# Package Overview

 I General Introduction II Arithmetic and Algebra III Combinatorial Algorithms IV Geometry Kernels V Convex Hull Algorithms VI Polygons VII 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

CGAL Editorial Board

This chapter explains how the manual is organized, presents a ``Hello World'' program, and gives recommendations for further readings.

Introduced in: Cgal 1.0
Citation Entry
User Manual   Reference Manual

### Preliminaries

CGAL Editorial Board

This chapter lists the licenses under which the Cgal datastructures and algorithms are distributed, and the third party libraries on which Cgal depends, or for which Cgal provides interfaces. The chapter further explains how to control inlining, thread safety, code deprecation, checking of pre- and postconditions, and how to alter the failure behavior.

Finally, 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. Note that on computer that does not have Visual C++ 2005 installed, you probably also need the Microsoft Visual C++ 2005 SP1 Redistributable Package (x86).

Introduced in: Cgal 1.0
Citation Entry
Demo: dlls used by all demos
User Manual   Reference Manual

### Installation

Joachim Reichel and Fernando Cacciola

This chapter describes how to install CGAL.

Introduced in: Cgal 3.6
Citation Entry
User Manual   Reference Manual

## II   Arithmetic and Algebra

### Algebraic Foundations

Michael Hemmer

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).

Introduced in: Cgal 3.3
Citation Entry
User Manual   Reference Manual

### Number Types

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

This package provides number type concepts as well as number type classes and wrapper classes for third party number type libraries.

Introduced in: Cgal 1.0
Citation Entry
User Manual   Reference Manual

### Modular Arithmetic

Michael Hemmer

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.

Introduced in: Cgal 3.4
Citation Entry
User Manual   Reference Manual

### Polynomial

Michael Hemmer

This package introduces a concept Polynomial_d, a concept for 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 Polynomial_d. The concept covers univariate polynomials as well.

Introduced in: Cgal 3.3
Citation Entry
User Manual   Reference Manual

### Algebraic Kernel

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

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.

Introduced in: Cgal 3.6
Depends on: Some models depend on RS.
Citation Entry
User Manual   Reference Manual

## III   Combinatorial Algorithms

### Monotone and Sorted Matrix Search

Michael Hoffmann

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.

Introduced in: Cgal 1.1
Citation Entry
User Manual   Reference Manual

### Linear and Quadratic Programming Solver

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

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.

Introduced in: Cgal 3.3
Citation Entry
User Manual   Reference Manual

## IV   Geometry Kernels

### 2D and 3D Linear Geometry Kernel

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

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.

Introduced in: Cgal 0.9
Citation Entry
Demo: Robustness
User Manual   Reference Manual

### dD Geometry Kernel

Michael Seel

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.

Introduced in: Cgal 1.1
Citation Entry
User Manual   Reference Manual

### 2D Circular Geometry Kernel

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

This package is an extension of the linear Cgal kernel. It offers functionalities on circles, circular arcs and line segments in the plane.

Introduced in: Cgal 3.2
Citation Entry
Demo: Arrangement of Circular Arcs
User Manual   Reference Manual

### 3D Spherical Geometry Kernel

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

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.

Introduced in: Cgal 3.4
Citation Entry
User Manual   Reference Manual

## V   Convex Hull Algorithms

### 2D Convex Hulls and Extreme Points

Susan Hert and Stefan Schirra

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.

Introduced in: Cgal 1.0
Citation Entry
Demo: See Bounding Volumes
User Manual   Reference Manual

### 3D Convex Hulls

Susan Hert and Stefan Schirra

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.

Introduced in: Cgal 1.1
Depends on: All algorithms produce as output a 3D Polyhedron. The dynamic algorithms depend on 3D Triangulations
Citation Entry
Demo: See Polyhedral Surface
User Manual   Reference Manual

### dD Convex Hulls and Delaunay Triangulations

Susan Hert and Michael Seel

This package provides functions for computing convex hulls and Delaunay triangulations in d-dimensional Euclidean space.

Introduced in: Cgal 2.3
Citation Entry
User Manual   Reference Manual

## VI   Polygons

### 2D Polygon

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.

Introduced in: Cgal 0.9
Citation Entry
Demo: Operations on Polygons
User Manual   Reference Manual

### 2D Regularized Boolean Set-Operations

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

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.

Introduced in: Cgal 3.2
Depends on: 2D Arrangements
Citation Entry
Demo: Boolean operations
User Manual   Reference Manual

### 2D Boolean Operations on Nef Polygons

Michael Seel

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.

Introduced in: Cgal 2.3
Citation Entry
Demo: 2D Nef Polygons
User Manual   Reference Manual

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

Peter Hachenberger, Lutz Kettner, and Michael Seel

This package offers the equivalent to 2D Nef Polygons in the plane. Here halfplanes correspond to half spheres delimited by great circles.

