3D Triangulations in Pierre Alliez

3D Triangulations in3D Triangulations in

http://www.cgal.orghttp://www.cgal.org

Pierre AlliezPierre Alliez

http://http://www.cgal.orgwww.cgal.org

3D Triangulation3D Triangulation

TetrahedronTetrahedron


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• DefinitionsDefinitions• RepresentationRepresentation• Types of TriangulationsTypes of Triangulations• Software DesignSoftware Design

– Geometric TraitsGeometric Traits– CustomizationCustomization

• ExamplesExamples• ApplicationsApplications


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DefinitionsDefinitions

3D Triangulation3D Triangulation of a point set A: of a point set A: partition of the convex hull of A partition of the convex hull of A into tetrahedrainto tetrahedra..


DefinitionsDefinitions

• The cells (3-faces) are such that The cells (3-faces) are such that two cells either do not intersect or two cells either do not intersect or share a common facet (2-face), share a common facet (2-face), edge (1-face) or vertex (0-face).edge (1-face) or vertex (0-face).


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RepresentationRepresentation

• CGAL Triangulation: CGAL Triangulation: partition of partition of IRIR3 3 (add unbounded cell).(add unbounded cell).

• To deal only with Tetrahedra:To deal only with Tetrahedra:– subdivide unbounded cell into subdivide unbounded cell into

tetrahedra.tetrahedra."infinite"infinite" vertex" vertex


RepresentationRepresentation

• Dealing only with Dealing only with TetrahedraTetrahedra::– each facet incident to exactly two each facet incident to exactly two

cells.cells.


3D Triangulation in 3D Triangulation in CGALCGAL

Input point setInput point set

TriangulationTriangulation– Finite tetrahedraFinite tetrahedra– One infinite vertexOne infinite vertex– Infinite tetrahedraInfinite tetrahedra


Infinite Vertex?Infinite Vertex?

• No valid coordinatesNo valid coordinates– no predicates applicableno predicates applicable


Internal Internal RepresentationRepresentation

Collection of Collection of verticesvertices and and cellscells linked linked together through incidence and together through incidence and adjacency relations (faces and edges adjacency relations (faces and edges are not explicitly represented).are not explicitly represented).

verticesvertices cellcell


CellCell

Access to:Access to:• 4 incident 4 incident verticesvertices• 4 adjacent 4 adjacent cellscells


VertexVertex

Access to:Access to:• 1 1 incident cellincident cell


Orientation of a CellOrientation of a Cell

4 vertices indexed in 4 vertices indexed in positive positive orientationorientation (induced by (induced by underlying underlying Euclidean space Euclidean space IRIR33).).

00

11

22

33

xx

yy

zz


Adjacent Cell IndexingAdjacent Cell Indexing

• Neighboring Neighboring cell indexed by cell indexed by ii oppositeopposite to to vertex vertex ii..

ii

iithth neighboring neighboringcellcell

cellcell


ii

FaceFace

Represented as a Represented as a pairpair

{cell, index {cell, index ii}}


ii

jj

EdgeEdge

Represented as a Represented as a triplettriplet

{cell,index {cell,index ii,index ,index jj}}


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3 Types of 3 Types of TriangulationsTriangulations

• DelaunayDelaunay• RegularRegular• HierarchyHierarchy


Delaunay TriangulationDelaunay Triangulation

• Empty circleEmpty circle property: property:– the circumscribing sphere of each the circumscribing sphere of each

cell does not contain any other cell does not contain any other vertex in its interior. vertex in its interior.

• Uniquely definedUniquely defined except in except in degenerate cases where five points are degenerate cases where five points are cospherical (CGAL computes a unique cospherical (CGAL computes a unique triangulation even in these cases).triangulation even in these cases).



Fully dynamic:Fully dynamic:• point insertionpoint insertion• vertex removalvertex removal




Regular TriangulationRegular Triangulation (weighted Delaunay triangulation)(weighted Delaunay triangulation)

Weighted pointWeighted point::– pp(w)(w) = (p,w = (p,wpp), p ), p IR IR33, w, wpp IR. IR.

pwp



Power productPower product:: (p(p(w)(w), z, z(w)(w)) = ||p-z||) = ||p-z||2 2 - w- wp p - w- wzz

– pp(w)(w) & z & z(w) (w) said to be said to be orthogonalorthogonal iff = 0 iff = 0



4 weighted points have a unique 4 weighted points have a unique common orthogonal weighted common orthogonal weighted point called point called power spherepower sphere. .


RegularRegular Triangulation Triangulation (weighted Delaunay triangulation)(weighted Delaunay triangulation)

• Let SLet S(w) (w) be a set of weighted points.be a set of weighted points.• A sphere zA sphere z(w) (w) is said to be regular if is said to be regular if

for all pfor all p(w)(w) S S(w)(w), , (p(p(w)(w), z, z(w)(w)) >= 0) >= 0

• A triangulation is A triangulation is regularregular if the if the power spheres of all simplices are power spheres of all simplices are regularregular..


