taken over all tetrahedra of the triangulation and % denotes the average value of % on. In order to relate n f to the density function, we note that there may exist boundary triangles with dierent areas, depending on the tetrahedron T chosen (for our choice all boundary triangles have the same area). Since in general all triangles should occur with roughly the same frequency we have % 2 F0n f ; (5) where F 0 is the average area of the boundary triangles. Similarly we estimate the number of edge vertices by denoting by E 0 the average edge length of the tetrahedron T. E0ne; (6) % Combining equations (3), (4), (5) and (6) we obtain for the number of vertices Z Z Z n 6V 0 % + 3 4F 0 % + 2 4E 0 % : (7) We can now use equation (7) to scale the density function, %! s %, so that equation (7) is maintained for the number of vertices constructed with our algorithm. This estimate was derived for regular triangulations only. However, it remains fairly accurate for an ART, in particular when the number of irregular vertices is small. In any case, a slight deviation from the optimal number of points has no major impact on the quality of the triangulation as will become apparent from the next section. Merely the length of the edges will not correspond exactly to the density and the triangulation might be slightly distorted at 4 Adjusting the Vertices Adjusting the vertices to improve the geometric characteristics of a triangulation is a standard procedure [, 6]. Unlike to smoothing in previous work (cf. e.g. [7, 8, 6]), the new feature of our approach is to we optimize the position of the vertices, before assigning any connections to them. There associated Delaunay triangulation should be almost regular. This can be achieved by minimizing a functional f which penalizes congurations producing irregular vertices. The choice of our functional is based on the following conjecture. A vector H is a permutation of D (H; D 2 R n ) if the equations (h + + h n) = = (d + + d n) = hold. (h h 2 n) =2 = (d d 2 n) =2. (8) (h n + + h n n) =n = (d n + + d n n) =n A proof for this assumption has yet not been found, but we can prove the following stronger theorem.

7 Theorem 2 A vector H is a permutation of a vector D i the equations hold. (h + + h n) = = (d + + d n) = (h h 2 n) =2 = (d d 2 n) =2. (9) (h 2n + + h 2n n ) =(2n) = (d 2n + + d 2n n ) =(2n) To proof this, we need the following lemma. Lemma Equations (9) imply for every polynom p(x) of degree 2n p(h ) + p(h 2) + + p(h n) = p(d ) + p(d 2) + + p(d n) : (0) Proof. Let p(x) = a 2nx 2n + a 2n?x 2n? + + a 0 be the monomial representation of the polynom, then the following holds p(h ) + + p(h n) and thus (0) holds. = a 2n(h 2n + + h 2n n ) + a n?(h 2n? + + h 2n? n ) + + n a 0 = a 2n (h 2n + + h 2n n ) =(2n) 2n + an? (h 2n? + + h 2n? n ) =(2n?) 2n? + + n a0 (because of (9)) = a 2n (d 2n + + d 2n n ) =(2n) 2n + an? (d 2n? + + d 2n? n ) =(2n?) 2n? + + n a0 = a 2n(d 2n + + d 2n n ) + a n?(d 2n? + + d 2n? n ) + + n a 0 = p(d ) + + p(d n) Having this lemma, the proof of Theorem 2 is quite simple. Proof of Theorem 2. \)" : If H is a permutation of D, then the equations (9) are trivially fullled. \(" : (complete induction) For n = we have h = d, and thus H a permutation of D. Now use Lemma with the polynom p(x) := (x? d ) 2 (x? d 2) 2 (x? d n) 2 : Then for a solution of equations (9) we have p(h ) + + p(h n) = p(d ) + + p(d n) = 0 : Because p(x) 0 it has to be p(h i) = 0 and thus 8h i9d j : h i = d j. Without loss of generality let now be h n = d n, then we can reduce the dimension of (9) by one. Now we can dene a functional, which penalizes the error of equations (8). We denote by d d 2 the distances of the vertices of a regular triangulation to a xed vertex. For an almost regular triangulation, the corresponding distances kx i? X k k; k = ; 2; : : : ; n should be close to a permutation of D. This means that 0 =j 0 B kx i? X k C B C k j? n k= k6=i n X k= k6=i d j ka=j is small for each i and j. This motivates the following denition of our penalty functional f:

