UPDATING ALGORITHMS FOR CONSTRAINT DELAUNAY TIN

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1 UPDATING ALGORITMS FOR CONSTRAINT DELAUNAY TIN WU LiXin a,b WANG YanBing a, c a Institute for GIS/RS/GPS & Subsidence Engineering Research, China Uni Mining & Technology, Beijing, China, awulixin@263.net, awulixin@yahoo.com b Institute for GIS/RS/GPS & Digital Mine Research, Northeastern University, Shenyang, China, c Laboratory of 3D Information Acquisition & Alication, Caital Normal University, Beijing, China - cnuwyb@yahoo.com.cn, wybing@cumtb.edu.cn KEY WORDS: Udating Algorithm, Constraint Delaunay TIN, Integral Ear Elimination, Influence Domain Re-Triangulation, Polymorhism ABSTRACT: Present algorithms for D-TIN are far insufficient for the dynamic udating of CD-TIN in different alications. This aer, based on imrovement to resent insertion and deletion algorithms for D-TIN, a grou of algorithms for CD-TIN udating are resented. 1) According to the olymorhism of the constraints in CD-TIN, virtual oint is adoted to describe the cross between constraint edges in CD-TIN; 2) Imroved from the EE algorithm for D-TIN, an algorithm namely Integral Ear Elimination (IEE) is resented for oint deletion in CD-TIN; 3) Considering the olymorhism of constraint edges in CD-TIN, an algorithm namely Influence Domain Re-triangulation for Virtual oint (IDRVP) is also resented for deleting constraint lines. An examle is shown to test the algorithms. 1. INTRODUCTION Delaunay triangular irregular networks (D-TIN) and nonstraint delaunay triangular irregular networks (CD-TIN) are two basic concets in comutational geometry, and have many ractical alications in several fields including geograhic information systems (GIS), finite element methods (FEM), comuter grahics and 3D reconstruction (Aurenhammer, 1987; Mostafavi, et al. 2003; Zhu, 2000). The method to construct a D-TIN based on a set of given oints has been widely studied, and many construction algorithms for D-TIN were roosed, such as divide-and-conquer (Dwyer, 1987; Guibas, 1985), sweeline (Fortune, 1987), incremental (Prearata, 1985; Green, 1977; Guibas, 1992; Ohya, 1984), asymtotically otimal algorithm (Chew, 1987). Furthermore, the udating algorithms for D-TIN insertion and deletion are studied. Devillers (1999) resented ear elimination (EE) algorithm to delete single oint in 2D delaunay triangulations (DT). Mostafavi (2003) imroved the EE algorithm for deleting oint in D-TIN and Voronoi diagram. Marc (2002) resented a method for a set of oints to be dynamically udated in regular triangulations (RT). Li (2000) suggested delete some inside oints during cliing the boundary of a D-TIN. Giving a grou of endoints and some non-crossing edges to a D-TIN, a CD-TIN following the criterion of Visibility and Constraint Emty Circle is obtained (ao, 2003). The resented udating algorithms for inserting vertices and segments in CD-TIN are constraint grah algorithm (Lee, 1986), divide-and-conquer (Dwyer, 1987; Guibas, 1985), incremental (Prearata, 1985; Mostafavi, 2003) and shell triangulation (Piegl, 1993). owever, the udating of CD-TIN not only includes the insertion of contained oint and constraint edges, but also includes the deletion of constraint edges. Based on the imrovement and extension to resent algorithms, this aer roosed integral ear elimination (IEE) algorithm for oint deletion, and influence domain re-triangulation for virtual oint (IDRVP) algorithm for the insertion and deletion of constraint edges. 2 ALGORITM FOR CONSTRAINT EDGE INSERTION The insertion of constraint edges is a basic oeration for CD-TIN construction. Tsai and Vonderohe resented Convex ull algorithm that inserting constraint edges before Convex ull calculating and data dividing for D-TIN (Li, 2000). Marc (2000) imroved the asymtotically otimal algorithm for direction constraint CD-TIN construction, which recursive re-triangulated the influence domain of constraint segments. owever, the mentioned algorithms can be alied for the intersection of two intersected constraint edges. Virtual oint is introduced to solve the roblem. 2.1 The Introduction of Virtual Point If a new constraint segment A 1 intersects with A 2 in CD-TIN, the oint v of intersection would be added in A 1 and A 2 as a new oint for udating CD-TIN. The intersection oint v is called virtual oint, and it will be deleted while delete A 1 or A 2, i.e., virtual oint v is only an interolated inner oint of the constraint segments. The broken lines are comosed of several segments, and constraint olygons are comosed of constraint segments end to end. The insertion and deletion of broken lines and olygon can be conducted as segments insertion and deletion in order. 2.2 Influence Domain and CD-TIN Re-Triangulation Without the consideration for olymorhism, the influence domain re-triangulation resented by Marc (2000) for inserting constraint segments cannot be alied directly for the intersection of constraint. Based on the introduction of virtual oint, it is extended for constraint edge with olymorhism being considered. 135

