3D Polyhedral Surfaces in Pierre Alliez.

3D Polyhedral Surfaces in3D Polyhedral Surfaces in

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

Pierre AlliezPierre Alliez

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

OutlineOutline• MotivationsMotivations• DefinitionDefinition• Halfedge Data StructureHalfedge Data Structure• TraversalTraversal• Euler OperatorsEuler Operators• CustomizationCustomization• Incremental BuilderIncremental Builder• File I/OFile I/O• ExamplesExamples• ApplicationsApplications• ExercisesExercises


MotivationsMotivations

From From renderingrendering......– StaticStatic– Compact storage in arrayCompact storage in array– Traversal over all facetsTraversal over all facets– Attributes per facet/vertexAttributes per facet/vertex


MotivationsMotivations

...to ...to algorithmsalgorithms on meshes on meshes– Dynamic pointer updatesDynamic pointer updates– Dynamic storage in listsDynamic storage in lists– Traversal over incidencesTraversal over incidences


Design GoalDesign Goal

Method: paradigm of Method: paradigm of Generic ProgrammingGeneric Programming

Example: STLExample: STL


DefinitionDefinition

Polyhedral SurfacePolyhedral Surface: boundary : boundary representation of a polyhedron in IRrepresentation of a polyhedron in IR33..


Polyhedral SurfacePolyhedral Surface

Represented by three sets Represented by three sets VV,,EE,,FF and an incidence and an incidence relation on them, restricted to orientable 2-relation on them, restricted to orientable 2-manifolds with boundary. manifolds with boundary.

VV = Vertices in IR = Vertices in IR33

EE = Edges, straight line segments. = Edges, straight line segments.

FF = Facets, simple, planar polygons without holes. = Facets, simple, planar polygons without holes.

Can be extended: edges to curves, facets to Can be extended: edges to curves, facets to curved surfaces.curved surfaces.



Represented by three sets Represented by three sets VV,,EE,,FF and an and an incidence relationincidence relation on them, restricted to on them, restricted to orientable 2-manifolds with boundary. orientable 2-manifolds with boundary.

Edge-based data structure ?Edge-based data structure ?


Edge-centered Data Edge-centered Data StructuresStructures


Winged-Edge Data Winged-Edge Data StructureStructure


Quad-Edge Data Quad-Edge Data StructureStructure


Halfedge Data Halfedge Data StructureStructure


VE-StructureVE-Structure


HalfedgesHalfedges

Require:Require:Oriented Oriented

surfacesurface

Idea:Idea:Consider 2/4 Consider 2/4

ways of ways of accessing an accessing an edgeedge


HalfedgeHalfedgeAssociated with:Associated with:• 1 vertex1 vertex• 1 edge1 edge• 1 facet1 facet

3 references:3 references:• vertexvertex• opposite opposite

halfedgehalfedge• facetfacet


HalfedgesHalfedgesGeometry:Geometry:• verticesvertices

Attributes:Attributes:• on verticeson vertices• on halfedgeson halfedges• on facetson facets

Connectivity:Connectivity:• halfedges onlyhalfedges only


Building BlocksBuilding Blocks


Polyhedral SurfacesPolyhedral Surfaces

Building blocks assembled with C+Building blocks assembled with C++ templates+ templates


Default PolyhedronDefault Polyhedron


Default PolyhedronDefault Polyhedron

typedef CGAL::Simple_cartesian<double> typedef CGAL::Simple_cartesian<double> KernelKernel;;typedef typedef KernelKernel::Point_3 Point_3;::Point_3 Point_3;typedef typedef CGAL::Polyhedron_3CGAL::Polyhedron_3<<KernelKernel> > PolyhedronPolyhedron;;typedef typedef PolyhedronPolyhedron::Vertex_iterator Vertex_iterator;::Vertex_iterator Vertex_iterator;

intint main() { main() { Point_3 p( 1.0, 0.0, 0.0);Point_3 p( 1.0, 0.0, 0.0); Point_3 q( 0.0, 1.0, 0.0);Point_3 q( 0.0, 1.0, 0.0); Point_3 r( 0.0, 0.0, 1.0);Point_3 r( 0.0, 0.0, 1.0); Point_3 s( 0.0, 0.0, 0.0);Point_3 s( 0.0, 0.0, 0.0);