Triangulation HierarchyTriangulation Hierarchy

Triangulation augmented with a Triangulation augmented with a hierarchical data structurehierarchical data structure to allow to allow for efficient for efficient point location queriespoint location queries..


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Derivation DiagramDerivation Diagram

(operate on vertex indices in cells)


Software DesignSoftware Design

Separation between Separation between geometrygeometry & & combinatoricscombinatorics..

Triangulation_3Triangulation_3<<

TriangulationTraits_3, TriangulationTraits_3, TriangulationDataStructure_3>TriangulationDataStructure_3>


Geometric TraitsGeometric Traits

Defines Defines geometric objectsgeometric objects & & predicatespredicates..

Objects:Objects:• pointpoint• segmentsegment• triangletriangle• tetrahedrontetrahedron• [weighted point][weighted point]

Predicates:Predicates:• orientation in spaceorientation in space• orientation of coplanar orientation of coplanar

pointspoints• order of collinear pointsorder of collinear points• empty sphere propertyempty sphere property• [power test][power test]


Triangulation Data Triangulation Data StructureStructure


Customization Customization (cells & (cells & vertices)vertices)


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ConstructionConstruction#include#include <CGAL/Simple_cartesian.h> <CGAL/Simple_cartesian.h>#include#include <CGAL/Triangulation_3.h> <CGAL/Triangulation_3.h>

typedeftypedef CGAL::Simple_cartesian< CGAL::Simple_cartesian<floatfloat> > kernelkernel;;typedeftypedef CGAL::Triangulation_vertex_base_3< CGAL::Triangulation_vertex_base_3<kernelkernel> Vb;> Vb;typedeftypedef CGAL::Triangulation_cell_base_3< CGAL::Triangulation_cell_base_3<kernelkernel> Cb;> Cb;typedeftypedef CGAL::Triangulation_data_structure_3<Vb,Cb> CGAL::Triangulation_data_structure_3<Vb,Cb> TdsTds;;typedeftypedef CGAL::Triangulation_3< CGAL::Triangulation_3<kernelkernel,,TdsTds> Triangulation;> Triangulation;typedeftypedef kernelkernel::Point_3 Point;::Point_3 Point;

Triangulation triangulation;Triangulation triangulation;

// insertion// insertiontriangulation.insert(Point(0,0,0));triangulation.insert(Point(0,0,0));triangulation.insert(Point(0,1,0));triangulation.insert(Point(0,1,0));triangulation.insert(Point(0,0,1));triangulation.insert(Point(0,0,1));

// insertion from a list of points// insertion from a list of pointsstd::list<Point> points;std::list<Point> points;points.push_front(Point(2,0,0));points.push_front(Point(2,0,0));points.push_front(Point(3,2,0));points.push_front(Point(3,2,0));points.push_front(Point(0,1,4));points.push_front(Point(0,1,4));triangulation.insert(points.begin(),points.end());triangulation.insert(points.begin(),points.end());


Point LocationPoint Location#include#include <CGAL/Simple_cartesian.h> <CGAL/Simple_cartesian.h>#include#include <CGAL/Triangulation_3.h> <CGAL/Triangulation_3.h>

typedeftypedef CGAL::Simple_cartesian< CGAL::Simple_cartesian<floatfloat> > kernelkernel;;typedeftypedef CGAL::Triangulation_vertex_base_3< CGAL::Triangulation_vertex_base_3<kernelkernel> Vb;> Vb;typedeftypedef CGAL::Triangulation_cell_base_3< CGAL::Triangulation_cell_base_3<kernelkernel> Cb;> Cb;typedeftypedef CGAL::Triangulation_data_structure_3<Vb,Cb> CGAL::Triangulation_data_structure_3<Vb,Cb> TdsTds;;typedeftypedef CGAL::Triangulation_3< CGAL::Triangulation_3<kernelkernel,,TdsTds> Triangulation;> Triangulation;typedeftypedef kernelkernel::Point_3 Point;::Point_3 Point;

Triangulation triangulation;Triangulation triangulation;

// insertion// insertiontriangulation.insert(Point(0,0,0));triangulation.insert(Point(0,0,0));triangulation.insert(Point(0,1,0));triangulation.insert(Point(0,1,0));

// locate// locate

Locate_type lt;Locate_type lt;

intint li, lj; li, lj;

Point p(0,0,0);Point p(0,0,0);

Cell_handle cell = triangulation.locate(p, lt, li, lj);Cell_handle cell = triangulation.locate(p, lt, li, lj);

assertassert(lt == Triangulation::VERTEX);(lt == Triangulation::VERTEX);

assertassert(cell->vertex(li)->point() == p);(cell->vertex(li)->point() == p);


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ApplicationsApplications

• MeshingMeshing– simulationsimulation– numerical engineeringnumerical engineering

• Reverse EngineeringReverse Engineering– surface reconstructionsurface reconstruction

• Efficient point locationEfficient point location• Etc.Etc.

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3D Triangulations in Pierre Alliez