2 Let, be the two endoints of constraint segment (, V), the rocedures for insertion into CD-TIN can be summarized as the following: 1) to insert constraint segment in original CD-TIN: all edges e i (i=1, 2, n) which are crossed by are saved in an edge stack. If an edge e i is a constraint segment, then e i is marked e' i and the virtual oint v i intersected between e' i and is inserted. 2) to delete all the non-constraint edges e i crossed by so that a olygon that we called Influence Domain of constraint edge without triangulation is left. Clearly, the influence domain contains some constraint edges intersected with. 3) to recursively re-triangulate the uer and lower influence domain (V Ui, V Di, i=1, 2, n) that is cut by constraint segment and e' i following constraint emty circle criterion. Figure 1 shows the insertion rocess of a constraint segment in original CD-TIN. For the intersection of constraint segment, and, the intersecting oint and are interolated into as virtual oints so that is comosed of three constraint segments, and. To search all the edges crossed by and to remove all the non-constraint edges to form the influence domain olygon, and then to re-triangulate the uer and lower olygons (V Ui, V Di, i=1, 2, 6) cut by, and. In each recursive re-triangulation, the emty circle criterion with resect to the bounding oints of the olygon is tested, and the constraint segments,,,,,, is visible. ence, anyone of the olygons is re-triangulated according to the constraint emty circle criterion. V U2 V U3 V D2 V D3 6 V U1 1 V D1 Figure 1. Virtual oint insertion and re-triangulation. original CD-TIN before inserting constrained edge. Influence domain of and the intersect oint between and other constrained edges. udated CD-TIN after inserting constrained edge. The interior of the influence domain V Ui and V Di are re-triangulated by using of recursive algorithm ( as in figure 2). As for the re-triangulation of V U1, the triangulation of V U1 must contain a triangle with constraint segment being its edges. The recursive algorithm searches a oint ' 0 in the bounding of V U1 that forms a triangle T (,, ' 0 ) with vertices,. The triangle meets the constraint emty circle criterion, that is to say, the circumcircle of T (,, ' 0 ) cannot contain any oint of V U1 excet for its three vertices. After inserting T (,, ' 0 ), the V U1 is divided into two sub-domains, F E (, ' 1, ' 0 ) and F D (' 0, ' i,, ). To aly the algorithm recursively to re-triangulate the sub-domains of F E and F D until the entire domain be triangulated. ' 1 F E ' 2 ' i ' i+1 ' 0 F D V U1 Figure 2. The reconstruct of influence domain 3. ALGORITM FOR CONSTRAINT POINT DELETION oints in CD-TIN must take the constraint edges into consideration. In other words, the oint to be deleted refers to both non-constraint oint and constraint oint. To delete a non-constraint oint, there may be constraint edges, which should be ket invariable during fliing edges, in the influence domain boundary of this oint. While to delete a constraint oint, the constraint segment that contains the oint would be deleted while the other endoint of this segment should ket be invariable. 3.1 Integral Ear Elimination (IEE) Algorithms The first rocess of non-constraint oint deletion can be shown in figure 3, where is a non-constraint oint and is a constraint segment. When deleting, is one edge of the influence domain boundary of, and the constraint roerty of must be ket in re-triangulating the influence domain. owever, the deletion of constraint oint is different as in figure 4, where the constraint segment must be deleted and kee oint invariable. Furthermore, a constraint condition is reduced because of the deletion of, and the diagonal fliing test has to be done according to constraint Delaunay criterion for. As comared to the oint deletion in D-TIN, the deletion of 136

