Package Overview


I  General Introduction
II  Arithmetic and Algebra
III  Combinatorial Algorithms
IV  Geometry Kernels
V  Convex Hull Algorithms
VI  Polygons
VII  Cell Complexes and Polyhedra
VIII  Arrangements
IX  Triangulations and Delaunay Triangulations
X  Voronoi Diagrams
XI  Mesh Generation
XII  Geometry Processing
XIII  Spatial Searching and Sorting
XIV  Geometric Optimization
XV  Interpolation
XVI  Kinetic Data Structures
XVII  Support Library
XVIII  Visualization


I   General Introduction

Introduction

Introduction Illustration

CGAL Editorial Board

Introduced in: Cgal 1.0
BibTeX: [cgal:eb-gi-12]
User Manual   Reference Manual
This chapter explains how the manual is organized, presents a ``Hello World'' program, and gives recommendations for further readings.

Preliminaries

Preliminaries Illustration

CGAL Editorial Board

Introduced in: Cgal 1.0
BibTeX: [cgal:eb-gi-12]
User Manual   Reference Manual
This chapter lists the licenses under which the Cgal datastructures and algorithms are distributed, how to control inlining, thread safety, code deprecation, checking of pre- and postconditions, and how to alter the failure behavior.

Installation

Installation Illustration

Joachim Reichel and Fernando Cacciola

Introduced in: Cgal 3.6
BibTeX: [cgal:cr-i-12]
Demo: dlls used by all demos
User Manual   Reference Manual
This chapter describes how to install Cgal, and lists the third party libraries on which Cgal depends, or for which Cgal provides interfaces.

If you want to download and run any of the precompiled demos, here you can find the dlls that are shared by all of them.


II   Arithmetic and Algebra

Algebraic Foundations

Algebraic Foundations Illustration

Michael Hemmer

Introduced in: Cgal 3.3
License: LGPL
BibTeX: [cgal:h-af-12]
User Manual   Reference Manual
This package defines what algebra means for Cgal, in terms of concepts, classes and functions. The main features are: (i) explicit concepts for interoperability of types (ii) separation between algebraic types (not necessarily embeddable into the reals), and number types (embeddable into the reals).

Number Types

Number Types Illustration

Michael Hemmer, Susan Hert, Sylvain Pion, and Stefan Schirra

Introduced in: Cgal 1.0
License: LGPL
BibTeX: [cgal:hhkps-nt-12]
User Manual   Reference Manual
This package provides number type concepts as well as number type classes and wrapper classes for third party number type libraries.

Modular Arithmetic

Modular Arithmetic  Illustration

Michael Hemmer

Introduced in: Cgal 3.4
License: LGPL
BibTeX: [cgal:h-ma-12]
User Manual   Reference Manual
This package provides arithmetic over finite fields. The provided tools are in particular useful for filters based on modular arithmetic and algorithms based on Chinese remainder.

Polynomial

Polynomial  Illustration

Michael Hemmer

Introduced in: Cgal 3.3
License: LGPL
BibTeX: [cgal:h-p-12]
User Manual   Reference Manual
This package introduces a concept for univariate and multivariate polynomials in d variables. Though the concept is written for an arbitrary number of variables, the number of variables is considered as fixed for a particular model of this concept.

Algebraic Kernel

Algebraic Kernel  Illustration

Eric Berberich, Michael Hemmer, Michael Kerber, Sylvain Lazard, Luis Peñaranda, and Monique Teillaud

Introduced in: Cgal 3.6
Depends on: Some models depend on RS.
License: LGPL
BibTeX: [cgal:bht-ak-12]
User Manual   Reference Manual
Real solving of polynomials is a fundamental problem with a wide application range. This package is targeted to provide black-box implementations of state-of-the-art algorithms to determine, compare and approximate real roots of univariate polynomials and bivariate polynomial systems. Such a black-box is called an Algebraic Kernel. So far the package only provides models for the univariate kernel. Nevertheless, it already defines concepts for the bivariate kernel, since this settles the interface for upcoming implementations.


III   Combinatorial Algorithms

Monotone and Sorted Matrix Search

Monotone and Sorted Matrix Search  Illustration

Michael Hoffmann

Introduced in: Cgal 1.1
License: GPL
BibTeX: [cgal:h-msms-12]
User Manual   Reference Manual
This package provides a matrix search framework, which is the underlying technique for the computation of all furthest neighbors for the vertices of a convex polygon, maximal k-gons inscribed into a planar point set, and computing rectangular p-centers.

Linear and Quadratic Programming Solver

Linear and Quadratic Programming Solver
 Illustration

Kaspar Fischer, Bernd Gärtner, Sven Schönherr, and Frans Wessendorp

Introduced in: Cgal 3.3
License: GPL
BibTeX: [cgal:fgsw-lqps-12]
User Manual   Reference Manual
This package contains algorithms for minimizing linear and convex quadratic functions over polyhedral domains, described by linear equations and inequalities. The algorithms are exact, i.e. the solution is computed in terms of multiprecision rational numbers.

The resulting solution is certified: along with the claims that the problem under consideration has an optimal solution, is infeasible, or is unbounded, the algorithms also deliver proofs for these facts. These proofs can easily (and independently from the algorithms) be checked for correctness.

The solution algorithms are based on a generalization of the simplex method to quadratic objective functions.


