Trivariate polynomials of maximum degree 15. More...

#include <algebraic.h>

Inheritance diagram for Algebraic:

Public Member Functions
	Algebraic ()
	Creates a null algebraic expression.

	Algebraic (const Box &, int, const QVector< double > &)
	Creates an algebraic implicit surface.

	Algebraic (const Box &, int)
	Creates an algebraic implicit surface.

	Algebraic (int)
	Creates an algebraic implicit surface.

virtual	~Algebraic ()
	Empty.

double &	operator() (int, int, int)
	Returns the value of term x_ijk in the array of coefficients.

double	operator() (int, int, int) const
	Return the value of term x_ijk in the array of coefficients.

double &	operator() (int)
	Returns the value of the term.

double	operator() (int) const
	Returns the value of the term.

int	Terms () const
	Number of terms.

double	Value (const Vector &) const
	Compute the intensity at a point in space.

Vector	Gradient (const Vector &) const
	Compute the gradient of the algebraic field function at a point in space.

int	Intersect (const Ray &, double *, const double &) const
	Computes the intersection between a ray and an algebraic surface.

virtual Box	GetBox () const
	Returns the bounding box of the algebraic surface.

virtual double	K () const
	Return the global Lipschitz constant of the algebraic functions.

virtual double	K (const Box &) const
	Compute the Lipschitz constant of the algebraic function in a box.

int	Term (int, int, int) const
	Compute the position of the term x_abc in the array of coefficients.

Public Member Functions inherited from AnalyticScalarField
	AnalyticScalarField (bool=true)
	Constructor.

Matrix	Hessian (const Vector &) const
	Compute the Hessian symmetric matrix of the field function.

virtual Vector	Normal (const Vector &) const
	Compute the normal to the surface.

bool	Inside (const double &) const
	Check if the value is considered as inside or outside.

virtual Color	GetMaterial (const Vector &, const Vector &=Vector::Null) const
	Compute the color.

void	Curvature (const Vector &, double &, double &) const
	Compute the gaussian and mean curvatures.

Vector	Cast (const Vector &, const double &, const double &=1e-6, int n=100) const
	Given a starting vertex x, project it onto the implicit surface following the gradient.

Vector	Dichotomy (Vector, Vector, double, double, double, const double &=1.0e-4) const
	Compute the intersection between a segment and an implicit surface.

virtual bool	GetSample (Vector &, const Box &, Random &=Random::R239) const
	Find a point sample on the implicit surface.

virtual int	GetSamples (QVector< Vector > &, const Box &, int, Random &=Random::R239) const
	Find a set of random sample points on the implicit surface.

virtual QVector< Vector >	Poisson (double, int, Random &=Random::R239) const
	Compute a Poisson sphere distribution on an implicit surface.

virtual ScalarField2	Sample (const Rectangles &, int, int) const
	Compute the field function at the vertices a lattice.

virtual ScalarField2	Sample (const Quadrangle &, int) const
	Sample the scalar field on a quadrangle.

virtual ScalarField	Sample (const Box &, int) const
	Sample the implicit surface to get a scalar field.

int	Roots (const Ray &, const double &, const double &, const double &, double *, int=1, const double &=1.0e-4) const
	Find the intersections between a ray and an implicit surface on a given interval.

int	Roots (const Ray &, const double &, const double &, const double &, const double &, const double &, double *, int=1, const double &=1.0e-4) const
	This functions searches the intersections between a ray and an implicit surface on a given interval.

virtual double	K (const Sphere &) const
	Compute the local Lipschitz constant inside a box.

virtual double	K (const Segment &) const
	Compute the local Lipschitz constant along a segment.

virtual void	Polygonize (int, Mesh &, const Box &, const double &=1e-4) const
	Compute the polygonal mesh approximating the implicit surface.

virtual void	PolygonizeLucie (const Box &, Mesh &, bool, bool) const
	Implementation of the original continuation polygonization method.

virtual void	PolygonizeLucie (const double &, Mesh &, bool, bool) const
	Wyvill's original marching cubes algorithm.

virtual void	Dual (int, Mesh &, const Box &) const
	Dual polygonization.

virtual bool	SphereTrace (const Ray &, const double &, const double &, const double &, double &, int &, const double &=1e-4) const
	Find the first intersection between a ray and an implicit surface on a given interval, using sphere tracing.