8 f(x ; ; X n) := nx i= j= 00 mx n k= k6=i '(h i;k C ) j A =j? 0 n k= k6=i '(d k C ) j A =j2 C A ; h i;k := kx i? X k k (%(X i) + %(X k ))=2 : We have replaced kx i? X k k by the normalized distance h i;k which is a good approximation to the corresponding component of D for neighbouring points since (%(X i) + %(X j))=2 describes the average side length; h i;k is approximately equal to the number of tetrahedra between the i-th and k-th vertex. For a constant density %, which results in an uniformal spacing of the vertices, this replacement is just a normalization of the distances. If we have a non constant density function, which results in a non-uniformal spacing of the vertices, then h i;k normalizes the distances of neighbouring vertices in the lattice. Of course, small distances are most relevant. Therefore, we introduce a cut-o function ' which vanishes for arguments larger than d l. d l+ d l d l d l+ d l d l+ Figure 8: Obvious but poor choice of '. Figure 9: Good choice of '. The obvious choice of ' is shown in Figure 8. The drawback of this choice is, that we cannot distinguish a distance in the interval [d l d l+ ] and [d l 0]. Therefore we prefer another (monotone) choice of ' as shown in Figure 9. We notice that we only have to compute l + distances in vector D, because '(x) = 0 for x d l+. In practice it suces to take one layer of neighbouring vertices, so that l = 4. For our choice of tetrahedron T we have D 2 = [d ; ; d l+ ] = [ 4=3 4=3 4=3 4=3 4=3 4=3 8=3]. Because of the cut-o function '() most of the terms in equations (8) are zero. Therefore it makes sense not to take the sum over j = ; ; n and to introduce a cut-o parameter m, having l m n. In practice we get good results for m = 2 l. One notices that, because of the local character of the 'cut-o' function ', the sum over k will contain only few nonzero terms. Therefore, with an appropriate data structure, the function f can be computed in O(n) operations. One also has to take into account, that the vector D diers for vertices inside and at the boundary of, thus the vertices X i have to be split into 2 sets. Of course there is great deal of heuristic in our choice of the penalty functional f. Strictly speaking, our analysis is valid only for a regular grid which is compatible with the boundary. However, because of the local character of ', one can expect that f will be nearly minimal for ARTs if the density function is smooth. In fact, in all numerical tests (cf. Section 6) we have obtained very good results. The numerical implementation of the minimization procedure is straightforward from a mathematical point of view. We employ a conjugate gradient method, and project the \search-direction" for vertices on the boundary. The actual C++-Code ( lines) is rather complex. More generally than described in this paper, the algorithm can handle free an xed vertices on the boundary as well as internal constraints. A number of additional strategies are employed to improve the eciency of the program. To speed up the convergence of the iteration we can compute

9 rst a coarse triangulation and obtain improved start congurations via subdivision. Also, we can parallelize most of the computations. Figure 0 shows a few typical steps of the smoothing process. Start conguration Point conguration after one step of CG 5 steps of CG 0 steps of CG Figure 0: Smoothing Process 5 Triangulation A Delaunay tessellation can be obtained as dual graph of the Voronoi-diagram, which consists of the sets of closest points to a given set of vertices [9]. To determine the Delaunay tetrahedra we use the following criterion. Denition (Delaunay Tetrahedron) The vertices P,Q,R,S form a Delaunay tetrahedron, if there is no further vertex in its circumscribed ball. Methods for a sequential construction of the Delaunay tetrahedra sequentially are well known (cf. e.g. [9]). Since our vertices are constructed with respect to a density function, we can implement an ecient parallel algorithm. This algorithm will be described in detail in []. In order to keep the resulting triangulation conform to the boundary, we use a constrained Delaunay triangulation (cf. e.g. [3]). Of course, the placement of the edges is not essential on smooth parts of the boundary, where any polygon is just an approximation. For the example considered in the previous section, Figures and 2 show the resulting Delaunay triangulation. For complex domains or domains with a lot of interior faces we can construct a density function, which \recognizes" the local complexity of the domain, and adapts the density. This has already been done for two dimensions (cf. Figure 3) and will be described for three dimensions in a forthcoming paper.

10 Figure : View from above Figure 2: Cut through the domain 2d-net zoomed net Figure 3: Two dimensional net with interior edges and automatically adapted density 6 Examples In the following we discuss various examples, illustrating the performance of our algorithm. For evaluating the quality of the results we use the following criteria which compare the ARTs with regular triangulations. Irregular vertices By n irr we denote the number of interior vertices which do not have 4 neighbours. For an ART, n irr should be small compared to n i, the total number of interior vertices. Dihedral angles To emphasize the distribution of the dihedral angles, we give the minimum ( min) and maximum ( max) angle, as well as the percentage of angles lower than 6 ; 2 and bigger than 68 ; 74.

11 Condition of FEM matrices We denote by C the condition number of the stiness matrix for the Galerkin approximation of the Poisson problem using piecewise linear basis functions. This condition number is compared with the corresponding condition numbers C o and C i for regular triangulations of a subset of a bounding domain. More precisely, we choose for C o the tetrahedra which have an intersection with the domain (cf. Figure 4) and for C i those which lie fully inside the domain (cf. Figure 5). The edge length is chosen as the average edge length of the triangulation. Figure 4: Outbounding Net Figure 5: Inbounding Net Figure 6 shows the triangulation of a cube with a hole. Since for this example the percentage of boundary vertices is rather high, there is not much exibility to position the interior vertices. Hence, there are many irregular vertices. Nevertheless, the minimum and maximum angle is very good ( min = 7:6, max = 68:2 ). Also the condition number for the FE-matrix is comparable to the one for regular meshes. Vertices Angle % Dihedral angles Condition Total Inner Boundary Irreg. min max < 6 < 2 > 68 > 74 C C o C i :6 68: Figure 6: Cube with a Hole

12 Figure 7 shows the triangulation of a ball. Surprisingly the triangulation consists entirely of regular vertices. Vertices Angle % Dihedral angles Condition Total Inner Boundary Irreg. min max < 6 < 2 > 68 > 74 C C o C i :4 48: Figure 7: Triangulation of a ball with cross section In Figure 9 we show a constraint triangulation. Since for this case most of the interior vertices have to be positioned on the inner plane, the number of irregular vertices is rather high. Nevertheless, the minimum ( min = 6:9 ) and maximum ( max = 69:3 ) dihedral angle is acceptable and Figure 8 shows the distribution of the inner vertices and angles for this triangulation n inner vertices angles Figure 8: Distribution of inner vertices and angles.