Polyhedron PPolyhedron P;; P.make_tetrahedron( p, q, r, s);P.make_tetrahedron( p, q, r, s); for (Vertex_iterator v = for (Vertex_iterator v = P.vertices_begin()P.vertices_begin(); ; v != v != P.vertices_end()P.vertices_end(); ++v); ++v) std::cout << v->point() << std::endl;std::cout << v->point() << std::endl;}}


Flexible Data StructureFlexible Data Structure


Flexible Polyhedral Flexible Polyhedral SurfacesSurfaces

template <template <

class class PolyhedronTraits_3PolyhedronTraits_3,,

class class PolyhedronItems_3PolyhedronItems_3 = CGAL::Polyhedron_items_3, = CGAL::Polyhedron_items_3,

template < class T, class I>template < class T, class I>

class class HalfedgeDSHalfedgeDS = CGAL::HalfedgeDS_default, = CGAL::HalfedgeDS_default,

class class AllocAlloc = CGAL_ALLOCATOR(int)> = CGAL_ALLOCATOR(int)>

class Polyhedron_3;class Polyhedron_3;


Default PolyhedronDefault Polyhedrontypedef CGAL::Simple_cartesian<double> Kernel;typedef CGAL::Simple_cartesian<double> Kernel;typedef Kernel::Point_3 Point_3;typedef Kernel::Point_3 Point_3;typedef typedef CGAL::Polyhedron_3CGAL::Polyhedron_3<Kernel> <Kernel> PolyhedronPolyhedron;;typedef typedef PolyhedronPolyhedron::Vertex_iterator Vertex_iterator;::Vertex_iterator Vertex_iterator;

int main() {int main() { Point_3 p( 1.0, 0.0, 0.0);Point_3 p( 1.0, 0.0, 0.0); Point_3 q( 0.0, 1.0, 0.0);Point_3 q( 0.0, 1.0, 0.0); Point_3 r( 0.0, 0.0, 1.0);Point_3 r( 0.0, 0.0, 1.0); Point_3 s( 0.0, 0.0, 0.0);Point_3 s( 0.0, 0.0, 0.0);

Polyhedron PPolyhedron P;; P.make_tetrahedron( p, q, r, s);P.make_tetrahedron( p, q, r, s); for ( Vertex_iterator v = for ( Vertex_iterator v = P.vertices_begin()P.vertices_begin(); ; v != v != P.vertices_end()P.vertices_end(); ++v); ++v) std::cout << v->point() << std::endl;std::cout << v->point() << std::endl;}}


Euler OperatorsEuler Operators

• Preserve the Euler-PoincarPreserve the Euler-Poincaré equationé equation• Abstract from direct pointer manipulations Abstract from direct pointer manipulations

Halfedge_handle P.split_facet(Halfedge_handle h,Halfedge_handle g)



Halfedge_handle P.split_vertex ( Halfedge_handle h, Halfedge_handle gHalfedge_handle P.split_vertex ( Halfedge_handle h, Halfedge_handle g

Halfedge_handle P.join_vertex ( Halfedge_handle h)Halfedge_handle P.join_vertex ( Halfedge_handle h)


Modifying the GenusModifying the Genus


Modifying Facets & Modifying Facets & HolesHoles


Create a Cube with Euler Create a Cube with Euler OperatorOperator

Halfedge_handle h = P.make_tetrahedron( Halfedge_handle h = P.make_tetrahedron(

Point(1,0,0), Point(0,0,1),Point(1,0,0), Point(0,0,1),

Point(0,0,0), Point(0,1,0));Point(0,0,0), Point(0,1,0));

Halfedge_handle g = h->next()->opposite()->next();Halfedge_handle g = h->next()->opposite()->next(); (a)(a)



Halfedge_handle h = P.make_tetrahedron( Halfedge_handle h = P.make_tetrahedron(

Point(1,0,0), Point(0,0,1),Point(1,0,0), Point(0,0,1),

Point(0,0,0), Point(0,1,0));Point(0,0,0), Point(0,1,0));

Halfedge_handle g = h->next()->opposite()->next(); Halfedge_handle g = h->next()->opposite()->next(); (a)(a)

P.split_edge( h->next());P.split_edge( h->next());

P.split_edge( g->next());P.split_edge( g->next());

