Abstract
The problem of representing 2 1/2 dimensional surfaces defined at a set of randomly located points by means of triangular grids is considered. Such representations approximate a surface as a network of planar, triangular faces with vertices at the data points. In the paper we describe different models and data structures for encoding triangular grids. Since Delaunay triangulation provides a common basis for many models of 2 1/2 D surfaces, we review its basic properties and we describe the most important approaches to its construction. Hierarchical surface models are also presented, which are based on nested triangulations of the surface domain and provide variable resolution surface representations. An algorithm is described for building a hierarchical description of a nested triangulation at different levels of abstraction. Finally, the 3D surface reconstruction problem is briefly discussed.
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Agiu J, Binford TO (1976) Representation and description of curved objects. IEEE Trans Comput C-25:439–449
Aho AV, Hopcroft JE, Ullman JD (1984) The design and analysis of computer algorithms. Addison-Wesley, Reading, Massachusetts
Akima H (1978) A method of bivariate interpolation and smooth surface fitting for values given at irregularly distributed points. ACM Trans Math Software 4(2):148–159
Ancona M, De Floriani L, Deogun JS (1986) Path problems in structured graphs. The Computer Journal 29(6):553–563
Babuska I, Aziz AK (1976) On the angie condition in the fimte element method. SIAM J Numer Anal 13(2):214–226
Ballard DH (1981) A hierarchical representation for curves Commun ACM 24(5):310–321
Ballard DH, Brown CM (1982) Computer vision Prentice Hall, Englewood Cliffs, NJ
Barnhill RE (1977) Representation and approximation of surfaces In: Rice JR (ed) Mathematical Software III. Academic Press. pp 69–120
Bartera R, Vazques AM (1984) A hierarchical method for representing terrain relief. Proc Pecora IX Symp Spatial Information Technologies for Remote Sensing Today and Tomorrow, Siour Falls. South Dakota, pp 87–92
Baumgardt MG (1972) Winged-edge polyhedron representation. Tech Rep CS 320. Stanford University
Bell SMM, Diaz BM, Holroyd F, Jackson MJ (1983) Spatially referenced methods of processing raster and vector data. Image and Vision Computing 1 (9):211–220
Bonssomat JD (1982) Representation of objects by triangulating points in 3-D space. Proc Sixth Internat Conf on Pattern Recognition, Munich, pp 830–832
Boissonat JD (1984). Geometric structures for three-dimensional shape representation. ACM Trans Graphics 3(4):266–286
Boissonat JD, Tellaud M (1986) A hierarchical representation of objects: the Delaunay tree. Proc Second ACM Symp on Computational Geometry. Yorktowa Heights, pp 260–268
Bowyer A (1984) Computing Dirichlet tessalations. The Computer Journal 27(2):165–171
Bramble JH, Zlarnal M (1970) Triangular eiements in the linite element method. Math Comput 24:809–820
Brassel KE, Ref D (1979) A procedare to generate Thiessen polygons. Geographical Analysis 11(3):289–303
Brown KQ (1979) Votonoi diagrams from convex hulls. Inf Proc Lett 9:223–228
Carlson WE (1982) An algorithin and data structure for 3D object synthesis using surface patch intersection. Computer Graphics 16(3):255–264
Cavedish, JC (1978) Automatic triangalation of arbitrary planar domaitts for the finite element method. Int J Numer Methods Eng 8:679–896
Chazelle BM (1982) A theorem on polygon cetting with application Proc 23rd IEEE Annuai Symp on the Foundations of Computer Science, pp 339–349
Chazelle BM, Incerpi J (1983) Triangulating a polygon by divide-and-conquer. Proc 21st Allerton Conf Commun Control Comput, pp 447–456
De Floriani L., Palcidieno B, Pienovi C (1982) Triangulated irregular networks in geographical data processing. In. Rinaldi S (ed) Environmental Systems Analysis and Management North-Holland, pp 801–811
De Floriani I., Falcicheno B, Nagy G, Pienovi C (1982) Yet another method for trangulation and contouring for automatral cartography. Proc American Congress on Surveying and Mapping, pp 101–110
De Floriani L, Falcidieno B, Pienovi C (1984) Graph representation of a hierarelncal surface model. Proc Seventh Int Conf on Pattern Recogninon, Montrical, pp 1093–1096
De Floriani L, Falcidieno B, Nagy G, Pienovi C (1984) A bierarchical structure for surface approxirnation Computer and Graphics 8(2):183–193
De Floriani L, Falcidieno B, Pienovi C (1985) A Delaunaybased representation of surfaces defined over arbitrarily-shaped domains. Computer Vision, Graphics and Image Processing 32:127–140
De Floriani L (1986) A hierarchical boundary model for variable resolation representation of three-dimensional objects. Proc Eight Int Conf on Pattern Recognition, Paris, 1986.