IV   Geometry Kernels

2D and 3D Linear Geometry Kernel

2D and 3D Linear Geometry Kernel Illustration

Hervé Brönnimann, Andreas Fabri, Geert-Jan Giezeman, Susan Hert, Michael Hoffmann, Lutz Kettner, Sylvain Pion, and Stefan Schirra

Introduced in: Cgal 0.9
License: LGPL
BibTeX: [cgal:bfghhkps-lgk23-12]
User Manual   Reference Manual
This package contains kernels each containing objects of constant size, such as point, vector, direction, line, ray, segment, circle as well as predicates and constructions for these objects. The kernels mainly differ in the way they handle robustness issues.

dD Geometry Kernel

dD Geometry Kernel Illustration

Michael Seel

Introduced in: Cgal 1.1
License: LGPL
BibTeX: [cgal:s-gkd-12]
User Manual   Reference Manual
The dD Kernel contains objects of constant size, such as point, vector, direction, line, ray, segment, circle in d dimensional Euclidean space, as well as predicates and constructions for these objects.

2D Circular Geometry Kernel

2D Circular Geometry Kernel  Illustration

Pedro Machado Manhães de Castro, Sylvain Pion, and Monique Teillaud

Introduced in: Cgal 3.2
License: GPL
BibTeX: [cgal:cpt-cgk2-12]
Demo: Arrangement of Circular Arcs
User Manual   Reference Manual
This package is an extension of the linear Cgal kernel. It offers functionalities on circles, circular arcs and line segments in the plane.

3D Spherical Geometry Kernel

3D Spherical Geometry Kernel  Illustration

Pedro Machado Manhães de Castro, Frédéric Cazals, Sébastien Loriot, and Monique Teillaud

Introduced in: Cgal 3.4
License: GPL
BibTeX: [cgal:cclt-sgk3-12]
User Manual   Reference Manual
This package is an extension of the linear Cgal Kernel. It offers functionalities on spheres, circles, circular arcs and line segments, in the 3D space or restricted on a reference sphere.


V   Convex Hull Algorithms

2D Convex Hulls and Extreme Points

2D Convex Hulls and Extreme Points  Illustration

Susan Hert and Stefan Schirra

Introduced in: Cgal 1.0
License: GPL
BibTeX: [cgal:hs-chep2-12]
Demo: See Bounding Volumes
User Manual   Reference Manual
This package provides functions for computing convex hulls in two dimensions as well as functions for checking if sets of points are strongly convex are not. There are also a number of functions described for computing particular extreme points and subsequences of hull points, such as the lower and upper hull of a set of points.

3D Convex Hulls

3D Convex Hulls Illustration

Susan Hert and Stefan Schirra

Introduced in: Cgal 1.1
Depends on: All algorithms produce as output a 3D Polyhedron. The dynamic algorithms depend on 3D Triangulations
License: GPL
BibTeX: [cgal:hs-ch3-12]
Demo: See Polyhedral Surface
User Manual   Reference Manual
This package provides functions for computing convex hulls in three dimensions as well as functions for checking if sets of points are strongly convex or not. One can compute the convex hull of a set of points in three dimensions in one of three ways: using a static algorithm, using an incremental construction algorithm, or using a triangulation to get a fully dynamic computation.

dD Convex Hulls and Delaunay Triangulations

 

Susan Hert and Michael Seel

Introduced in: Cgal 2.3
License: GPL
BibTeX: [cgal:hs-chdt3-12]
User Manual   Reference Manual
This package provides functions for computing convex hulls and Delaunay triangulations in d-dimensional Euclidean space.


VI   Polygons

2D Polygon

2D Polygon Illustration

Geert-Jan Giezeman and Wieger Wesselink

Introduced in: Cgal 0.9
License: LGPL
BibTeX: [cgal:gw-p2-12]
Demo: Operations on Polygons
User Manual   Reference Manual

This package provides a 2D polygon class and operations on sequences of points, like bounding box, extremal points, signed area, simplicity and convexity test, orientation, and point location.

The demo includes operations on polygons, such as computing a convex partition, and the straight skeleton.

2D Regularized Boolean Set-Operations

2D Regularized Boolean Set-Operations Illustration

Efi Fogel, Ophir Setter, Ron Wein, Guy Zucker, Baruch Zukerman, and Dan Halperin

Introduced in: Cgal 3.2
Depends on: 2D Arrangements
License: GPL
BibTeX: [cgal:fwzh-rbso2-12]
Demo: Boolean operations
User Manual   Reference Manual
This package consists of the implementation of Boolean set-operations on point sets bounded by weakly x-monotone curves in 2-dimensional Euclidean space. In particular, it contains the implementation of regularized Boolean set-operations, intersection predicates, and point containment predicates.

2D Boolean Operations on Nef Polygons

2D Boolean Operations on Nef Polygons  Illustration

Michael Seel

Introduced in: Cgal 2.3
License: GPL
BibTeX: [cgal:s-bonp2-12]
Demo: 2D Nef Polygons
User Manual   Reference Manual
A Nef polygon is any set that can be obtained from a finite set of open halfspaces by set complement and set intersection operations. Due to the fact that all other binary set operations like union, difference and symmetric difference can be reduced to intersection and complement calculations, Nef polygons are also closed under those operations. Apart from the set complement operation there are more topological unary set operations that are closed in the domain of Nef polygons interior, boundary, and closure.

2D Boolean Operations on Nef Polygons Embedded on the Sphere

2D Boolean Operations on Nef Polygons Embedded on the Sphere  Illustration

Peter Hachenberger, Lutz Kettner, and Michael Seel

Introduced in: Cgal 3.1
Depends on: 2D Nef Polygon
License: GPL
BibTeX: [cgal:hk-bonpes2-12]
User Manual   Reference Manual
This package offers the equivalent to 2D Nef Polygons in the plane. Here halfplanes correspond to half spheres delimited by great circles.

2D Polygon Partitioning

2D Polygon Partitioning  Illustration

Susan Hert

Introduced in: Cgal 2.3
License: GPL
BibTeX: [cgal:h-pp2-12]
Demo: 2D Polygon Partition
Demo: Operations on Polygons
User Manual   Reference Manual
This package provides functions for partitioning polygons in monotone or convex polygons. The algorithms can produce results with the minimal number of polygons, as well as approximations which have no more than four times the optimal number of convex pieces but they differ in their runtime complexities.