3 Figure 3. Deletion of non-constraint oint 7 7 Figure 4. Deletion of constraint oint q 3.2 Integral Ear Elimination (IEE) Algorithm Integral ear elimination (IEE) algorithm must not only meet Delaunay criterion, but also kee toological comleteness. The rocedures are: 1) to search the influence domain of oint to be deleted, and to identify whether is constraint or not; 2) to select three oints in the boundary of as a otential triangle T so as to test if it is an ear or not. If T is an ear, to fli the diagonals of quadrilateral formed by T and and to delete T; if not, to return to the second ste. Note that in the ste of fliing diagonal, it is needed to test the ear and its adjoining triangles referring to constraint emty criterion if is constraint. Let CD-TIN grah be G = (V { }, A), to delete. The algorithm goes as in the following: 1) to identify and the other constraint oint ' connected with along the constraint segment (as in Figure 4) if oint to be deleted is a constraint oint, else go to next ste. 2) to search out all the oints {,, n-1, n } connected with to form the influence domain = {,, n-1, n = } of in anticlockwise order. 3) to check if T( i, i+1, i+2 ) is an ear. Each trile of oints of is considered as a otential ear T( i, i+1, i+2 ). If T meets any one of the following conditions: a) D( i, i+1, i+2 ) is negative (means that T forms a re-entrant, not an ear, of the olygon formed by the adjoining of ); b) D( i, i+2, ) is negative (means that T encloses ) and c) ( i, i+1, i+2, j ) is ositive (means that one or more remaining oints j of fall inside the circumcircle of T), T is not an ear and is rejected immediately. If T meets all of these conditions, T is an ear, and may be removed by switching the diagonals i+1 and i i+2 of quadrilateral formed by T and. 4) if the oint i+1 of T( i, i+1, i+2 ) equal to ', e.g., i+1 =' (means that i+1 ' is a constraint segment), it is needed to test whether T meets the constraint emty circle criterion. That is to say, if another oint q of the triangles adjoining to segment i i+1 or i+1 i+2 is in the circumcircle of T, it is needed to switch the diagonals of quadrilateral formed by T and q (as in figure 4). 5) to reeat ste) and ste) until only three adjoining oints in remained, in which case the three oints are merged into one triangle and is deleted. 4. ALGORITM FOR CONSTRAINT EDGE DELETION The deletion of constraint edge in CD-TIN is an inverse oeration of constraint edge insertion, i.e. to remove all the segments of the constraint edge one by one. If delete a constraint edge with a virtual oint, the deletion must meet the constraint emty circle criterion, and another constraint edge crossing the virtual oint cannot be deleted in considering the toological comleteness of CD-TIN. Therefore, the deletion of constraint edge falls into two kinds of situations that with virtual oint or without virtual oint. 4.1 Constraint Edge Deletion without Virtual Point If there is no virtual oint in constraint edge, the constraint oint of this edge can be deleted one by one by using of IEE algorithm (as in figure 5). Figure 5. To delete constraint edges without virtual oint delete constraint broken line ; delete constraint triangular T( ); the reconstructed CD-TIN 137

4 4.2 Constraint Edge Deletion with Virtual oint If there is virtual oint in the edge, another algorithm called influence domain re-triangulation for virtual oint (IDRVP) is resented. Given a CD-TIN grah G={V, A A'}, to delete a constraint edge C with virtual oint v. Firstly, the influence domain of v is searched out, which is a olygon of the boundary of all the triangles connected with v. Secondly, is divided into two arts, L and R, by the other constraint edge C i crossing v. Thirdly, L and R are re-triangulated individually referring to the constraint emty circle criterion. Finally, the virtual oint v is deleted. As shown in figure 6, is a constraint edge to be deleted, and are two oints where ' 1 ' 2 and ' 3 ' 4 intersected with. Edge is comosed of three constraint segments, and. When delete, and will be deleted too, but ' 1 ' 2 and ' 3 ' 4 are left in the CD-TIN. The detailed oerations are: 1) to delete the constraint oint and by using of IEE algorithm, and the constraint segments and will be deleted simultaneously (as in figure 6a); 2) to search the influence domain of and to divide it into two arts L and R by ' 1 ' 2, and then L and R are re-triangulated individually, and to delete (as in figure 6b); 3) to reeat ste) to delete. Finally, a new CD-TIN is constructed where the constraint edge is deleted (Figure 6c). ' 3 ' 1 ' 3 ' 1 ' 3 ' 1 ' 4 ' 2 ' 4 ' 2 ' 4 ' 2 Figure 6. To delete constraint edges with virtual oint the original CD-TIN and constraint edge with virtual oints and ; the reconstructed CD-TIN after deleting constraint oints, ; the final CD-TIN after deleting virtual oints and. The algorithm rocedures for the deletion of constraint edges are as in the following: 1) to search all the segments e i (i=1,2, n-1) of constraint edge C = { n }; 2) to identify the constraint oints i (i=1,2, n) and virtual oints ' j (j=1,2, m) and to store it into two oint lists L C and L V ; 3) to delete all the oints i (i=0,1, n) in L C by of using IEE algorithms; 4) to delete all the oints ' j (j=0,1, m) in L V by using of IDRVP algorithms. 5. A CASE FOR ALGORITMS TESTING A rototye system with VC++ is develoed to test roerties of the algorithms. Figure 7 shows the test results for shae reconstruction, where the dynamic deletion and insertion of constraint edges in CD-TIN is illustrated with the letters modification for RS to GIS. Shae of letter RS as constraint edge Delete R and insert GI as constraint edge Figure 7. The shae reconstruction with dynamic deletion and insertion of letters in a CD-TIN 6. CONCLUSION The CD-TIN is a widely used structure in GIS and comuter geometry, and the insertion and deletion oeration of oints and constraint edges are the basis of CD-TIN dynamic udating. By introducing virtual oint, this aer has made imrovements and extensions to resent udating algorithms for D-TIN and had reached a grou of new algorithms in consideration of olymorhism. The new algorithms include the integral ear elimination (IEE) algorithms for constraint oint deletion in CD-TIN and the influence domain re-triangulation for virtual oint (IDRVP) algorithms for constraint edge deletion in CD-TIN. The test shows that the algorithms are efficient and could be widely alied for the udating of DEM, DTM and other integral vector models of digital city and digital mine. 138

5 ACKNOWLEDGMENTS The author is grateful for the jointly suort of the National Outstanding Youth Funds of China (No ), the National Natural Science Funds of China (No ), National igh-tech Project (No.2006AA12Z216) and the Science Develoment Plan Awarded by Beijing Municial Commission of Education Science (No. KM ). REFERENCES Aurenhammer, F., 1987, Power diagrams: criterion algorithms and alications. SIAM Journal of Comuting, 16: Chew, L. P., 1987, Constraint Delaunay Triangulations. In: Proceedings of the Annual Symosium on Comutational Geometry ACM: Devillers, O., 1999, On deletion in Delaunay triangulations. In: 15th Annual ACM Symosium on Comutational Geometry: Dwyer, R. A., 1987, A fast divide-and-conquer algorithm for constructing delaunay triangulations. Algorithmica, 2: Fortune, S., 1987, A Sweeline Algorithm for Voronoi Diagrams. Algorithmica, 2: Green, E., R. Sibson, 1977, Comuting Dirichlet tessellations in the lane, Comut. J. 21: Guibas, L., D. Knuth, M. Sharir, 1992, Randomized incremental construction of Delaunay and Voronoi diagrams, Algorithmica. 7: Guibas, L., J. Stolfi, 1985, Primitives for the maniulation of general subdivisions and the comutation of Voronoi diagrams, ACM Trans. Grahics. 4 (2): ao,. S., Wu LiXin, 2003, 3D-Geosciences Modeling and Visualization Based on Strongly Constraint Delaunay TIN. Geograhy and Geo-Information Science(in Chinese), 19(2): Lee, D. T., Lin. K., 1986, Genaeralized Delaunay Triangulation for lanar grahs. Discrete Comutational Geometry. 1: Li, Z. L., Zhu Qing, 2000, Digital Elevation Model (in Chinese). Wuhan, Wuhan Surveying And Maing University Press. Marc, V., Nuria Pla., 2000, Comuting directional constraint Delaunay triangulations. Comuters & Grahics, 24: Marc, V., Núria Pla, Jose Cotrina., 2002, Regular triangulations of dynamic sets of oints. Comuter Aided Geometric Design, 19: Mostafavi, M. A., Christoher Gold, Maciej Dakowicz, 2003, Deletion and insert oerations in Voronoi/Delaunay methods and alications. Comuters & Geosciences, 23: Ohya, T., M. Iri and K. Murota, 1984, Imrovements of the incremental method for the Voronoi diagram with comutational comarison of various algorithms, J. Oer. Res. Soc. Jaan 27: Piegl, L. A., Arnaud M. Richard, 1993, Algorithm and Data Structure for Triangulation Multily Connected Polygonal Domains [J]. Comuter and Grahics, 17(5): Prearata, F. P., Shamos M. I., (editors), 1985, Comutational Geometry, An Introduction. (New Yerk: Sringer-Verlag). Zhu, X. X., 2000, Free curved lines and face modeling technology(in Chinese). Beijing, Science Press,