13 Vertices Angle % Dihedral angles Condition Total Inner Boundary Irreg. min max < 6 < 2 > 68 > 74 C C o C i :9 69: Figure 9: Cube with an inner plane

14 The last example (Figure 20) shows a graded triangulation. Vertices Angle % Dihedral angles Condition Total Inner Boundary Irreg. min max < 6 < 2 > 68 > 74 C C o C i :9 74: Figure 20: Graded Triangulation 7 Conclusion We have presented a method for constructing almost regular triangulations (ARTs). The key idea is to adjust the vertices of Delaunay triangulations using a functional which penalizes irregular vertices. Our actual implementation of the triangulation algorithm is slightly more exible than described in this paper. For example, one may choose features like interior faces and one has a number of options to control the optimization process. For such details we refer to [2] and the online description of our program which is available via anonymous ftp from ftp.mathematik.uni-stuttgart.de or via WWW from A corresponding algorithm for 2-dimensional problems is described in [0]. An advantage of our method compared to many existing algorithms is, that the smoothing and optimization of the point conguration is done before any connections are assigned. This provides greater exibility, in particular when propagating irregularities of the boundary towards the interior of the domain. We intend to generalize our algorithm to NURBS solids and trimmed surfaces. Acknowledgements This work was performed at the University of Stuttgart. This work was supported by the SFB 404, University of Stuttgart. References [] Scott A. Canann, Micheal B. Stephenson, and Ted Blacker. Optismoothing: An optimization-driven approach to mesh smoothing. Finite Elements in Analysis and Design, 3:85{90, 993.

15 [2] Susanne C. Brenner and Scott L. Ridgway. The Mathematical Theory of Finite Element Methods. Springer- Verlag, 994. [3] H. Borouchaki, F. Hecht, E. Saltel and P.L. George. Reasonably Ecient Delaunay Based Mesh Generator in 3 Dimensions. In Proceedings, 4th International Meshing Roundtable, pages 3{4. Sandia National Laboratories, 995. [4] G. Farin. Geometric Modelling: Algorithms and new Trends. SIAM, 987. [5] G. Farin. Curves and Surfaces for Computer Aided Geometric Design. Academic Press Inc. (London) Ltd., edition, 988. [6] Lori A. Freitag, Mark Jones, and Paul Plassmann. An Ecient Parallel Algorithm for Mesh Smoothing. In Proceedings, 4th International Meshing Roundtable, pages 47{58. Sandia National Laboratories, 995. [7] Lori A. Freitag and Carl Ollivier-Gooch. A Comparison of Tetrahedral Mesh Improvement Techniques. In Proceedings, 5th International Meshing Roundtable, pages 87{00. Sandia National Laboratories, 996. [8] Lori A. Freitag. Tetrahedral Mesh Improvement Using Swapping and Smoothing. To appear in International Journal for Numerical Methods in Engineering. [9] A. Fuchs. Optimierung von Delaunay-Triangulierungen. Master's thesis, Universitat Stuttgart, 996. [0] A. Fuchs. Almost regular Delaunay-Triangulations. Int. J. for Numerical Methods in Engineering, 40:4595{460, 997. [] A. Fuchs. Parallel Delaunay triangulation. in preparation, University of Stuttgart, 998. [2] NAGD Research Group. Software. Report 96-5, University of Stuttgart, 996. [3] J. Horner. Condition Numbers of FE-Matrices. private communication. [4] K. Hollig and H. Mogerle. G-splines. Computer Aided Geometric Design, 7:97{207, 990. [5] J. Hoschek and D. Lasser. Grundlagen der Geometrischen Datenverarbeitung. B.G. Teubner, Stuttgart, 989. [6] N. A. Golias and T. D. Tsiboukis. An Approach to Rening Three-Dimensional Tetrahedral Meshes Based on Delaunay Transformations. Int. J. for Numerical Methods in Engineering, 37:793{82, 994. [7] D. T. Lee and B. J. Schachter. Two Algorithms for Constructing a Delaunay Triangulation. Int. J. Comput. Inf. Sci. 9, 9:29{242, 980. [8] D. Liu and J. Hoschek. GC Continuity Conditions between Adjacent Rectangular and Triangular Bezier Surface Patches. CAD, 2(4):94{200, May 989. [9] Franco P. Preparata and Michael Shamos. Computational Geometry: An Introduction. Texts and Monographs in Computer Science. Springer-Verlag, 985.