2D Straight Skeleton and Polygon Offsetting

2D Straight Skeleton and Polygon Offsetting  Illustration

Fernando Cacciola

Introduced in: Cgal 3.2
Depends on: Halfedge Data Structure
License: GPL
BibTeX: [cgal:c-sspo2-12]
Demo: 2D Straight Skeleton
Demo: Operations on Polygons
User Manual   Reference Manual
This package implements an algorithm to construct a halfedge data structure representing the straight skeleton in the interior of 2D polygons with holes and an algorithm to construct inward offset polygons at any offset distance given a straight skeleton.

2D Minkowski Sums

2D Minkowski Sums Illustration

Ron Wein

Introduced in: Cgal 3.3
Depends on: 2D Arrangements
License: GPL
BibTeX: [cgal:w-rms2-12]
User Manual   Reference Manual
This package consists of functions that compute the Minkowski sum of two simple straight-edge polygons in the plane. It also contains functions for computing the Minkowski sum of a polygon and a disc, an operation known as offsetting or dilating a polygon. The package can compute the exact representation of the offset polygon, or provide a guaranteed approximation of the offset.


VII   Cell Complexes and Polyhedra

3D Polyhedral Surface

3D Polyhedral Surface  Illustration

Lutz Kettner

Introduced in: Cgal 1.0
Depends on: Halfedge Data Structure
License: GPL
BibTeX: [cgal:k-ps-12]
Demo: Operations on Polyhedra
User Manual   Reference Manual
Polyhedral surfaces in three dimensions are composed of vertices, edges, facets and an incidence relationship on them. The organization beneath is a halfedge data structure, which restricts the class of representable surfaces to orientable 2-manifolds - with and without boundary. If the surface is closed we call it a polyhedron.

Halfedge Data Structures

Halfedge Data Structures   Illustration

Lutz Kettner

Introduced in: Cgal 1.0
License: LGPL
BibTeX: [cgal:k-hds-12]
User Manual   Reference Manual
A halfedge data structure is an edge-centered data structure capable of maintaining incidence information of vertices, edges and faces, for example for planar maps, polyhedra, or other orientable, two-dimensional surfaces embedded in arbitrary dimension. Each edge is decomposed into two halfedges with opposite orientations. One incident face and one incident vertex are stored in each halfedge. For each face and each vertex, one incident halfedge is stored. Reduced variants of the halfedge data structure can omit some of these information, for example the halfedge pointers in faces or the storage of faces at all.

Combinatorial Maps

Combinatorial Maps Illustration

Guillaume Damiand

Introduced in: Cgal 3.9
License: LGPL
BibTeX: [cgal:d-cm-12]
User Manual   Reference Manual
This package implements Combinatorial Maps in d-dimension. A combinatorial map is a data structure allowing to represent an orientable subdivided object by describing all the cells of the subdivision (for example in 3D vertices, edges, faces, volumes) and all the incidence and adjacency relationships between these cells. Information can be associated to cells thanks to attributes.

In 2D, a combinatorial map is equivalent to an halfedge data structure.

The package provides basic creations, modification operations, and several iterators allowing to run through some specific part of the object.

Linear Cell Complex

Linear Cell Complex Illustration

Guillaume Damiand

Introduced in: Cgal 4.0
Depends on: Combinatorial Maps
License: LGPL
BibTeX: [cgal:d-lcc-12]
Demo: 3D Linear Cell Complex
User Manual   Reference Manual
This package implements linear cell complexes, objects in d-dimension with linear geometry. The combinatorial part of objects is described by a combinatorial map, representing all the cells of the object plus the incidence and adjacency relations between cells. Geometry is added to combinatorial maps simply by associating a point to each vertex of the map.

Taking a 2D combinatorial map, and using 3D points, gives a linear cell complex equivalent to a Polyhedron_3.

3D Boolean Operations on Nef Polyhedra

3D Boolean Operations on  Nef Polyhedra Illustration

Peter Hachenberger, Lutz Kettner, and Michael Seel

3D Nef polyhedra, are a boundary representation for cell-complexes bounded by halfspaces that supports Boolean operations and topological operations in full generality including unbounded cells, mixed dimensional cells (e.g., isolated vertices and antennas). Nef polyhedra distinguish between open and closed sets and can represent non-manifold geometry.

Convex Decomposition of Polyhedra

Convex Decomposition of Polyhedra Illustration

Peter Hachenberger

This packages provides a function for decomposing a bounded polyhedron into convex sub-polyhedra. The decomposition yields O(r2) convex pieces, where r is the number of edges, whose adjacent facets form an angle of more than 180 degrees with respect to the polyhedron's interior. This bound is worst-case optimal.

3D Minkowski Sum of Polyhedra

3D Minkowski Sum of Polyhedra Illustration

Peter Hachenberger

This package provides a function, which computes the Minkowski sum of two point sets in 3. These point sets may consist of isolated vertices, isolated edges, surfaces with convex facets without holes, and open and closed solids. Thus, it is possible to compute the configuration space of translational robots (even in tight passage scenarios) as well as several graphics operations, like for instance the glide operation, which computes the point set swept by a polyhedron that moves along a polygonal line.


VIII   Arrangements

2D Arrangement

2D Arrangement Illustration

Ron Wein, Efi Fogel, Baruch Zukerman, Dan Halperin, Eric Berberich, and Oren Zalzman

Introduced in: Cgal 2.1
License: GPL
BibTeX: [cgal:wfzh-a2-12]
User Manual   Reference Manual
This package can be used to construct, maintain, alter, and display arrangements in the plane. Once an arrangement is constructed, the package can be used to obtain results of various queries on the arrangement, such as point location. The package also includes generic implementations of two algorithmic frameworks, that are, computing the zone of an arrangement, and line-sweeping the plane, the arrangements is embedded on. These frameworks are used in turn in the implementations of other operations on arrangements. Computing the overlay of two arrangements, for example, is based on the sweep-line framework.

