Subdivision Methods

A subdivision method recursively refines a coarse mesh and generates an ever closer approximation to a smooth surface.

Subdivision_method_3 consists of four subdivision methods and their refinement hosts. Each refinement host is a template function of a polyhedron class and a geometry policy class. It refines the connectivity of the control mesh and computes the geometry of the refined mesh. The geometry computation is dedicated to the custom geometry policy. A geometry policy consists of functions that compute the new point based on the subdivision stencil. A stencil defines the footprint (a submesh of the control mesh) of a new point.

The four supported refinement hosts are the primal quadrilateral quadrisection (PQQ), the primal triangle quadrisection (PTQ), the dual quadrilateral quadrisection (DQQ), and the \( \sqrt{3}\) triangulation. These refinements are respectively used in Catmull-Clark, Loop, Doo-Sabin and \( \sqrt{3}\) subdivision.

Refinement Host

A refinement host is a template function of a polyhedron class and a geometry mask class. It refines the input polyhedron, and computes new points through the geometry masks. Subdivision_method_3 supports four refinement hosts: PQQ, PTQ, DQQ and Sqrt3.

Example

This example program subdivides a polyhedral mesh with Catmull-Clark subdivision.

File Subdivision_method_3/CatmullClark_subdivision.cpp

#include <CGAL/Simple_cartesian.h>
#include <CGAL/Subdivision_method_3.h>
#include <iostream>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/IO/Polyhedron_iostream.h>

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

using namespace std;
using namespace CGAL;

int main(int argc, char** argv) {
  if (argc != 2) {
    cout << "Usage: CatmullClark_subdivision d < filename" << endl;
    cout << "       d: the depth of the subdivision (0 < d < 10)" << endl;
    cout << "       filename: the input mesh (.off)" << endl;
    return 0;
  }

  int d = argv[1][0] - '0';

  Polyhedron P;
  cin >> P; // read the .off

  Subdivision_method_3::CatmullClark_subdivision(P,d);

  cout << P; // write the .off

  return 0;
}

See Also

CGAL::CatmullClark_mask_3<Polyhedron_3>

CGAL::Loop_mask_3<Polyhedron_3>

CGAL::Sqrt3_mask_3<Polyhedron_3>

Functions
template<class Polyhedron_3 , template< typename > class Mask>
void	CGAL::Subdivision_method_3::PQQ (Polyhedron_3 &p, Mask< Polyhedron_3 > mask, int step=1)
	applies the PQQ refinement on the control mesh p step times. More...
template<class Polyhedron_3 , template< typename > class Mask>
void	CGAL::Subdivision_method_3::PTQ (Polyhedron_3 &p, Mask< Polyhedron_3 > mask, int step=1)
	applies the PTQ refinement on the control mesh p step times, where p contains only triangle facets. More...
template<class Polyhedron_3 , template< typename > class Mask>
void	CGAL::Subdivision_method_3::DQQ (Polyhedron_3 &p, Mask< Polyhedron_3 > mask, int step=1)
	applies the DQQ refinement on the control mesh p step times. More...
template<class Polyhedron_3 , template< typename > class Mask>
void	CGAL::Subdivision_method_3::Sqrt3 (Polyhedron_3 &p, Mask< Polyhedron_3 > mask, int step=1)
	applies the \( \sqrt{3}\) triangulation on the control mesh p step times, where p contains only triangle facets. More...
template<class Polyhedron_3 >
void	CGAL::Subdivision_method_3::CatmullClark_subdivision (Polyhedron_3 &p, int step=1)
	applies Catmull-Clark subdivision step times on the control mesh p. More...
template<class Polyhedron_3 >
void	CGAL::Subdivision_method_3::Loop_subdivision (Polyhedron_3 &p, int step=1)
	applies Loop subdivision step times on the control mesh p. More...
template<class Polyhedron_3 >
void	CGAL::Subdivision_method_3::DooSabin_subdivision (Polyhedron_3 &p, int step=1)
	applies Doo-Sabin subdivision step times on the control mesh p. More...
template<class Polyhedron_3 >
void	CGAL::Subdivision_method_3::Sqrt3_subdivision (Polyhedron_3 &p, int step=1)
	applies \( \sqrt{3}\) subdivision step times on the control mesh p. More...