Introduced in: Cgal 3.1
Depends on: 2D Nef Polygon
Citation Entry
User Manual   Reference Manual

### 2D Polygon Partitioning

Susan Hert

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.

Introduced in: Cgal 2.3
Citation Entry
Demo: 2D Polygon Partition
Demo: Operations on Polygons
User Manual   Reference Manual

### 2D Straight Skeleton and Polygon Offsetting

Fernando Cacciola

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.

Introduced in: Cgal 3.2
Depends on: Halfedge Data Structure
Citation Entry
Demo: 2D Straight Skeleton
Demo: Operations on Polygons
User Manual   Reference Manual

### 2D Minkowski Sums

Ron Wein

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.

Introduced in: Cgal 3.3
Depends on: 2D Arrangements
Citation Entry
User Manual   Reference Manual

## VII   Polyhedra

### 3D Polyhedral Surface

Lutz Kettner

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.

Introduced in: Cgal 1.0
Depends on: Halfedge Data Structure
Citation Entry
Demo: Operations on Polyhedra
User Manual   Reference Manual

### Halfedge Data Structures

Lutz Kettner

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.

Introduced in: Cgal 1.0
Citation Entry
User Manual   Reference Manual

### 3D Boolean Operations on Nef Polyhedra

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.

Introduced in: Cgal 3.1
Depends on: 2D Nef Polygons, Nef Polygons on the Sphere
Citation Entry
Demo: Operations on Polyhedra
User Manual   Reference Manual

### Convex Decomposition of Polyhedra

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.

Introduced in: Cgal 3.5
Depends on: 2D Nef Polygons, Nef Polygons on the Sphere 3D Nef Polyhedra,
Citation Entry
User Manual   Reference Manual

### 3D Minkowski Sum of Polyhedra

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.

Introduced in: Cgal 3.5
Depends on: 2D Nef Polygons, Nef Polygons on the Sphere 3D Nef Polyhedra, Convex Decomposition of Polyhedra
Citation Entry
Demo: Operations on Polyhedra
User Manual   Reference Manual

## VIII   Arrangements

### 2D Arrangement

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

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.

Introduced in: Cgal 2.1
Citation Entry
User Manual   Reference Manual

### 2D Intersection of Curves

Baruch Zukerman and Ron Wein

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.

Introduced in: Cgal 2.4
Depends on: 2D Arrangements
Citation Entry
User Manual   Reference Manual

### 2D Snap Rounding

Eli Packer

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.

Introduced in: Cgal 3.1
Depends on: Arrangements
Citation Entry
Demo: 2D Snap Rounding
User Manual   Reference Manual

### 2D Envelopes

Ron Wein

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.

Introduced in: Cgal 3.3
Depends on: 2D Arrangements
Citation Entry
User Manual   Reference Manual

### 3D Envelopes

Dan Halperin, Michal Meyerovitch, Ron Wein and Baruch Zukerman

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.

Introduced in: Cgal 3.3
Depends on: 2D Arrangements
Citation Entry
Demo: 3D Envelopes
User Manual   Reference Manual

## IX   Triangulations and Delaunay Triangulations

### 2D Triangulation

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.

Introduced in: Cgal 0.9
Depends on: 2D Triangulation Data Structure
Citation Entry
Demo: Delaunay Triangulation
Demo: Regular Triangulation
Demo: Constrained Delaunay Triangulation
User Manual   Reference Manual

### 2D Triangulation Data Structure

Sylvain Pion and Mariette Yvinec

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.

Introduced in: Cgal 2.2
Citation Entry
User Manual   Reference Manual

### 3D Triangulations

Sylvain Pion and Monique Teillaud

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.

Introduced in: Cgal 2.1
Citation Entry
User Manual   Reference Manual

### 3D Triangulation Data Structure

Sylvain Pion and Monique Teillaud

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.

Introduced in: Cgal 2.1
Citation Entry
User Manual   Reference Manual

### 3D Periodic Triangulations

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.

Introduced in: Cgal 3.5
Depends on: 3D Triangulation and 3D Triangulation Data Structure
Citation Entry
Demo: Periodic Delaunay Triangulation
User Manual   Reference Manual

### 2D Alpha Shapes

Tran Kai Frank Da

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).

Introduced in: Cgal 2.1
Depends on: 2D Triangulation
Citation Entry
Demo: 2D Alpha Shapes
User Manual   Reference Manual

### 3D Alpha Shapes

Tran Kai Frank Da and Mariette Yvinec

This package offers a data structure encoding the whole family of alpha-complexes related to a given 3D 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).

Introduced in: Cgal 2.3
Depends on: 2D Triangulation
Citation Entry
Demo: 3D Alpha Shapes
User Manual   Reference Manual

## X   Voronoi Diagrams

### 2D Segment Delaunay Graphs

Menelaos Karavelas

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.