Arrangements and arrangement components can also be extended to store additional data. An important extension stores the construction history of the arrangement, such that it is possible to obtain the originating curve of an arrangement subcurve.

2D Intersection of Curves

2D Intersection of Curves Illustration

Baruch Zukerman and Ron Wein

Introduced in: Cgal 2.4
Depends on: 2D Arrangements
License: GPL
BibTeX: [cgal:wfz-ic2-12]
User Manual   Reference Manual
This package provides three free functions implemented based on the sweep-line paradigm: given a collection of input curves, compute all intersection points, compute the set of subcurves that are pairwise interior-disjoint induced by them, and check whether there is at least one pair of curves among them that intersect in their interior.

2D Snap Rounding

2D Snap Rounding  Illustration

Eli Packer

Introduced in: Cgal 3.1
Depends on: Arrangements
License: GPL
BibTeX: [cgal:p-sr2-12]
Demo: 2D Snap Rounding
User Manual   Reference Manual
Snap Rounding is a well known method for converting arbitrary-precision arrangements of segments into a fixed-precision representation. In the study of robust geometric computing, it can be classified as a finite precision approximation technique. Iterated Snap Rounding is a modification of Snap Rounding in which each vertex is at least half-the-width-of-a-pixel away from any non-incident edge. This package supports both methods.

2D Envelopes

2D Envelopes Illustration

Ron Wein

Introduced in: Cgal 3.3
Depends on: 2D Arrangements
License: GPL
BibTeX: [cgal:w-e2-12]
User Manual   Reference Manual
This package consits of functions that computes the lower (or upper) envelope of a set of arbitrary curves in 2D. The output is represented as an envelope diagram, namely a subdivision of the x-axis into intervals, such that the identity of the curves that induce the envelope on each interval is unique.

3D Envelopes

3D Envelopes Illustration

Dan Halperin, Michal Meyerovitch, Ron Wein and Baruch Zukerman

Introduced in: Cgal 3.3
Depends on: 2D Arrangements
License: GPL
BibTeX: [cgal:mwz-e3-12]
Demo: 3D Envelopes
Demo: L1 Voronoi Diagram
User Manual   Reference Manual
This package consits of functions that compute the lower (or upper) envelope of a set of arbitrary surfaces in 3D. The output is represented as an 2D envelope diagram, namely a planar subdivision such that the identity of the surfaces that induce the envelope over each diagram cell is unique.


IX   Triangulations and Delaunay Triangulations

2D Triangulation

2D Triangulation  Illustration

Mariette Yvinec

This package allows to build and handle various triangulations for point sets two dimensions. Any Cgal triangulation covers the convex hull of its vertices. Triangulations are build incrementally and can be modified by insertion or removal of vertices. They offer point location facilities.

The package provides plain triangulation (whose faces depend on the insertion order of the vertices) and Delaunay triangulations. Regular triangulations are also provided for sets of weighted points. Delaunay and regular triangulations offer nearest neighbor queries and primitives to build the dual Voronoi and power diagrams.

Finally, constrained and Delaunay constrained triangulations allows to force some constrained segments to appear as edges of the triangulation. Several versions of constrained and Delaunay constrained triangulations are provided: some of them handle intersections between input constraints segment while others do not.

2D Triangulation Data Structure

2D Triangulation Data Structure  Illustration

Sylvain Pion and Mariette Yvinec

Introduced in: Cgal 2.2
License: GPL
BibTeX: [cgal:py-tds2-12]
User Manual   Reference Manual
This package provides a data structure to store a two-dimensional triangulation that has the topology of a two-dimensional sphere. The package acts as a container for the vertices and faces of the triangulation and provides basic combinatorial operation on the triangulation.

3D Triangulations

3D Triangulations Illustration

Sylvain Pion and Monique Teillaud

Introduced in: Cgal 2.1
License: GPL
BibTeX: [cgal:pt-t3-12]
Demo: 3D Triangulation
User Manual   Reference Manual
This package allows to build and handle triangulations for point sets in three dimensions. Any Cgal triangulation covers the convex hull of its vertices. Triangulations are build incrementally and can be modified by insertion, displacements or removal of vertices. They offer point location facilities.

The package provides plain triangulation (whose faces depends on the insertion order of the vertices) and Delaunay triangulations. Regular triangulations are also provided for sets of weighted points. Delaunay and regular triangulations offer nearest neighbor queries and primitives to build the dual Voronoi and power diagrams.

3D Triangulation Data Structure

3D Triangulation Data Structure  Illustration

Sylvain Pion and Monique Teillaud

Introduced in: Cgal 2.1
License: GPL
BibTeX: [cgal:pt-tds3-12]
User Manual   Reference Manual
This package provides a data structure to store a three-dimensional triangulation that has the topology of a three-dimensional sphere. The package acts as a container for the vertices and cells of the triangulation and provides basic combinatorial operations on the triangulation.

3D Periodic Triangulations

3D Periodic Triangulations Illustration

Manuel Caroli and Monique Teillaud

This package allows to build and handle triangulations of point sets in the three dimensional flat torus. Triangulations are built incrementally and can be modified by insertion or removal of vertices. They offer point location facilities.

The package provides Delaunay triangulations and offers nearest neighbor queries and primitives to build the dual Voronoi diagrams.