P.split_edge( g);P.split_edge( g); (b)(b)



h->next()->vertex()->point() = Point( 1, 0, 1);h->next()->vertex()->point() = Point( 1, 0, 1);

g->next()->vertex()->point() = Point( 0, 1, 1);g->next()->vertex()->point() = Point( 0, 1, 1);

g->opposite()->vertex()->point() = Point( 1, 1, 0);g->opposite()->vertex()->point() = Point( 1, 1, 0); (c)(c)





g->opposite()->vertex()->point() = Point( 1, 1, 0); g->opposite()->vertex()->point() = Point( 1, 1, 0); (c)(c)

Halfedge_handle f = P.split_facet( g->next(), Halfedge_handle f = P.split_facet( g->next(),

g->next()->next()->next());g->next()->next()->next()); (d)(d)







g->next()->next()->next()); g->next()->next()->next()); (d)(d)

Halfedge_handle e = P.split_edge( f);Halfedge_handle e = P.split_edge( f);

e->vertex()->point() = Point( 1, 1, 1);e->vertex()->point() = Point( 1, 1, 1); (e)(e)







g->next()->next()->next()); g->next()->next()->next()); (d)(d)

Halfedge_handle e = P.split_edge( f);Halfedge_handle e = P.split_edge( f);

e->vertex()->point() = Point( 1, 1, 1); e->vertex()->point() = Point( 1, 1, 1); (e)(e)

P.split_facet( e, f->next()->next());P.split_facet( e, f->next()->next()); (f)(f)


Extending PrimitivesExtending Primitives

typedef CGAL::Polyhedron_3< typedef CGAL::Polyhedron_3< Traits,Traits,

CGAL::Polyhedron_items_3,CGAL::Polyhedron_items_3,

CGAL::HalfedgeDS_default> Polyhedron;CGAL::HalfedgeDS_default> Polyhedron;

classclass Polyhedron_items_3 { Polyhedron_items_3 {

public:public:

template < class Refs, class Traits>template < class Refs, class Traits>

struct struct Vertex_wrapperVertex_wrapper { {

typedef typename Traits::Point_3 Point;typedef typename Traits::Point_3 Point;

typedef CGAL::HalfedgeDS_vertex_base<Refs, CGAL::Tag_true, Point> Vertex;typedef CGAL::HalfedgeDS_vertex_base<Refs, CGAL::Tag_true, Point> Vertex;

};};


struct struct Halfedge_wrapperHalfedge_wrapper { {

typedef CGAL::HalfedgeDS_halfedge_base<Refs> Halfedge;typedef CGAL::HalfedgeDS_halfedge_base<Refs> Halfedge;

};};


struct struct Face_wrapperFace_wrapper { {

typedef typename Traits::Plane_3 Plane;typedef typename Traits::Plane_3 Plane;

typedef CGAL::HalfedgeDS_face_base<Refs, CGAL::Tag_true, Plane> Face;typedef CGAL::HalfedgeDS_face_base<Refs, CGAL::Tag_true, Plane> Face;

};};

};};


Add Color to FacetsAdd Color to Facetstemplate <class Refs>template <class Refs>struct struct CFaceCFace : public CGAL::HalfedgeDS_face_base<Refs>{ : public CGAL::HalfedgeDS_face_base<Refs>{ CGAL::Color colorCGAL::Color color;;};};

// ...// ...

typedef CGAL::Simple_cartesian<double> Kernel;typedef CGAL::Simple_cartesian<double> Kernel;typedef CGAL::Polyhedron_3<Kernel, typedef CGAL::Polyhedron_3<Kernel, ......> Polyhedron;> Polyhedron;typedef Polyhedron::Halfedge_handle Halfedge_handle;typedef Polyhedron::Halfedge_handle Halfedge_handle;

int main() {int main() { Polyhedron P;Polyhedron P; Halfedge_handle h = P.make_tetrahedron();Halfedge_handle h = P.make_tetrahedron(); h->facet()->color = CGAL::RED;h->facet()->color = CGAL::RED; return 0;return 0;}}