Introduced in: Cgal 3.1
Depends on: 2D Triangulation Data Structure
Citation Entry
Demo: 2D Segment Voronoi Diagram
User Manual   Reference Manual

### 2D Apollonius Graphs (Delaunay Graphs of Disks)

Menelaos Karavelas and Mariette Yvinec

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.

Introduced in: Cgal 3.0
Depends on: 2D Triangulation Data Structure
Citation Entry
Demo: 2D Apollonius Graph
User Manual   Reference Manual

Menelaos Karavelas

The 2D Voronoi diagram adaptor package provides an adaptor that adapts a 2-dimensional triangulated Delaunay graph to the corresponding Voronoi diagram, represented as a doubly connected edge list (DCEL) data structure. The adaptor has the ability to automatically eliminate, in a consistent manner, degenerate features of the Voronoi diagram, that are artifacts of the requirement that Delaunay graphs should be triangulated even in degenerate configurations. Depending on the type of operations that the underlying Delaunay graph supports, the adaptor allows for the incremental or dynamic construction of Voronoi diagrams and can support point location queries.

Introduced in: Cgal 3.2
Citation Entry
User Manual   Reference Manual

## XI   Mesh Generation

### 2D Conforming Triangulations and Meshes

Laurent Rineau

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.

Introduced in: Cgal 3.1
Depends on: 2D Delaunay Triangulation
Citation Entry
Demo: 2D Mesh Generator
User Manual   Reference Manual

### 3D Surface Mesh Generation

Laurent Rineau and Mariette Yvinec

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.

Introduced in: Cgal 3.2
Citation Entry
Demo: Surface Mesh Generator
User Manual   Reference Manual

### Surface Reconstruction from Point Sets

Pierre Alliez, Laurent Saboret, Gael Guennebaud

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.

Introduced in: Cgal 3.5
Depends on: Taucs
Citation Entry
Demo: Point Set Processing
User Manual   Reference Manual

### 3D Skin Surface Meshing

Nico Kruithof

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.

Introduced in: Cgal 3.3
Depends on: 3D Triangulation and 3D Polyhedral Surface
Citation Entry
User Manual   Reference Manual

### 3D Mesh Generation

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

The generated meshes are 3 dimensional isotropic simplicial meshes. The discretized region may be a monodomain, i.e. a single connected component, or a multidomain subdivided into different subdomains. The domain is input as an oracle able to answer queries, of a few different types, on the domain. In the current version, the mesh generator is able to discretize domains described through implicit functions, 3D images or polyhedral boundaries. The output is a 3D mesh of the domain volume and conformal surface meshes for all the boundary and subdividing surfaces.

Introduced in: Cgal 3.5
Depends on: 3D Triangulations, 2D Meshes, 3D Surface Mesh Generation
Citation Entry
User Manual   Reference Manual

## XII   Geometry Processing

### 3D Surface Subdivision Methods

Le-Jeng Andy Shiue

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.

Introduced in: Cgal 3.2
Citation Entry
Demo: Operations on Polyhedra
User Manual   Reference Manual

### Triangulated Surface Mesh Simplification

Fernando Cacciola

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.

Introduced in: Cgal 3.3
Depends on: Polyhedron
Citation Entry
Demo: Operations on Polyhedra
User Manual   Reference Manual

### Planar Parameterization of Triangulated Surface Meshes

Laurent Saboret, Pierre Alliez and Bruno Lévy

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.

Introduced in: Cgal 3.2
Depends on: Solvers as OpenNL or Taucs.
Citation Entry
Demo: Operations on Polyhedra
User Manual   Reference Manual

### 2D Placement of Streamlines

Abdelkrim Mebarki

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.

Introduced in: Cgal 3.2
Depends on: 2D Delaunay triangulation
Citation Entry
Demo: 2D Stream Lines
User Manual   Reference Manual

### Approximation of Ridges and Umbilics on Triangulated Surface Meshes

Marc Pouget and Frédéric Cazals

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.

Introduced in: Cgal 3.3
Depends on: Solvers as Lapack and Blas.
Citation Entry
User Manual   Reference Manual

### Estimation of Local Differential Properties

Marc Pouget and Frédéric Cazals

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.

Introduced in: Cgal 3.3
Depends on: Solvers as Lapack and Blas.
Citation Entry
Demo: Operations on Polyhedra
User Manual   Reference Manual

### Point Set Processing

Pierre Alliez, Laurent Saboret, Nader Salman

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.

Introduced in: Cgal 3.5
Depends on: Lapack
Citation Entry
Demo: Point Set Processing
User Manual   Reference Manual

## XIII   Spatial Searching and Sorting

### 2D Range and Neighbor Search

Matthias Bäsken

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.