2D Alpha Shapes

2D Alpha Shapes Illustration

Tran Kai Frank Da

Introduced in: Cgal 2.1
Depends on: 2D Triangulation
License: GPL
BibTeX: [cgal:d-as2-12]
Demo: 2D Alpha Shapes
User Manual   Reference Manual
This package offers a data structure encoding the whole family of alpha-complexes related to a given 2D Delaunay or regular triangulation. In particular, the data structure allows to retrieve the alpha-complex for any alpha value, the whole spectrum of critical alpha values and a filtration on the triangulation faces (this filtration is based on the first alpha value for which each face is included on the alpha-complex).

3D Alpha Shapes

3D Alpha Shapes Illustration

Tran Kai Frank Da, Sébastien Loriot, and Mariette Yvinec

Introduced in: Cgal 2.3
Depends on: 2D Triangulation
License: GPL
BibTeX: [cgal:dy-as3-12]
Demo: 3D Alpha Shapes
User Manual   Reference Manual
This package offers a data structure encoding either one alpha-complex or the whole family of alpha-complexes related to a given 3D Delaunay or regular triangulation. In the latter case, the data structure allows to retrieve the alpha-complex for any alpha value, the whole spectrum of critical alpha values and a filtration on the triangulation faces (this filtration is based on the first alpha value for which each face is included on the alpha-complex).


X   Voronoi Diagrams

2D Segment Delaunay Graphs

2D Segment Delaunay Graphs  Illustration

Menelaos Karavelas

Introduced in: Cgal 3.1
Depends on: 2D Triangulation Data Structure
License: GPL
BibTeX: [cgal:k-sdg2-12]
Demo: 2D Segment Voronoi Diagram
User Manual   Reference Manual
An algorithm for computing the dual of a Voronoi diagram of a set of segments under the Euclidean metric. It is a generalization of the standard Voronoi diagram for points. The algorithms provided are dynamic.

2D Apollonius Graphs (Delaunay Graphs of Disks)

2D Apollonius Graphs (Delaunay Graphs of Disks) Illustration

Menelaos Karavelas and Mariette Yvinec

Introduced in: Cgal 3.0
Depends on: 2D Triangulation Data Structure
License: GPL
BibTeX: [cgal:ky-ag2-12]
Demo: 2D Apollonius Graph
User Manual   Reference Manual
Algorithms for computing the Apollonius graph in two dimensions. The Apollonius graph is the dual of the Apollonius diagram, also known as the additively weighted Voronoi diagram. The latter can be thought of as the Voronoi diagram of a set of disks under the Euclidean metric, and it is a generalization of the standard Voronoi diagram for points. The algorithms provided are dynamic.


XI   Mesh Generation

2D Conforming Triangulations and Meshes

2D Conforming Triangulations and Meshes Illustration

Laurent Rineau

Introduced in: Cgal 3.1
Depends on: 2D Delaunay Triangulation
License: GPL
BibTeX: [cgal:r-ctm2-12]
Demo: 2D Mesh Generator
User Manual   Reference Manual
This package implements a Delaunay refinement algorithm to construct conforming triangulations and 2D meshes.

Conforming Delaunay triangulations are obtained from constrained Delaunay triangulations by refining constrained edges until they are Delaunay edges. Conforming Gabriel triangulations are obtained by further refining constrained edges until they become Gabriel edges.

The package provides also a 2D mesh generator that refines triangles and constrained edges until user defined size and shape criteria on triangles are satisfied. The package can handle intersecting input constraints and set no restriction on the angle formed by two constraints sharing an endpoint.

3D Surface Mesh Generation

3D Surface Mesh Generation Illustration

Laurent Rineau and Mariette Yvinec

Introduced in: Cgal 3.2
License: GPL
BibTeX: [cgal:ry-smg-12]
Demo: Surface Mesh Generator
User Manual   Reference Manual
This package provides functions to generate surface meshes that interpolate smooth surfaces. The meshing algorithm is based on Delaunay refinement and provides some guarantees on the resulting mesh: the user is able to control the size and shape of the mesh elements and the accuracy of the surface approximation. There is no restriction on the topology and number of components of input surfaces. The surface mesh generator may also be used for non smooth surfaces but without guarantee.

Currently, implementations are provided for implicit surfaces described as the zero level set of some function and surfaces described as a gray level set in a three-dimensional image.

Surface Reconstruction from Point Sets

Surface Reconstruction from Point Sets Illustration

Pierre Alliez, Laurent Saboret, Gael Guennebaud

Introduced in: Cgal 3.5
Depends on: Eigen or Taucs
License: GPL
BibTeX: [cgal:asg-srps-12]
Demo: Surface Reconstruction
User Manual   Reference Manual
This Cgal package implements a surface reconstruction method: Poisson Surface Reconstruction. It takes as input a set of points with oriented normals and computes an implicit function. The Cgal surface mesh generator can then be used to extract an iso-surface from this function.

3D Skin Surface Meshing

3D Skin Surface Meshing  Illustration

Nico Kruithof

Introduced in: Cgal 3.3
Depends on: 3D Triangulation and 3D Polyhedral Surface
License: GPL
BibTeX: [cgal:k-ssm3-12]
User Manual   Reference Manual
This package allows to build a triangular mesh of a skin surface. Skin surfaces are used for modeling large molecules in biological computing. The surface is defined by a set of balls, representing the atoms of the molecule, and a shrink factor that determines the size of the smooth patches gluing the balls together.

The construction of a triangular mesh of a smooth skin surface is often necessary for further analysis and for fast visualization. This package provides functions to construct a triangular mesh approximating the skin surface from a set of balls and a shrink factor. It also contains code to subdivide the mesh efficiently.

