GEOMETRIC LIBRARY. Maharavo Randrianarivony
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1 GEOMETRIC LIBRARY Maharavo Randrianarivony During the last four years, I have maintained a numerical geometric library. The constituting routines, which are summarized in the following list, are implemented in C/C++ and the visualization is done with the help of OpenGL and GLUT. They are not necessarily of strict geometric nature but they are all used for geometric purpose. 1. POLYGON: (a) Intersection of two convex polygons (b) Triangulation of a multiply connected polygon by using only boundary nodes: i. Ear cutting algorithm ii. Stack storage method (c) Decomposition of a polygon into monotone ones (d) Decomposition of a polygon into simply connected ones (e) Decomposition of a polygon into convex ones 2. QUADRANGULATION: (a) Quadrangulation of hexagons without boundary Steiner points (b) Quadrangulation of octagons (c) Removing triangular-shaped quadrilaterals (d) Converting a planar triangular mesh into a quadrilateral one with moderate number of Steiner points (e) Quadrangulation of a multiply connected polygon by repeatedly removing quadrilaterals (f) Converting nonconvex quadrangulations into convex ones (g) Quadrangulation of circular domains 1
2 3. MESH: (a) Mesh simplification (LOD) using quadric metrics (Garland) (b) Fast point location using hierarchy in a 2D-mesh (c) Mesh manipulations: slicing along curves, merging (d) Segmentation of a free-form closed smooth 3D-surface mesh into i. Four-sided segments ii. Three-sided segments (e) Visualization (f) Interactive coarse triangular 2D-mesh generation (g) Interactive coarse quadrilateral 2D-mesh generation (h) Convertion from RAW-format to standard format (i) Converting a scene into a POV-RAY script 4. (NON)LINEAR LEAST-SQUARE/OPTIMIZATION: (a) Normal equation (for mesh) (b) LS-CG: Least Square Conjugate Gradient (c) Levenberg-Marquardt for nonlinear least-square (d) Gauss-Newton for nonlinear least-square (e) Newton iteration for nonlinear optimization 5. C.A.D. INTERFACE AND CSG/B-REP: (a) Loading an IGES file (uncompressed format) (b) Parsing some IGES entities into C/C++ data structures (c) Evaluation routines for some selected IGES entities (d) Visualizing a surface stored in an IGES file (e) Treating objects with multiple SHELLS in B-Rep scheme (f) Segmentation of exact CAD surfaces (g) Output in ascii/binary (big and little endian) (h) Loading of molecular data from PDB files (Protein Data Bank) (i) Automatic specification of Van der Walls radii (j) Convertion from CSG to B-rep of molecular surfaces 2
3 6. LINEAR SOLVERS: (a) Gaussian elimination (b) CG (Conjugate Gradient) (c) PCG (Preconditioned Conjugate Gradient) with various preconditioners (d) Jacobi iterative solver (e) Gauss-Seidel iterative solver (f) Using Harwell-Boeing structure for efficient sparse storage 7. CURVE/SURFACE REPARAMETERIZATION: (a) Chord length reparametrization with cubic Bézier splines (b) Conformal surface mesh reparametrization (c) Floater surface mesh reparametrization (d) MIPS (Most Isometric ParameterizationS) 8. TRIANGULATION: (a) Delaunay triangulation: i. Only internal node insertion ii. Only edge-node insertion iii. With respect to an edge size function (b) Generalized Delaunay triangulation. (c) Mesh generation on a piecewise Riemannian surface: i. Boundary node determination (smoothness, harmonicity) ii. Edge size determination (d) Triangulation of a monotone polygon 9. RECONSTRUCTION: (a) Bézier approximation and interpolation (Curve/surface) (b) Bézier approximation with boundary interpolation (Curve/surface) (c) B-spline approximation (Curve/surface) (d) NURBS approximation (Curve/surface) (e) Approximation with basic curves (circle, line, etc) (f) Approximation with basic surfaces (sphere, cylinder, plane, etc) (g) Reconstruction of closed orientable free-form surfaces with arbitrary genera. 3
4 10. GRAPH ALGORITHMS: (a) Weighted and simple graphs operations (b) Dijkstra (shortest path) (c) BFS (Breadth First Search) (d) Prim s algorithm for MST (Minimal Spanning Tree) (e) RLS (Recursive Largest First) for graph coloring 11. NUMERICAL TOPOLOGY: (a) Smith normal form (b) Incidence matrices (c) Homology computation (d) Computing line integrals from saddles to extrema (e) Morse complex on a surface mesh embedded in R 3 (f) Reidemeister moves of curves on surfaces (g) Computing canonical curves on orientable surfaces with non-zero genera and without boundaries (h) Singularity (mimimum, maximum, saddle) w.r.t. the eigenvectors of the Laplace operator on a surface mesh (i) Handle decomposition: split a mesh along the canonical curves and find a parametrization from a 2D polygonal disk 12. SPECIAL CURVES, SURFACES AND OPERATORS: (a) Stereographic/hyperbolic projections in homogeneous coordinates (b) Conics as rational Bézier curves (c) Spherical patches as rational Bézier surfaces (d) Evaluation of Bézier curve/surface (e) Bézier degree elevation (f) Bézier subdivision (g) Evaluation of B-spline curve/surface (h) B-Spline subdivision (i) Evaluation of NURBS curve/surface (j) Composite curve (k) Surface of revolution 4
5 13. EIGENVALUES: (a) Power method (b) Shift-and-invert iteration (c) QR-decomposition (d) Hessenberg (e) Givens rotation 14. CURVED ENTITIES: (a) Splitting into simply-connected subregions (b) Splitting into four-sided subregions (c) Generation of diffeomorphism (d) Amalgamation (e) Hanging node removal (f) Visualization 15. EDGE DETECTION: (a) 1D-FWT and 2D-FWT (b) MRS curve (c) MRS surface (d) Resampling (for nonuniform samples) (e) Application to 3D-surfaces 16. TRANSFINITE INTERPOLATION: (a) Treatment of Coons and Gordon patches (b) Checking diffeomorph Coons (c) Adaptivity using Bézier subdivisions (d) Automatic internal curve insertion for Gordon (e) Diffeomorph Gordon patches (f) Rational spherical transfinite interpolants 17. RIEMANNIAN GEOMETRY: (a) Entities (Christoffel symbols, Riemannian tensors,...) (b) Geodesic in general (c) Hierarchical geodesic (d) Geodesics w.r.t. Bézier/Hierarchical Bézier (e) Laplace solver with FEM (f) Laplace-Beltrami solver with FEM 5
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