Introduced in: Cgal 2.1
Depends on: 2D Delaunay triangulation
Citation Entry
User Manual   Reference Manual

### Interval Skip List

Andreas Fabri

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.

Introduced in: Cgal 3.0
Citation Entry
User Manual   Reference Manual

### dD Spatial Searching

Hans Tangelder and Andreas Fabri

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.

Introduced in: Cgal 3.0
Citation Entry
Demo: 2D Spatial Searching
User Manual   Reference Manual

### dD Range and Segment Trees

Gabriele Neyer

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.

Introduced in: Cgal 0.9
Citation Entry
User Manual   Reference Manual

### Intersecting Sequences of dD Iso-oriented Boxes

Lutz Kettner, Andreas Meyer, and Afra Zomorodian

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.

Introduced in: Cgal 3.1
Citation Entry
Demo: Operations on Polyhedra
User Manual   Reference Manual

### AABB Tree

Pierre Alliez, Stéphane Tayeb, Camille Wormser

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.

Introduced in: Cgal 3.5
Citation Entry
Demo: AABB Tree
User Manual   Reference Manual

### Spatial Sorting

Christophe Delage

This package provides functions for sorting geometric objects in two and three dimensions, in order to improve efficiency of incremental geometric algorithms.

Introduced in: Cgal 3.3
Citation Entry
User Manual   Reference Manual

## XIV   Geometric Optimization

### Bounding Volumes

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

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.

Introduced in: Cgal 1.1
Citation Entry
Demo: 2D Bounding Volumes
User Manual   Reference Manual

### Inscribed Areas

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.

Introduced in: Cgal 1.1
Citation Entry
Demo: 2D Inscribed k-gon
Demo: 2D Largest Empty Rectangle
User Manual   Reference Manual

### Optimal Distances

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

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.

Introduced in: Cgal 1.1
Citation Entry
User Manual   Reference Manual

### Principal Component Analysis

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.

Introduced in: Cgal 3.2
Citation Entry
Demo: Operations on Polygons
Demo: Operations on Polyhedra
User Manual   Reference Manual

## XV   Interpolation

### 2D and Surface Function Interpolation

Julia Flötotto

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.

Introduced in: Cgal 3.1
Citation Entry
User Manual   Reference Manual

## XVI   Kinetic Data Structures

### Kinetic Data Structures

Daniel Russel

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.

Introduced in: Cgal 3.2
Depends on: KDS Framework. Two dimensional visualization depends on Qt, three dimensional visualization depends on Coin.
Citation Entry
User Manual   Reference Manual

### Kinetic Framework

Daniel Russel

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.

Introduced in: Cgal 3.2
Depends on: Two dimensional visualization depends on Qt, three dimensional visualization depends on Coin.
Citation Entry
User Manual   Reference Manual

## XVII   Support Library

### STL Extensions for CGAL

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

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

Introduced in: Cgal 1.0
Citation Entry
User Manual   Reference Manual

### CGAL and the Boost Graph Library

Andreas Fabri, Fernando Cacciola, and Ron Wein

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.

Introduced in: Cgal 3.3
Citation Entry
User Manual   Reference Manual

### CGAL and Boost Property Maps

Andreas Fabri and Laurent Saboret

This package provides a framework for interfacing Cgal data structures with algorithms expecting Boost Property Maps.

Introduced in: Cgal 3.5
Citation Entry
User Manual   Reference Manual

### Handles and Circulators

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

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.

Introduced in: Cgal 1.0
Citation Entry
User Manual   Reference Manual

### Geometric Object Generators

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

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.

Introduced in: Cgal 1.0
Citation Entry
User Manual   Reference Manual

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

Lutz Kettner, Sylvain Pion, and Michael Seel

This package provides classes for profiling time and memory consumption, a hash map, a union find data structure and a modifier.

Introduced in: Cgal 3.2
Citation Entry
User Manual   Reference Manual

### IO Streams

Andreas Fabri, Geert-Jan Giezeman, and Lutz Kettner

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.

Introduced in: Cgal 1.0
Citation Entry
User Manual   Reference Manual

## XVIII   Visualization

### Geomview

Andreas Fabri and Sylvain Pion

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

Introduced in: Cgal 2.0
Citation Entry
User Manual   Reference Manual

### CGAL and the Qt Graphics View Framework

Andreas Fabri and Laurent Rineau

This package provides classes for displaying Cgal objects and data structures in the Qt 4 Graphics View Framework.

Introduced in: Cgal 3.4
Depends on: Qt 4
Citation Entry
User Manual   Reference Manual

### CGAL Ipelets

Sébastien Loriot and Sylvain Pion

This package provides a generic framework to easily write ipelets (plug-in's) using Cgal for the the Ipe extensible drawing editor.

Introduced in: Cgal 3.5