3D Mesh Generation

3D Mesh Generation  Illustration

Pierre Alliez, Laurent Rineau, Stéphane Tayeb, Jane Tournois, Mariette Yvinec

This package is devoted to the generation of isotropic simplicial meshes discretizing 3D domains. The domain to be meshed is a region of 3D space that has to be bounded. The region may be connected or composed of multiple components and/or subdivided in several subdomains. The domain is input as an oracle able to answer queries, of a few different types, on the domain. Boundary and subdivision surfaces are either smooth or piecewise smooth surfaces, formed with planar or curved surface patches. Surfaces may exhibit 1-dimensional features (e.g. crease edges) and 0-dimensional features (e.g. singular points as corners tips, cusps or darts), that have to be fairly approximated in the mesh.


XII   Geometry Processing

3D Surface Subdivision Methods

3D Surface Subdivision Methods Illustration

Le-Jeng Andy Shiue

Introduced in: Cgal 3.2
License: LGPL
BibTeX: [cgal:s-ssm2-12]
Demo: Operations on Polyhedra
User Manual   Reference Manual
Subdivision methods recursively refine a control mesh and generate points approximating the limit surface. This package consists of four popular subdivision methods and their refinement hosts. Supported subdivision methods include Catmull-Clark, Loop, Doo-Sabin and sqrt(3) subdivisions. Their respective refinement hosts are Pqq, Ptq, Dqq and sqrt(3) refinements. Variations of those methods can be easily extended by substituting the geometry computation of the refinement host.

Triangulated Surface Mesh Simplification

Triangulated Surface Mesh Simplification Illustration

Fernando Cacciola

Introduced in: Cgal 3.3
Depends on: Polyhedron
License: GPL
BibTeX: [cgal:c-tsms-12]
Demo: Operations on Polyhedra
User Manual   Reference Manual
This package provides an algorithm to simplify a triangulated surface mesh by edge collapsing. It is an implementation of the Turk/Lindstrom memoryless mesh simplification algorithm.

Planar Parameterization of Triangulated Surface Meshes

Planar Parameterization of Triangulated Surface Meshes Illustration

Laurent Saboret, Pierre Alliez and Bruno Lévy

Introduced in: Cgal 3.2
Depends on: Solvers as Eigen or OpenNL or Taucs.
License: GPL
BibTeX: [cgal:sal-pptsm2-12]
Demo: Operations on Polyhedra
User Manual   Reference Manual
Parameterizing a surface amounts to finding a one-to-one mapping from a suitable domain to the surface. In this package, we focus on triangulated surfaces that are homeomorphic to a disk and on piecewise linear mappings into a planar domain. This package implements some of the state-of-the-art surface mesh parameterization methods, such as least squares conformal maps, discrete conformal map, discrete authalic parameterization, Floater mean value coordinates or Tutte barycentric mapping.

2D Placement of Streamlines

2D Placement of Streamlines Illustration

Abdelkrim Mebarki

Introduced in: Cgal 3.2
Depends on: 2D Delaunay triangulation
License: GPL
BibTeX: [cgal:m-ps-12]
Demo: 2D Stream Lines
User Manual   Reference Manual
Visualizing vector fields is important for many application domains. A good way to do it is to generate streamlines that describe the flow behavior. This package implements the "Farthest Point Seeding" algorithm for placing streamlines in 2D vector fields. It generates a list of streamlines corresponding to an input flow using a specified separating distance. The algorithm uses a Delaunay triangulation to model objects and address different queries, and relies on choosing the centers of the biggest empty circles to start the integration of the streamlines.

Approximation of Ridges and Umbilics on Triangulated Surface Meshes

Approximation of Ridges and Umbilics on
    Triangulated Surface Meshes  Illustration

Marc Pouget and Frédéric Cazals

Introduced in: Cgal 3.3
Depends on: Solvers as Eigen, or Lapack and Blas
License: GPL
BibTeX: [cgal:cp-arutsm-12]
User Manual   Reference Manual
Global features related to curvature extrema encode important informations used in segmentation, registration, matching and surface analysis. Given pointwise estimations of local differential quantities, this package allows the approximation of differential features on a triangulated surface mesh. Such curvature related features are curves: ridges or crests, and points: umbilics.

Estimation of Local Differential Properties of Point-Sampled Surfaces

Estimation of Local Differential Properties  of Point-Sampled Surfaces Illustration

Marc Pouget and Frédéric Cazals

Introduced in: Cgal 3.3
Depends on: Solvers as Eigen, or Lapack and Blas
License: GPL
BibTeX: [cgal:pc-eldp-12]
User Manual   Reference Manual
For a surface discretized as a point cloud or a mesh, it is desirable to estimate pointwise differential quantities. More precisely, first order properties correspond to the normal or the tangent plane; second order properties provide the principal curvatures and directions, third order properties provide the directional derivatives of the principal curvatures along the curvature lines, etc. This package allows the estimation of local differential quantities of a surface from a point sample.

Point Set Processing

Point Set Processing Illustration

Pierre Alliez, Laurent Saboret, Nader Salman

Introduced in: Cgal 3.5
Depends on: Lapack
License: GPL
BibTeX: [cgal:ass-psp-12]
Demo: Surface Reconstruction
User Manual   Reference Manual
This Cgal component implements methods to analyze and process unorganized point sets. The input is an unorganized point set, possibly with normal attributes (unoriented or oriented). The point set can be analyzed to measure its average spacing, and processed through functions devoted to the simplification, outlier removal, smoothing, normal estimation and normal orientation.


XIII   Spatial Searching and Sorting

2D Range and Neighbor Search

2D Range and Neighbor Search Illustration

Matthias Bäsken

Introduced in: Cgal 2.1
Depends on: 2D Delaunay triangulation
License: GPL
BibTeX: [cgal:b-ss2-12]
User Manual   Reference Manual
This package supports circular, triangular, and isorectangular range search queries as well as (k) nearest neighbor search queries on 2D point sets. In contrast to the spatial searching package, this package uses a Delaunay triangulation as underlying data structure.

