A core analytic three-dimensional scalar field. More...

#include <scalarfield.h>

Inheritance diagram for AnalyticScalarField:

Public Member Functions
	AnalyticScalarField (bool=true)
	Constructor. More...

virtual double	Value (const Vector &) const
	Compute the value of the field. More...

virtual Vector	Gradient (const Vector &) const
	Compute the gradient of the field. More...

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

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

virtual Box	GetBox () const
	Compute the bounding box of the scalar field. More...

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

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

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. More...

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

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

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

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

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

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

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

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. More...

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. More...

virtual double	K () const
	Compute the Lipschitz constant of the field function. More...

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

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

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

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

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

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

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

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. More...

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

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

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. More...

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

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

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

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

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

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

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

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

Detailed Description

A core analytic three-dimensional scalar field.

Constructor & Destructor Documentation

◆ AnalyticScalarField()

AnalyticScalarField::AnalyticScalarField ( bool s = true )

inline

Constructor.

Parameters

s	Sign, by default set sign convention to true, i.e., negative values inside and positive outside.

Member Function Documentation

◆ Cast()

Vector AnalyticScalarField::Cast	(	const Vector &	p,
		const double &	step,
		const double &	epsilon = `1e-6`,
		int	n = `100`
	)		const

Given a starting vertex x, project it onto the implicit surface following the gradient.

This function returns a vertex on the implicit surface. It updates a pair of points that are moved so that they straddle the surface, before invoking a bisection step.

Parameters

p	Starting vertex.
step	Stepping distance.
epsilon	Precision.
n	Maximum number of iterations when marching in the direction of the gradient.

Returns: Point on the implicit surface.

◆ Colorize()

void AnalyticScalarField::Colorize	(	const Mesh &	m,
		MeshColor &	c
	)		const

virtual

Colorize the vertexes of a mesh according to the analytic scalar field.

Parameters

m	Mesh.
c	Colored mesh.

◆ Curvature()

void AnalyticScalarField::Curvature	(	const Vector &	p,
		double &	gaussian,
		double &	mean
	)		const

Compute the gaussian and mean curvatures.

Parameters

p	Point.
gaussian	Returned gaussian curvature.
mean	Returned mean curvature.

◆ Dichotomy()

Vector AnalyticScalarField::Dichotomy	(	Vector	a,
		Vector	b,
		double	va,
		double	vb,
		double	length,
		const double &	epsilon = `1.0e-4`
	)		const

Compute the intersection between a segment and an implicit surface.

This method iteratively converges to the solution by performing a bisection on the segment. It avoids the internal computation if the length of the segment if it happens to be known already.

The segment should straddle the implicit surface only once, otherwise should several intersections exist, there is no guarantee that the bissection will converge to an intersection.

Parameters

a,b	End vertices of the segment straddling the surface.
va,vb	Field function value at those end vertices.
length	Distance between vertices.
epsilon	Precision.

Returns: Point on the implicit surface.

◆ Dual()

void AnalyticScalarField::Dual	(	int	n,
		Mesh &	g,
		const Box &	box
	)		const

virtual

Dual polygonization.

Compute a scalar field with function values sampled at the vertices of the grid, and invoque the dual scalar field polygonization algorithm.

See also: ScalarField::Dual

Parameters

n	Subdivision.
g	Mesh.
box	The box.

◆ Find()

bool AnalyticScalarField::Find	(	Vector &	p,
		bool	s,
		const Box &	box,
		int	n = `10000`,
		Random &	random = `Random::R239`
	)		const

protected

Find a random sample point inside or outside the surface.

Parameters

p	Returned point.
s	Prescribed position with respect to the surface, true if inside, false if outside.
box	Box domain where the point is to be found.
n	Maximum number of random points being evaluated.
random	Random number generator.

Returns: Return true if a sample has been found, false otherwise.

◆ GetBox()

Box AnalyticScalarField::GetBox ( ) const

virtual

Compute the bounding box of the scalar field.

Simply return Box::Infinity.

Reimplemented in Algebraic.

◆ GetMaterial()

Color AnalyticScalarField::GetMaterial	(	const Vector &	p,
		const Vector &	n = `Vector::Null`
	)		const

virtual

Compute the color.

