Pierre Alliez The Computational Geometry Algorithm Library

Goals Promote Research in Computational Geometry (CG)Promote Research in Computational Geometry (CG) “make the large body of geometric algorithms developed in the field of CG available for industrial applications”“make the large body of geometric algorithms developed in the field of CG available for industrial applications” -> robust programs

CG Impact Task Force Report, 1996 Among the key recommendations: Production and distribution of usable (and useful) geometric codesProduction and distribution of usable (and useful) geometric codes Reward structure for implementations in academiaReward structure for implementations in academia

The CGAL Project Started in 1996 as joined project of: ETH Zurich ETH Zurich INRIA INRIA MPI für Informatik MPI für Informatik Tel-Aviv U Tel-Aviv U Utrecht U Utrecht U Trier U Trier U FU Berlin FU Berlin

Structure of CGAL Basic Library Support Library: Configuration, Assertions Visualization File I/O Number Types Generators STL extensions Point, Segment,... Predicates Kernel TriangulationsArrangements Halfedge Datastructure Convex HullOptimisation...

The Basic Library

Convex Hull 2D Convex Hull2D Convex Hull –5 algorithms for points –CH of a simple polyline Convexity TestConvexity Test Extremal PointsExtremal Points 3D Convex Hull3D Convex Hull –static, incremental, dynamic

Planar Subdivisions Framework for arrangements of 2D curvesFramework for arrangements of 2D curves –based on planar maps –based on topological maps –Models exist for polylines, circles, conic arcs OperationsOperations –point location, overlay, ray shooting,..

Search Structures Multi dimensional data structuresMulti dimensional data structures –range tree, segment tree –kD tree OperationsOperations –window queries –nearest neighbor

Triangulations in 2D and 3D Delaunay, regular, constrained, conformal DelaunayDelaunay, regular, constrained, conformal Delaunay Rich APIsRich APIs –fast point location, –insertion, removal of points and constraints –traversal along a line –traversal of the triangulation

Triangulations 2D, 3D Alpha shapes2D, 3D Alpha shapes 2D, 3D Voronoi diagram, power diagram2D, 3D Voronoi diagram, power diagram 2D Delaunay mesh, 3D Delaunay mesh2D Delaunay mesh, 3D Delaunay mesh Natural neighbors*Natural neighbors* k-order Voronoi diagram 2*k-order Voronoi diagram 2* Voronoi of segments *Voronoi of segments * Voronoi of circlesVoronoi of circles (*) on-going

Optimization Based on LP/QP solversBased on LP/QP solvers –Smallest enclosing circle/ellipse in 2D –Smallest enclosing sphere in dD –Rectangular p center, for p = 2, 3, 4 –Polytope distances in dD –Smallest enclosing annulus in dD –Smallest enclosing sphere of spheres *

Nef Polyhedra 2,3*

and... dD Delaunay/VoronoidD Delaunay/Voronoi Polygon decompositionPolygon decomposition Polyhedral surfacesPolyhedral surfaces Halfedge data structureHalfedge data structure

Library Design Let’s play Lego

int min(int a, int b) float min(float a, float b) template CompType min(CompType a, CompType b) { return (a<b) ? a : b; } BigInt n(9), m(8), r; r = min( n, m ); BigInt is a model for the concept CompType Generic Programming [Musser 89]

Parameterization of Kernel Classes Point_3 CartesianHomogeneous double Gmpq leda::Real int Gmpz Concepts: NumberType and Kernel core::Expr

template template class Delaunay_triangulation_2 { void insert(Geometry::Point t) void insert(Geometry::Point t) { if(Geometry::orientation(p,q,t)==..) if(Geometry::orientation(p,q,t)==..) if(Geometry::incircle(p,q,r,t)) if(Geometry::incircle(p,q,r,t)) }}; Parameterization of Datastructures works with CGAL kernelsworks with CGAL kernels works with projections kernelsworks with projections kernels write thin glue layer for your kernelwrite thin glue layer for your kernel

Value for Users High quality of most componentsHigh quality of most components Interoperability of componentsInteroperability of components Rich functionalityRich functionality –allows rapid prototyping –allows focus on application domain –Reduces time to market (publication)