Interval Skip List

Interval Skip List Illustration

Andreas Fabri

Introduced in: Cgal 3.0
License: GPL
BibTeX: [cgal:f-isl-12]
User Manual   Reference Manual
An interval skip list is a data structure for finding all intervals that contain a point, and for stabbing queries, that is for answering the question whether a given point is contained in an interval or not. For a triangulated terrain, this allows to quickly identify the triangles which intersect an iso line.

dD Spatial Searching

dD Spatial Searching Illustration

Hans Tangelder and Andreas Fabri

Introduced in: Cgal 3.0
License: GPL
BibTeX: [cgal:tf-ssd-12]
Demo: 2D Spatial Searching
User Manual   Reference Manual
This package implements exact and approximate distance browsing by providing exact and approximate algorithms for range searching, k-nearest and k-furthest neighbor searching, as well as incremental nearest and incremental furthest neighbor searching, where the query items are points in dD Euclidean space.

dD Range and Segment Trees

dD Range and Segment Trees  Illustration

Gabriele Neyer

Introduced in: Cgal 0.9
License: GPL
BibTeX: [cgal:n-rstd-12]
User Manual   Reference Manual
Range and segment trees allow to perform window queries on point sets, and to enumerate all ranges enclosing a query point. The provided data structures are static and they are optimized for fast queries.

Intersecting Sequences of dD Iso-oriented Boxes

Intersecting Sequences of dD Iso-oriented Boxes Illustration

Lutz Kettner, Andreas Meyer, and Afra Zomorodian

Introduced in: Cgal 3.1
License: GPL
BibTeX: [cgal:kmz-isiobd-12]
Demo: Operations on Polyhedra
User Manual   Reference Manual
An efficient algorithm for finding all intersecting pairs for large numbers of iso-oriented boxes, in order to apply a user defined callback on them. Typically these boxes will be bounding boxes of more complicated geometries. The algorithm is useful for (self-) intersection tests of surfaces etc.

3D Fast Intersection and Distance Computation (AABB Tree)

3D Fast Intersection and Distance Computation (AABB Tree) Illustration

Pierre Alliez, Stéphane Tayeb, Camille Wormser

Introduced in: Cgal 3.5
License: GPL
BibTeX: [cgal:atw-aabb-12]
Demo: AABB Tree
User Manual   Reference Manual
The AABB (axis-aligned bounding box) tree component offers a static data structure and algorithms to perform efficient intersection and distance queries on sets of finite 3D geometric objects.

Spatial Sorting

Spatial Sorting  Illustration

Christophe Delage and Olivier Devillers

Introduced in: Cgal 3.3
License: LGPL
BibTeX: [cgal:dd-ss-12]
User Manual   Reference Manual
This package provides functions for sorting geometric objects in two, three and higher dimensions, in order to improve efficiency of incremental geometric algorithms.


XIV   Geometric Optimization

Bounding Volumes

Bounding Volumes  Illustration

Kaspar Fischer, Bernd Gärtner, Thomas Herrmann, Michael Hoffmann, and Sven Schönherr

Introduced in: Cgal 1.1
License: GPL
BibTeX: [cgal:fghhs-bv-12]
Demo: 2D Bounding Volumes
User Manual   Reference Manual
This package provides algorithms for computing optimal bounding volumes of point sets. In d-dimensional space, the smallest enclosing sphere, ellipsoid (approximate), and annulus can be computed. In 3-dimensional space, the smallest enclosing strip is available as well, and in 2-dimensional space, there are algorithms for a number of additional volumes (rectangles, parallelograms, k=2,3,4 axis-aligned rectangles). The smallest enclosing sphere algorithm can also be applied to a set of d-dimensional spheres.

Inscribed Areas

Inscribed Areas  Illustration

Michael Hoffmann and Eli Packer

This package provides algorithms for computing inscribed areas. The algorithms for computing inscribed areas are: the largest inscribed k-gon (area or perimeter) of a convex point set and the largest inscribed iso-rectangle.

Optimal Distances

Optimal Distances  Illustration

Kaspar Fischer, Bernd Gärtner, Thomas Herrmann, Michael Hoffmann, and Sven Schönherr

Introduced in: Cgal 1.1
License: GPL
BibTeX: [cgal:fghhs-od-12]
User Manual   Reference Manual
This package provides algorithms for computing the distance between the convex hulls of two point sets in d-dimensional space, without explicitely constructing the convex hulls. It further provides an algorithm to compute the width of a point set, and the furthest point for each vertex of a convex polygon.

Principal Component Analysis

Principal Component Analysis Illustration

Pierre Alliez, Sylvain Pion and Ankit Gupta

This package provides functions to compute global information about the shape of a set of 2D or 3D objects. It provides the computation of axis-aligned bounding boxes for point sets, and barycenters of weighted point sets. In addition, it provides computation of centroids (center of mass) and linear least squares fitting for point sets as well as for sets of other bounded objects. More specifically, it is possible to fit 2D lines to 2D segments, circles, disks, iso rectangles and triangles, as well as to fit 3D lines or 3D planes to 3D segments, triangles, iso cuboids, tetrahedra, spheres and balls. The common interface to these functions takes an iterator range of objects.


XV   Interpolation

2D and Surface Function Interpolation

2D and Surface Function Interpolation Illustration

Julia Flötotto

Introduced in: Cgal 3.1
License: GPL
BibTeX: [cgal:f-i-12]
User Manual   Reference Manual
This package implements different methods for scattered data interpolation: Given measures of a function on a set of discrete data points, the task is to interpolate this function on an arbitrary query point. The package further offers functions for natural neighbor interpolation.