Parameters

p	Point.
n	Normal at point.

Returns: Color.

◆ GetSample()

bool AnalyticScalarField::GetSample	(	Vector &	p,
		const Box &	box,
		Random &	random = `Random::R239`
	)		const

virtual

Find a point sample on the implicit surface.

Simply find two random points inside and outside of the surface, and perform bisection to converge to the surface.

Parameters

box	Sampling box.
p	Point sample on the surface, if succeded in finding one.
random	Random number generator.

◆ GetSamples()

int AnalyticScalarField::GetSamples	(	QVector< Vector > &	s,
		const Box &	box,
		int	n,
		Random &	random = `Random::R239`
	)		const

virtual

Find a set of random sample points on the implicit surface.

Simply call AnalyticScalarField::GetSample() several times.

Parameters

box	Sampling box.
s	Set of sample points.
n	Number of (desired) sampes.
random	Random number generator.

Returns: The number of samples found on the surface.

◆ Gradient()

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

virtual

Compute the gradient of the field.

Parameters

p Point.

Reimplemented in Algebraic.

◆ Hessian()

Matrix AnalyticScalarField::Hessian ( const Vector & p ) const

Compute the Hessian symmetric matrix of the field function.

This method uses a numerical approximation of the successive derivatives in each direction.

Parameters

p Point.

Returns: Hessian matrix.

◆ K() [1/4]

double AnalyticScalarField::K ( ) const

virtual

Compute the Lipschitz constant of the field function.

Compute the Lipschitz constant.

By default, return 1.0.

Reimplemented in Algebraic.

◆ K() [2/4]

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

virtual

Compute the local Lipschitz constant inside a box.

Parameters

box Box.

Reimplemented in Algebraic.

◆ K() [3/4]

double AnalyticScalarField::K ( const Segment & segment ) const

virtual

Compute the local Lipschitz constant along a segment.

Parameters

segment The segment.

◆ K() [4/4]

double AnalyticScalarField::K ( const Sphere & sphere ) const

virtual

Compute the local Lipschitz constant inside a box.

Parameters

sphere Sphere.

◆ Normal()

Vector AnalyticScalarField::Normal ( const Vector & p ) const

virtual

Compute the normal to the surface.

See also: AnalyticScalarField::Gradient(const Vector&) const

Parameters

p	Point (should be on the surface).

◆ Poisson()

QVector< Vector > AnalyticScalarField::Poisson	(	double	r,
		int	n,
		Random &	rng = `Random::R239`
	)		const

virtual

Compute a Poisson sphere distribution on an implicit surface.

This function uses a simple O(n²) dart throwing algorithm.

Parameters

r	Radius of the sphere.
n	Number of candidate points.
rng	Random number generator.

◆ Polygonize() [1/3]

void AnalyticScalarField::Polygonize	(	const Box &	box,
		QVector< Triangle > &	triangles,
		const double &	epsilon = `1e-4`,
		bool	tetrahedra = `false`
	)		const

virtual

Polygonize a cube.

The algorithm uses either a look-up table disambiguation scheme, or a decomposition into six tetrahedra.

The fundamental problem is to form a facet approximation to an isosurface through a scalar field sampled on a rectangular grid. Given one grid cell defined by its vertices and scalar values at each vertex, it is necessary to create planar facets that best represent the isosurface through that grid cell.

The isosurface may not be pass through the grid cell, it may cut off any one of the vertices, or it may pass through in any one of a number of more complicated ways. Each possibility will be characterised by the number of vertices that have values above or below the isosurface. If one vertex is above the isosurface say and an adjacent vertex is below the isosurface then we know the isosurface cuts the edge between these two vertices.

The position that it cuts the edge will be linearly interpolated, the ratio of the length between the two vertices will be the same as the ratio of the isosurface value to the values at the vertices of the grid cell.

Parameters

box	The box.
triangles	Set of triangles.
epsilon	Precision.
tetrahedra	Boolean specifying whether tetrahedral decomposition should be used.

◆ Polygonize() [2/3]

void AnalyticScalarField::Polygonize	(	const Tetrahedra &	tetra,
		QVector< Triangle > &	triangles,
		const double &	epsilon = `1e-4`
	)		const

virtual

Polygonize a tetrahedral cell.