Devijver PA, Maybank S (1982) Computation of the Delaunay triangulation of a convex polygon with a minimum space complexity constraint. Proc Sixth Int Conf on Pattern Recognition, Munich, pp 420–422
Dobkin DP, Kirkpatrick DG (1983) Fast detection of polyhedral intersection. Theor Comput Sci 27:241–253
Dutton G (1983) Geodesic modeling of planetary relief. Proc Sixth Int Conf on Automated Cartography, pp 186–201
Edelsbrunner H, Kirkpatrick DG, Seidel R (1983) On the shape of a set of points in the plane. IEEE Trans Inf Theory 29(4):551–559
Faugeras OD, Ponce J (1983) Prism trees: a hierarchical representation for 3D objects. Proc Eight Int Joint Conf on Artificial Intelligence. Karlsruhe, pp 982–988
Faugeras OD, Hebert M, Mussi P, Boissonat JD (1984) Polyhedral approximation of 3-D objects without holes. Computer Vision, Graphics and Image Processing 25:169–183
Fekete J, Davis LS (1984) Property spheres: a new represenlation for 3-D object recognition. Proc Workshop on Compater Vision: Representation and Control, pp 192–201
Field DA (1986) Implementing Watson's algorithm in three-dimensions. Proc Second ACM Symp on Computational Geometry. Yorktown Heights pp 246–259
Fowler RF, Little JJ (1979) Automatic extraction of irregular digital terrain models. Computer Graphics 13:199–207
Fuchs H, Kedem Z, Usenton SP (1977) Optimal surface reconstruction from planar contours. Commun ACM 20(10):693–702
Ganapathy S, Dennehy TG (1982) A new general triangulation method for planar contours. Computer Graphies 16(3):69–75
Garey MR, Johnson DS, Preparata FP, Tarjan RE (1978) Triangulating a simple polygon. Inf Proc Lett 7(4):175–179
Gold CM, Charters TD, Ramsden J (1977) Automated contour mapping using triangular element data structures and an interpolant over each irregular triangular domain. Computer Graphics 11(3):170–175
Gemez D, Guzman A. Digital model for three-dimensional surface representation. Geo-Processing 1:53–70
Green PJ, Sibson R (1978) Computing Dirichlet tesselations in the plane. Comput J 21(2):168–173
Gulbas LJ, Stolfi J (1983) Primitives for the manipulation of general subdivisions and the computation of Voronol diagrams. Proc 15th ACM Conf on the Theory of Computing, pp 221⅔4
Harary F (1969) Graph, theory. Addison Wesley, Reading, MA
Hermeline F (1982) Triangulation automatique d'un polyedre on dimension N. RAIRO Numerical Analysis. 16(3):211–242
Keppel E (1975) Approximating complex surfaces by triangulation of contour lines. IBM J Res Dev 19(1):2–11
Kirkpatrick DG (1983) Optimal search in planar subdivisions. SIAM J Comput 12(1):28–35
Kleinsteurer C, Haldeman JT (1980) A triangular finite element mesh generator for fluid systems of arbitrary geometry. Int J Numer Methods Eng 15:1325–1334
Klingsek GT (1980) Minimal triangulations of polygonal domains. Ann Discrete Math 9:121–123
Klucewicz MI (1978) A piecewise C1 interpolant to arbitrarily spaced data. Computer Graphics and Image Processing 8:92–112
Knuth DE (1973) The art of computer programming, vol 1. Fundamental algorithms and sorting and searching, vol 3. Addison-Wesley, Reading, MA
Lane JM, Riesenfeld RF (1980) A theoretical development for the computer generation of precewise polynomial surfaces. IEEE Trans Pattern Anal Mach Intell, PAMI-2 1:35–46
Lawson CL (1977) Software for CI surface interpolation. In: Rice JR (ed) Mathematical Software III. Academic Press pp 161–164
Lee DT, Preparata FP (1977) Location of a point, in a planar subdivision and its applications. SIAM J Comput 6(3):594–606
Lee DT (1978) Proximity and reachability in the plane. PhD Dissertation. Coordinated Science Laboratory Rep ACT-12. University of Illinois, Urbana
Lee DT (1980) Two dimensional Voronoi diagrams in the Lp-metric. J Assoc Comput Mach 27:604–618
Lee, DT, Schacter BJ (1980) Two algorithms for constructing a Delaunay Triangulation. Int J Comput Inf Sci 9(3):219–242
Lee DT, Drysdale II RL (1981) Generalization of Voronoi diagrams in the plane, SIAM J Comput 10:73–87