Add Color to FacetsAdd Color to Facetstemplate <class Refs>template <class Refs>struct CFace : public CGAL::HalfedgeDS_face_base<Refs>{struct CFace : public CGAL::HalfedgeDS_face_base<Refs>{ CGAL::Color color;CGAL::Color color;};};

struct struct CItemsCItems : public CGAL::Polyhedron_items_3 { : public CGAL::Polyhedron_items_3 { template <class Refs, class Traits>template <class Refs, class Traits> struct Face_wrapper {struct Face_wrapper { typedef CFace<Refs> Face;typedef CFace<Refs> Face; };};};};

typedef CGAL::Simple_cartesian<double> Kernel;typedef CGAL::Simple_cartesian<double> Kernel;typedef CGAL::Polyhedron_3<Kernel, typedef CGAL::Polyhedron_3<Kernel, CItemsCItems> Polyhedron;> Polyhedron;


Add Vertex_handle to Add Vertex_handle to FacetsFacets

template <class template <class RefsRefs>>

struct VFace : public CGAL::HalfedgeDS_face_base<Refs>{struct VFace : public CGAL::HalfedgeDS_face_base<Refs>{

typedef typename typedef typename RefsRefs::::Vertex_handle Vertex_handleVertex_handle Vertex_handle;;

Vertex_handleVertex_handle vertex_ref; vertex_ref;

};};


Incremental BuilderIncremental Builder• Uses the modifier design to access the internal Uses the modifier design to access the internal

HDSHDS


Make Triangle with Incremental Make Triangle with Incremental BuilderBuilder

template <class HDS>template <class HDS>structstruct Mk_triangleMk_triangle : public : public CGAL::Modifier_base<HDS>CGAL::Modifier_base<HDS> { {

voidvoid operator()( HDS& hds)operator()( HDS& hds) {// Postcond: ‘hds' valid{// Postcond: ‘hds' valid

CGAL::Polyhedron_incremental_builder_3<HDS> B(hds);CGAL::Polyhedron_incremental_builder_3<HDS> B(hds); B.B.begin_surfacebegin_surface( 3, 1, 6);( 3, 1, 6); typedeftypedef typenametypename HDS::Vertex Vertex; HDS::Vertex Vertex; typedeftypedef typenametypename Vertex::Point Point; Vertex::Point Point; B.B.add_vertexadd_vertex( Point( 0, 0, 0));( Point( 0, 0, 0)); B.B.add_vertexadd_vertex( Point( 1, 0, 0));( Point( 1, 0, 0)); B.B.add_vertexadd_vertex( Point( 0, 1, 0));( Point( 0, 1, 0)); B.B.begin_facetbegin_facet();(); B.B.add_vertex_to_facetadd_vertex_to_facet( 0);( 0); B.B.add_vertex_to_facetadd_vertex_to_facet( 1);( 1); B.B.add_vertex_to_facetadd_vertex_to_facet( 2);( 2); B.B.end_facetend_facet();(); B.B.end_surfaceend_surface();(); }}};};


Make Triangle with Incremental Make Triangle with Incremental BuilderBuilder

main() {main() { Polyhedron P;Polyhedron P; Mk_triangle<HalfedgeDS> triangle;Mk_triangle<HalfedgeDS> triangle; P.delegateP.delegate( ( triangletriangle);); return 0;return 0;}}


TraversalTraversal


IteratorsIterators


Iterators and Iterators and CirculatorsCirculators


IterationIteration

Vertex_iterator iter;Vertex_iterator iter;

forfor(( iter = polyhedron.vertices_begin();iter = polyhedron.vertices_begin();

iter != polyhedron.vertices_end();iter != polyhedron.vertices_end();

iter++)iter++)

{{

Vertex_handle hVertex = iter;Vertex_handle hVertex = iter;

// do something with hVertex// do something with hVertex

}}


CirculationCirculation

// circulate around hFacet// circulate around hFacet

Halfedge_around_facet_circulator circ = Halfedge_around_facet_circulator circ =

hFacet->facet_begin();hFacet->facet_begin();

Halfedge_around_facet_circulator end = circ;Halfedge_around_facet_circulator end = circ;

CGAL_For_all(circ,end)CGAL_For_all(circ,end)

{{

Halfedge_handle hHalfedge = circ;Halfedge_handle hHalfedge = circ;

// do something with hHalfedge// do something with hHalfedge

}}


CirculationCirculation

// circulate around hVertex// circulate around hVertex

Halfedge_around_vertex_circulator circ = Halfedge_around_vertex_circulator circ =

hVertex->vertex_begin();hVertex->vertex_begin();