Parameters

tetra	Tetrahedral cell.
triangles	Set of triangles.
epsilon	Precision.

◆ Polygonize() [3/3]

void AnalyticScalarField::Polygonize	(	int	n,
		Mesh &	g,
		const Box &	box,
		const double &	epsilon = `1e-4`
	)		const

virtual

Compute the polygonal mesh approximating the implicit surface.

This member function implements the marching cube tesselation algorithm, using a fixed look-up table. This technique does suffer from the ambiguities in the traditional marching cubes algorithms where some special cases result in unexpected holes in the resulting surface.

The first part of the algorithm uses a table (edgeTable) which maps the vertices under the isosurface to the intersecting edges. An 8 bit index is formed where each bit corresponds to a vertex.

Looking up the edge table returns a 12 bit number, each bit corresponding to an edge, 0 if the edge isn't cut by the isosurface, 1 if the edge is cut by the isosurface. If none of the edges are cut the table returns a 0, this occurs when cubeindex is 0 (all vertices below the isosurface) or 0xff (all vertices above the isosurface).

The last part of the algorithm involves forming the correct facets from the positions that the isosurface intersects the edges of the grid cell. Again a table is used which this time uses the same cubeindex but allows the vertex sequence to be looked up for as many triangular facets are necessary to represent the isosurface within the grid cell. There at most 5 triangular facets necessary.

Lets say vertex 0 and 3 are below the isosurface. cubeindex will then be 0000 1001==9. The 9th entry into the egdeTable is 905 = 1001 0000 0101: edges 11, 8, 2, and 0 are cut and so we work out the vertices of the intersection of the isosurface with those edges. Next 9 in the triangle table is 0, 11, 2, 8, 11, 0. This corresponds to 2 triangular facets, one between the intersection of edge 0 11 and 2; the other between the intersections along edges 8, 11, and 0.

Parameters

box	Box defining the region that will be polygonized.
n	Discretization parameter.
g	Returned geometry.
epsilon	Epsilon value for computing vertices on straddling edges.

◆ PolygonizeLucie() [1/2]

void AnalyticScalarField::PolygonizeLucie	(	const Box &	box,
		Mesh &	mesh,
		bool	dichotomy,
		bool	computeNormals
	)		const

virtual

Implementation of the original continuation polygonization method.

Parameters

box	Starting cube straddling the surface.
mesh	Mesh.
dichotomy	Boolean for computing vertexes using bisection.
computeNormals	Boolean for computing normals.

The original method is described in: Data structure forsoft objects. Geoff Wyvill, Craig McPheeters, Brian Wyvill. The Visual Computer, 2, 227�234, 1986.

◆ PolygonizeLucie() [2/2]

void AnalyticScalarField::PolygonizeLucie	(	const double &	size,
		Mesh &	mesh,
		bool	dichotomy,
		bool	computeNormals
	)		const

virtual

Wyvill's marching cubes algorithm.

Find a cube straddling the surface, and apply Wyvill's marching algorithm.

Parameters

size	Size of the cubes that will be straddling the surface.
mesh	Mesh.
dichotomy	Boolean for computing vertexes using bisection.
computeNormals	Boolean for computing normals.

◆ PolygonizeOctree()

Mesh AnalyticScalarField::PolygonizeOctree	(	const Box &	box,
		int	level,
		const double &	epsilon = `1e-4`
	)		const

virtual

Compute the polygonal mesh approximating the implicit surface using an octree decomposition of space.

Parameters

box	Box defining the region that will be polygonized.
level	Octree depth, the corresponding grid subdivision is 2ⁿ+1.
epsilon	Epsilon value for computing vertices on straddling edges.

◆ Roots() [1/2]

int AnalyticScalarField::Roots	(	const Ray &	ray,
		const double &	a,
		const double &	b,
		const double &	k,
		const double &	ya,
		const double &	yb,
		double *	x,
		int	n = `1`,
		const double &	epsilon = `1.0e-4`
	)		const

This functions searches the intersections between a ray and an implicit surface on a given interval.