Function Documentation

template<class Polyhedron_3 >

void CGAL::Subdivision_method_3::CatmullClark_subdivision	(	Polyhedron_3 &	p,
		int	step = 1
	)

applies Catmull-Clark subdivision step times on the control mesh p.

This function overwrites the control mesh p with the subdivided mesh.

#include <CGAL/Subdivision_method_3.h>

Examples:

Subdivision_method_3/CatmullClark_subdivision.cpp.

template<class Polyhedron_3 >

void CGAL::Subdivision_method_3::DooSabin_subdivision	(	Polyhedron_3 &	p,
		int	step = 1
	)

applies Doo-Sabin subdivision step times on the control mesh p.

This function overwrites the control mesh p with the subdivided mesh.

#include <CGAL/Subdivision_method_3.h>

Examples:

Subdivision_method_3/DooSabin_subdivision.cpp.

template<class Polyhedron_3 , template< typename > class Mask>

void CGAL::Subdivision_method_3::DQQ	(	Polyhedron_3 &	p,
		Mask< Polyhedron_3 >	mask,
		int	step = 1
	)

applies the DQQ refinement on the control mesh p step times.

The geometry of the refined mesh is computed by the geometry policy mask. This function overwrites the control mesh p with the refined mesh.

#include <CGAL/Subdivision_method_3.h>

template<class Polyhedron_3 >

void CGAL::Subdivision_method_3::Loop_subdivision	(	Polyhedron_3 &	p,
		int	step = 1
	)

applies Loop subdivision step times on the control mesh p.

This function overwrites the control mesh p with the subdivided mesh.

#include <CGAL/Subdivision_method_3.h>

template<class Polyhedron_3 , template< typename > class Mask>

void CGAL::Subdivision_method_3::PQQ	(	Polyhedron_3 &	p,
		Mask< Polyhedron_3 >	mask,
		int	step = 1
	)

applies the PQQ refinement on the control mesh p step times.

The geometry of the refined mesh is computed by the geometry policy mask. This function overwrites the control mesh p with the refined mesh.

#include <CGAL/Subdivision_method_3.h>

template<class Polyhedron_3 , template< typename > class Mask>

void CGAL::Subdivision_method_3::PTQ	(	Polyhedron_3 &	p,
		Mask< Polyhedron_3 >	mask,
		int	step = 1
	)

applies the PTQ refinement on the control mesh p step times, where p contains only triangle facets.

The geometry of the refined mesh is computed by the geometry policy mask. This function overwrites the control mesh p with the refined mesh. The result of a non-triangle mesh p is undefined.

#include <CGAL/Subdivision_method_3.h>

Examples:

Subdivision_method_3/Customized_subdivision.cpp.

template<class Polyhedron_3 , template< typename > class Mask>

void CGAL::Subdivision_method_3::Sqrt3	(	Polyhedron_3 &	p,
		Mask< Polyhedron_3 >	mask,
		int	step = 1
	)

applies the \( \sqrt{3}\) triangulation on the control mesh p step times, where p contains only triangle facets.

The geometry of the refined mesh is computed by the geometry policy mask. This function overwrites the control mesh p with the refined mesh. The result of a non-triangle mesh p is undefined.

#include <CGAL/Subdivision_method_3.h>

template<class Polyhedron_3 >

void CGAL::Subdivision_method_3::Sqrt3_subdivision	(	Polyhedron_3 &	p,
		int	step = 1
	)

applies \( \sqrt{3}\) subdivision step times on the control mesh p.

This function overwrites the control mesh p with the subdivided mesh.

#include <CGAL/Subdivision_method_3.h>