virtual void	Polygonize (const Box &, QVector< Triangle > &, const double &=1e-4, bool=false) const
	Polygonize a cube.

virtual void	Polygonize (const Tetrahedra &, QVector< Triangle > &, const double &=1e-4) const
	Polygonize a tetrahedral cell.

virtual Mesh	PolygonizeOctree (const Box &, int, const double &=1e-4) const
	Compute the polygonal mesh approximating the implicit surface using an octree decomposition of space.

virtual void	Voxelize (const Box &, int, Box &, QVector< Vector > &) const
	Voxelize the scalar field.

virtual void	Voxelize (int, Voxel &, const Box &) const
	Voxelization.

virtual double	Volume (int=5) const
	Computes the volume of the BlobTree.

virtual double	Volume (const Box &, int) const
	Compute the volume of a BlobTree object.

virtual double	Volume (const Box &, int, double &) const
	Compute the volume of an implicit surface.

virtual double	StochasticVolume (const Box &, int) const
	Compute the volume of an implicit surface.

virtual Sphere	Center (int) const
	Computes the gravity center of the BlobTree.

virtual Sphere	Center (const Box &, int) const
	Compute the center of gravity of the BlobTree.

virtual void	Colorize (const Mesh &, MeshColor &) const
	Colorize the vertexes of a mesh according to the analytic scalar field.

Static Public Member Functions
static double	Barth (const Vector &, const double &)
	Compute the Barth sextic value.

Static Public Member Functions inherited from AnalyticScalarField
static int	Cubes ()
	Get the number of cubes processed.

Protected Member Functions
Vector	LowGradient (const Vector &) const
	Gradient of a low degree polynomial. Quartics and lower degree polynomials are handled by this function. This function avoids loops.

Vector	HighGradient (const Vector &) const
	Gradient of a high degree (quadric, cubic and quartic) polynomial.

int	IntersectLinear (const Ray &, double *) const
	Computes the intersection between a ray and a linear algebraic form, i.e., a plane.

int	IntersectQuadratic (const Ray &, double *) const
	Computes the intersection between a ray and a quadratic algebraic form.

int	IntersectHigh (const Ray &, double *) const
	Compute the intersection between a ray and an arbitrary polynomial function.

Protected Member Functions inherited from AnalyticScalarField
bool	Find (Vector &, bool, const Box &, int=10000, Random &=Random::R239) const
	Find a random sample point inside or outside the surface.

Protected Attributes
int	n = 0
	Order of the algebraic surface.

int	size = 0
	Size of the array containing the coefficients.

Box	box = Box::Null
	Box.

double	k = 0
	Lipschitz constant.

QVector< double >	c
	Array of coefficients.

Protected Attributes inherited from AnalyticScalarField
bool	sign = true
	Sign convention, used for normal computation.

Static Protected Attributes
static const int	degree = 8
	Maximum order of algebraic polynomial.

Static Protected Attributes inherited from AnalyticScalarField
static int	ncubes = 0
	Number of cubes processed by polygonization algorithms.

static const double	Epsilon = 1e-6
	Epsilon value for partial derivatives.

Detailed Description

Trivariate polynomials of maximum degree 15.

The algebraic surface should be embedded in a bounding box.

Constructor & Destructor Documentation

◆ Algebraic() [1/3]

Algebraic::Algebraic	(	const Box &	b,
		int	n,
		const QVector< double > &	a )

explicit

Creates an algebraic implicit surface.

If the array pointer is either null or not specified, coefficients are initialized to 0.0, otherwise arguments are copied in an local array.

Parameters

b	The box.
n	Degree of the trivariate polynomial.
a	Array of coeffcients which will be copied.

◆ Algebraic() [2/3]

Algebraic::Algebraic	(	const Box &	b,
		int	n )

explicit

Creates an algebraic implicit surface.

If the array pointer is either null or not specified, coefficients are initialized to 0.0, otherwise arguments are copied in an local array.

Parameters

b	The box.
n	Degree of the trivariate polynomial.

◆ Algebraic() [3/3]

Algebraic::Algebraic ( int n )

explicit

Creates an algebraic implicit surface.

Parameters

n	Degree of the trivariate polynomial.