The original algorithm recursively isolate the roots by applying the Lipschitz criteria over the interval. It has been rewritten into an iterative form which is more efficient.

Parameters

ray	The ray.
a,b	Interval of analysis.
k	Lipschitz constant.
ya,yb	Field function values at the end-points of the interval of analysis.
x	Array for storing roots.
n	Search for at most n roots.
epsilon	Precision parameter.

◆ Roots() [2/2]

int AnalyticScalarField::Roots	(	const Ray &	ray,
		const double &	a,
		const double &	b,
		const double &	k,
		double *	s,
		int	n = `1`,
		const double &	epsilon = `1.0e-4`
	)		const

Find the intersections between a ray and an implicit surface on a given interval.

Parameters

ray	The ray.
a,b	Interval of analysis.
k	Lipschitz constant.
s	Array for storing roots.
n	Search for n roots at most.
epsilon	Precision parameter.

◆ Sample() [1/3]

ScalarField AnalyticScalarField::Sample	(	const Box &	box,
		int	n
	)		const

virtual

Sample the implicit surface to get a scalar field.

Parameters

n	Subdivision.
box	The box.

◆ Sample() [2/3]

ScalarField2 AnalyticScalarField::Sample	(	const Quadrangle &	quad,
		int	n
	)		const

virtual

Sample the scalar field on a quadrangle.

Returns the 2D scalarfield.

Parameters

quad	Quadrangle
n	discretization

◆ Sample() [3/3]

ScalarField2 AnalyticScalarField::Sample	(	const Rectangles &	r,
		int	x,
		int	y
	)		const

virtual

Compute the field function at the vertices a grid.

Parameters

r	Rectangle.
x,y	Discretization.

◆ SphereTrace()

bool AnalyticScalarField::SphereTrace	(	const Ray &	ray,
		const double &	k,
		const double &	a,
		const double &	b,
		double &	t,
		int &	s,
		const double &	epsilon = `1e-4`
	)		const

virtual

Find the first intersection between a ray and an implicit surface on a given interval, using sphere tracing.

Parameters

ray	The ray.
k	Lipschitz constant.
a,b	Interval where intersection is tracked.
t	Nearest root, if it exists.
s	Return number of steps.
epsilon	Precision on field function values for detecting ray-surface intersection.

Simply march along the ray with variable steps depending on the magnitude of the field function at point samples. Intersection occurs if the absolute value of the field function get lower than epsilon.

◆ Value()

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

virtual

Compute the value of the field.

Parameters

p Point.

Reimplemented in Algebraic.

◆ Voxelize() [1/2]

void AnalyticScalarField::Voxelize	(	const Box &	box,
		int	n,
		Box &	cell,
		QVector< Vector > &	cubes
	)		const

virtual

Voxelize the scalar field.

Parameters

box	The box.
n	Subdivision.
cell	Returned reference cube.
cubes	Set of vectors defining the position of the instantiated reference cube.

◆ Voxelize() [2/2]

void AnalyticScalarField::Voxelize	(	int	n,
		Voxel &	voxel,
		const Box &	box
	)		const

virtual

Voxelization.

Parameters

n	Subdivision.
voxel	Voxel.
box	The box.

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Static Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ AnalyticScalarField()

Member Function Documentation

◆ Cast()

◆ Colorize()

◆ Curvature()

◆ Dichotomy()

◆ Dual()

◆ Find()

◆ GetBox()

◆ GetMaterial()

◆ GetSample()

◆ GetSamples()

◆ Gradient()

◆ Hessian()

◆ K() [1/4]

◆ K() [2/4]

◆ K() [3/4]

◆ K() [4/4]

◆ Normal()

◆ Poisson()

◆ Polygonize() [1/3]

◆ Polygonize() [2/3]

◆ Polygonize() [3/3]

◆ PolygonizeLucie() [1/2]

◆ PolygonizeLucie() [2/2]

◆ PolygonizeOctree()

◆ Roots() [1/2]

◆ Roots() [2/2]

◆ Sample() [1/3]

◆ Sample() [2/3]

◆ Sample() [3/3]

◆ SphereTrace()

◆ Value()

◆ Voxelize() [1/2]

◆ Voxelize() [2/2]