Lee DT (1983) Visibility of a simple polygon. Computer Vision, Graphics and Image Processing 22:207–221
Lee DT, Preparata FP (1984) Computational geometry — a survey. IEEE Trans on Computers, C-33, 12:1072–1101
Lewis BA, Robinson JS (1979) Triangulation of planar regions with apphcanens. Comput J 21(4):324–332
Lloyd EL (1977) On triangulations of a set of points in the piane Proc IEEE 18th Annual Symp on the Foundations of Computer Science, pp 228–240
Manacher GK, Zobrist AL (1979) Neither the greedy not the Delaunay triangulation of a planar point set approximates the optimal triangulation. Inf Proc Lett 9:31–34
McCullagh MJ, Ross CG (1980) Delaunay triangulation of a random data set for isarithmic mapping. Cartographic Journal, vol 2
Mc Lain DH (1975) Two-dimensional interpolation from randem data. Comput J 19(2):178–181
Mirante A, Weingarten N (1982) The radial sweep algorithal for constructing triangulated irregular networks. IEEE Comput Graph Appl 2:11–21
Mueller DF, Preparata EP (1978) Finding the intersection of two convex polyhedra. Theor Comput Sci 7:217–236
Nagy G, Wagle S (1979) Geopraphical data processing. ACM Comput Surv 11:139–181
O'Rourke J (1981) Triangulation of minimal area as 3-D object models. Proc 7th Int Joint Conf on Artilical, Intelligence, pp 664–666
Peucker TK, Douglas DH (1975) Detection of surface-specific points by local parallel processing of discrete terrain elevation data. Computer Graphics and Image Processing 4:375–387
Preparata FP (1979) A note on locating a set of points in a planar subdivision. SIAM J Comput 8:542–545
Preparata FP, Shamos MI (1985) Computational geometry: an introduction, Springer
Requicha AAG (1980) Representations of rigid solids — Theory, methods and systems. ACM Comput Sury 12(4):437–463
Rhynsburger D (1973) Analytic delineation of Thiessen polygons. Geographical Analysis 5(2):133–144
Sadek EA (1980) A scheme for the autematic generation of triangular finite elements. Int J Numer Methods Eng 15:1813–1822
Samet H (1982) Neighbor finding techniques for images represented by quadtrees. Computer Graphics and Image Processing 18:37–57
Samet H (1984) The quadtree and related hierarchical data structures. ACM Comput Surv 16(2):187–260
Shamos MI (1975) Geometric complexity. Proc 7th annual ACM Symp on the Theory of Computing, pp 224–233
Shamos MI (1978) Computational Geometry. PhD Thesis. Yale University, New Haven, Connectient
Sibson R (1978) Locally equangular triangulation. Comput J 22:243–245
Sloan KR, Hrechanyk LD (1981) Surface reconstruction from sparse data. Proc Conf on Pattern Recognition and Image Processing, pp 45–48
Tamminen M (1981) The EXCELL method for efficient access to data. Acta Polytech Scand. Math Comput Sci Ser, n 34
Tarvydes A (1983) Terrain approximation by triangular facets. Proc Sixth Int, Conf on Automated Cartography, p 240
Terzopulos D (1980) Multilevel reconstruction of visual surfaces: variational principles and finite element representations. In: Rosenfeld A (ed) Multiresolution Image Processing and Analysis, Springer, pp 237–310
Toussaint GT, Avis D (1982) On a convex hull algorithm for polygon and its application to triangulation problems. Pattern Recognition 15:23–29
Watson DF (1981) Computing then-dimensional Delaunay tesselation with application to Voronoi polytopes. Comput J 24:167–171
Watson DF (1982) ACORD-Automatic contouring of raw data. Computer and Geosciences 8:97–101
Watson DF, Philip GM (1984) Systematic triangulations. Computer Vision. Graphics, and Image Processing 26:217–223
Weiler K (1985) Edge-based data structures for solid modeling in curved-surface environments. IEEE Comput Graph Appl 5(1):21–40
Woo TC (1985) A combinatorial analysis of boundary data structure schemata. IEEE Comput Graph Appl 5(3):19–27
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De Floriani, L. Surface representations based on triangular grids. The Visual Computer 3, 27–50 (1987). https://doi.org/10.1007/BF02153649
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DOI: https://doi.org/10.1007/BF02153649