Member Function Documentation

◆ Barth()

double Algebraic::Barth	(	const Vector &	p,
		const double &	w )

static

Compute the Barth sextic value.

Parameters

p	Point.
w	Parameter.

◆ GetBox()

Box Algebraic::GetBox ( ) const

inlinevirtual

Returns the bounding box of the algebraic surface.

As the field function is not bounded in space, this box was provided as a parameter in the constructor.

Reimplemented from AnalyticScalarField.

◆ Gradient()

Vector Algebraic::Gradient ( const Vector & p ) const

virtual

Compute the gradient of the algebraic field function at a point in space.

Parameters

p Point.

Reimplemented from AnalyticScalarField.

◆ HighGradient()

Vector Algebraic::HighGradient ( const Vector & q ) const

protected

Gradient of a high degree (quadric, cubic and quartic) polynomial.

Parameters

q Point.

◆ Intersect()

int Algebraic::Intersect	(	const Ray &	ray,
		double *	t,
		const double &	maxdepth ) const

Computes the intersection between a ray and an algebraic surface.

First perform the intersection test between the ray and the box, and escape if no intersections ocur. Intersections within the box interval are stacked in the depth stack array.

Parameters

ray	The ray.
t	Depth stack which should be defined as an array.
maxdepth	Maximum intersection depth.

◆ IntersectHigh()

int Algebraic::IntersectHigh	(	const Ray &	ray,
		double *	Depths ) const

protected

Compute the intersection between a ray and an arbitrary polynomial function.

Parameters

ray	The ray.
Depths	Depth stack which should be defined as an array.

◆ IntersectLinear()

int Algebraic::IntersectLinear	(	const Ray &	ray,
		double *	t ) const

protected

Computes the intersection between a ray and a linear algebraic form, i.e., a plane.

Parameters

ray	The ray.
t	Depth stack which should be defined as an array.

◆ IntersectQuadratic()

int Algebraic::IntersectQuadratic	(	const Ray &	ray,
		double *	t ) const

protected

Computes the intersection between a ray and a quadratic algebraic form.

Parameters

ray	The ray.
t	Depth stack which should be defined as an array.

◆ K()

double Algebraic::K ( const Box & box ) const

virtual

Compute the Lipschitz constant of the algebraic function in a box.

Parameters

box The box.

Reimplemented from AnalyticScalarField.

◆ LowGradient()

Vector Algebraic::LowGradient ( const Vector & p ) const

protected

Gradient of a low degree polynomial. Quartics and lower degree polynomials are handled by this function. This function avoids loops.

Parameters

p Point.

◆ operator()() [1/4]

double & Algebraic::operator() ( int t )

inline

Returns the value of the term.

Parameters

t	Term index, which could be computed from exponents using Term.

◆ operator()() [2/4]

double Algebraic::operator() ( int t ) const

inline

Returns the value of the term.

Parameters

t	Term index, which could be computed from exponents using Term.

◆ operator()() [3/4]

double & Algebraic::operator()	(	int	i,
		int	j,
		int	k )

Returns the value of term x_ijk in the array of coefficients.

Parameters

i,j,k Indexes.

◆ operator()() [4/4]

double Algebraic::operator()	(	int	i,
		int	j,
		int	k ) const

Return the value of term x_ijk in the array of coefficients.

Parameters

i,j,k Indexes.

◆ Term()

int Algebraic::Term	(	int	a,
		int	b,
		int	c ) const

Compute the position of the term x_abc in the array of coefficients.

Parameters

a,b,c Exponents of the coefficient x^a y^b z^c.

◆ Value()

double Algebraic::Value ( const Vector & p ) const

virtual

Compute the intensity at a point in space.

Parameters

p Point.

Reimplemented from AnalyticScalarField.

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Static Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ Algebraic() [1/3]

◆ Algebraic() [2/3]

◆ Algebraic() [3/3]

Member Function Documentation

◆ Barth()

◆ GetBox()

◆ Gradient()

◆ HighGradient()

◆ Intersect()

◆ IntersectHigh()

◆ IntersectLinear()

◆ IntersectQuadratic()

◆ K()

◆ LowGradient()

◆ operator()() [1/4]

◆ operator()() [2/4]

◆ operator()() [3/4]

◆ operator()() [4/4]

◆ Term()

◆ Value()