Halfedge_around_vertex_circulator end = circ;Halfedge_around_vertex_circulator end = circ;

CGAL_For_all(circ,end)CGAL_For_all(circ,end)

{{

Halfedge_handle hHalfedge = circ;Halfedge_handle hHalfedge = circ;

// do something with hHalfedge// do something with hHalfedge

}}


File I/OFile I/O

I/O: OFF indexed formatI/O: OFF indexed format

Output: vrml1-2, iv, geomviewOutput: vrml1-2, iv, geomview

OFFOFF6 6 06 6 00.000000 1.686000 0.0000000.000000 1.686000 0.0000001.192000 0.000000 -1.1920001.192000 0.000000 -1.192000-1.192000 0.000000 -1.192000-1.192000 0.000000 -1.192000-1.192000 0.000000 1.192000-1.192000 0.000000 1.1920001.192000 0.000000 1.1920001.192000 0.000000 1.1920000.000000 -1.68600 0.0000000.000000 -1.68600 0.0000003 0 4 1 3 0 4 1 3 1 5 2 3 1 5 2 3 2 3 0 3 2 3 0 3 1 2 0 3 1 2 0 3 3 4 0 3 3 4 0 3 3 2 53 3 2 5


ApplicationsApplications

• RenderingRendering• Subdivision SurfacesSubdivision Surfaces• Algorithms on MeshesAlgorithms on Meshes

– SimplificationSimplification– ApproximationApproximation– RemeshingRemeshing– SmoothingSmoothing– CompressionCompression

• Etc.Etc.


Warm-Up ExercicesWarm-Up Exercices


Highlight Boundary Highlight Boundary EdgesEdges

Notes:Notes:

boolbool halfedge->is_border(); halfedge->is_border();

// change color// change color

::glColor3f(r,g,b); ::glColor3f(r,g,b); // in [0.0f-1.0f]// in [0.0f-1.0f]

// Assemble primitive// Assemble primitive


Render all Facets as Render all Facets as PolygonsPolygons

typedeftypedef Polyhedron::Facet_iterator Polyhedron::Facet_iterator Facet_iteratorFacet_iterator;;

typedeftypedef Polyhedron::Halfedge_around_facet_circulator Polyhedron::Halfedge_around_facet_circulator

Halfedge_facet_circulatorHalfedge_facet_circulator;;

forfor ( (Facet_iteratorFacet_iterator i = P.facets_begin(); i = P.facets_begin();

i != P.facets_end(); i != P.facets_end();

++i) ++i)

{{

Halfedge_facet_circulatorHalfedge_facet_circulator j = i->facet_begin(); j = i->facet_begin();

::::glBeginglBegin(GL_POLYGON);(GL_POLYGON);

dodo

::glVertex3d(j->vertex()->point().x(),::glVertex3d(j->vertex()->point().x(),

j->vertex()->point().y(),j->vertex()->point().y(),

j->vertex()->point().z());j->vertex()->point().z());

whilewhile( ++j != i->facet_begin());( ++j != i->facet_begin());

::::glEndglEnd();();

}}


Exercices Around Exercices Around CombinatoricsCombinatorics


Combinatorial GenusCombinatorial Genus

• Enumerate connected Enumerate connected componentscomponents

• Enumerate boundariesEnumerate boundaries• Deduce genus using Euler Deduce genus using Euler

formulaformula


Exercices Around Exercices Around GeometryGeometry


Discrete LaplacianDiscrete Laplacian


valence(vertex)valence(vertex)


Discrete Discrete LaplacianLaplacian

intint nbv = size_of_vertices(); nbv = size_of_vertices();

std::vector<Vector_3> dx(nbv);std::vector<Vector_3> dx(nbv);

iterate on verticesiterate on vertices

dx[i] = Vector_3(0,0,0); dx[i] = Vector_3(0,0,0); // init // init

circulate around current vertexcirculate around current vertex

... ... // (compute displacements)// (compute displacements)

iterate on verticesiterate on vertices

apply displacementsapply displacements


AdvancedAdvanced


DualizationDualization

primalprimal dualdual


Dualization using Dualization using Incremental BuilderIncremental Builder

3D Polyhedral Surfaces in Pierre Alliez.

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