XVI   Kinetic Data Structures

Kinetic Data Structures

Kinetic Data Structures Illustration

Daniel Russel

Introduced in: Cgal 3.2
Depends on: KDS Framework. Two dimensional visualization depends on Qt, three dimensional visualization depends on Coin.
License: LGPL
BibTeX: [cgal:r-kds-12]
User Manual   Reference Manual
Kinetic data structures allow combinatorial structures to be maintained as the primitives move. The package provides implementations of kinetic data structures for Delaunay triangulations in two and three dimensions, sorting of points in one dimension and regular triangulations in three dimensions. The package supports exact or inexact operations on primitives which move along polynomial trajectories.

Kinetic Framework

 

Daniel Russel

Introduced in: Cgal 3.2
Depends on: Two dimensional visualization depends on Qt, three dimensional visualization depends on Coin.
License: LGPL
BibTeX: [cgal:r-kdsf-12]
User Manual   Reference Manual
Kinetic data structures allow combinatorial geometric structures to be maintained as the primitives move. The package provides a framework to ease implementing and debugging kinetic data structures. The package supports exact or inexact operations on primitives which move along polynomial trajectories.


XVII   Support Library

STL Extensions for CGAL

STL Extensions for CGAL Illustration

Michael Hoffmann, Lutz Kettner, Sylvain Pion, and Ron Wein

Introduced in: Cgal 1.0
License: LGPL
BibTeX: [cgal:hkpw-se-12]
User Manual   Reference Manual
Cgal is designed in the spirit of the generic programming paradigm to work together with the Standard Template Library (STL). This package provides non-geometric STL-like algorithms and datastructures that are not provided in the STL standard but in Cgal

CGAL and the Boost Graph Library

CGAL and the Boost Graph Library Illustration

Andreas Fabri, Fernando Cacciola, and Ron Wein

Introduced in: Cgal 3.3
License: LGPL
BibTeX: [cgal:cfw-cbgl-12]
User Manual   Reference Manual
This package provides a framework for interfacing Cgal data structures with the algorithms of the BGL. It allows to run graph algorithms directly on Cgal data structures which are model of the BGL graph concepts, for example the shortest path algorithm on a Delaunay triangulation in order to compute the Euclidean minimum spanning tree. Furthermore, it introduces a new graph concept, the HalfedgeEdgeGraph. This concept describes graphs which are embedded on surfaces.

CGAL and Boost Property Maps

CGAL and Boost Property Maps Illustration

Andreas Fabri and Laurent Saboret

Introduced in: Cgal 3.5
License: GPL
BibTeX: [cgal:fs-cbpm-12]
User Manual   Reference Manual
This package provides a framework for interfacing Cgal data structures with algorithms expecting Boost Property Maps.

Handles and Circulators

Handles and Circulators Illustration

Olivier Devillers, Lutz Kettner, Sylvain Pion, Michael Seel, and Mariette Yvinec

Introduced in: Cgal 1.0
License: LGPL
BibTeX: [cgal:dksy-hc-12]
User Manual   Reference Manual
This package descibes handles and circulators. They are related to iterators. Handles allow to dereference but neither to increment nor to decrement. Circulators have no notion of past-the-end, and they are used in Cgal whenever we have cyclic stuctures.

Geometric Object Generators

Geometric Object Generators Illustration

Olivier Devillers, Susan Hert, Michael Hoffmann, Lutz Kettner, and Sven Schönherr

Introduced in: Cgal 1.0
License: LGPL
BibTeX: [cgal:dhhk-gog-12]
Demo: Generators
User Manual   Reference Manual
This package provides a variety of generators for geometric objects. They are useful as synthetic test data sets, e.g. for testing algorithms on degenerate object sets and for performance analysis.

Profiling tools, Hash Map, Union-find, Modifiers

Profiling tools, Hash Map, Union-find, Modifiers  Illustration

Lutz Kettner, Sylvain Pion, and Michael Seel

Introduced in: Cgal 3.2
License: LGPL
BibTeX: [cgal:kps-pthum-12]
User Manual   Reference Manual
This package provides classes for profiling time and memory consumption, a hash map, a union find data structure and a modifier.

IO Streams

IO Streams Illustration

Andreas Fabri, Geert-Jan Giezeman, and Lutz Kettner

Introduced in: Cgal 1.0
License: LGPL
BibTeX: [cgal:fgk-ios-12]
User Manual   Reference Manual
All classes in the Cgal kernel provide input and output operators for IO streams. The basic task of such an operator is to produce a representation of an object that can be written as a sequence of characters on devices as a console, a file, or a pipe. In Cgal we distinguish between a raw ascii, a raw binary and a pretty printing format.


XVIII   Visualization

Geomview

Geomview Illustration

Andreas Fabri and Sylvain Pion

Introduced in: Cgal 2.0
License: LGPL
BibTeX: [cgal:fp-gv-12]
User Manual   Reference Manual

This package implements an interface to Geomview, an interactive 3D viewing program, originally developed at the Geometry Center in Minneapolis.

CGAL and the Qt Graphics View Framework

CGAL and the Qt Graphics View Framework  Illustration

Andreas Fabri and Laurent Rineau

Introduced in: Cgal 3.4
Depends on: Qt 4
License: GPL
BibTeX: [cgal:fr-cqgvf-12]
User Manual   Reference Manual
This package provides classes for displaying Cgal objects and data structures in the Qt 4 Graphics View Framework.

CGAL Ipelets

CGAL Ipelets  Illustration

Sébastien Loriot and Sylvain Pion

Introduced in: Cgal 3.5
License: LGPL
BibTeX: [cgal:lp-gi-12]
User Manual   Reference Manual
This package provides a generic framework to easily write ipelets (plug-in's) using Cgal for the the Ipe extensible drawing editor.