A simple tetrahedron. More...

#include <tetrahedra.h>

Public Member Functions
	Tetrahedra ()
	Empty.

	Tetrahedra (const Vector &, const Vector &, const Vector &, const Vector &)
	Creates a tetrahedron. More...

	Tetrahedra (const Box &, int)
	Returns the k-th tetrahedron of the box. More...

	~Tetrahedra ()
	Empty.

Vector	Vertex (int) const
	Returns the i-thvertex of the tetrahedron. More...

Triangle	GetTriangle (int) const
	Returns the i-th triangle of the tetrahedron. More...

Plane	GetPlane (int) const
	Returns the i-th plane of the tetrahedron. More...

Vector	Normal (int) const
	Returns the normal of the i-th triangle of the tetrahedron. More...

constexpr int	VertexIndex (int, int) const
	Return the index of the j-th vertex of the i-th triangle. More...

Vector	Normal (const Vector &) const
	Computes the vector distance between the tetrahedron and a point. More...

Vector	Normal2 (const Vector &) const
	Computes the vector distance between the tetrahedron and a point. More...

double	R (const Vector &) const
	Computes the squared distance between the tetrahedron and a point. More...

double	Signed (const Vector &) const
	Computes the signed distance between the tetrahedron and a point. More...

double	Volume () const
	Compute the volume of a tetrahedron. More...

double	Area () const
	Compute the surface area of a tetrahedron. More...

Box	GetBox () const
	Compute the bounding box of a tetrahedron.

Sphere	GetSphere () const
	Compute the sphere bounding the argument tetrahedron. More...

Tetrahedra	Scaled (const Vector &) const
	Scale the tetrahedron. More...

Tetrahedra	Scaled (const double &) const
	Uniform scaling. More...

Tetrahedra	Translated (const Vector &) const
	Translate the tetrahedron. More...

Tetrahedra	Rotated (const Matrix &) const
	Rotate the tetrahedron. More...

bool	Intersect (const Ray &, double &, double &) const
	Compute the intersection between tetrahedra and a ray. More...

bool	Intersect (const Ray &, double &, double &, Vector &, Vector &) const
	Compute the intersection between tetrahedra and a ray. More...

bool	Inside (const Vector &) const
	Check if a point lies inside the tetrahedron. More...

bool	Intersect (const Tetrahedra &) const
	Check if two tetrahedra overlap. More...

bool	Inside (const Plane &) const
	Compute if the tetrahedra is inside a half plane. More...

Vector	RandomInside (Random &=Random::R239) const
	Generate a random point inside the tetrahedron. More...

Static Public Attributes
static double	epsilon = 1e-7
	Internal epsilon constant.

Protected Attributes
Vector	p [4]
	Array of vertices.

Vector	n [4]
	Normals to the faces.

Static Protected Attributes
static const int	vertex [4][3]
	Array of indexes defining the faces of the tetrahedron.

static const int	split [6][4]
	Indexes of the vertexes of the sub-tetrahedron of a cube.

Friends
std::ostream &	operator<< (std::ostream &, const Tetrahedra &)
	Overloaded. More...

Detailed Description

A simple tetrahedron.

The class stores the vertices, the normals of the triangles, and the unit vectors of the edges.

This class also implements some useful functions, such as the intersection with a ray. The member function GetSphere() used to compute the bounding sphere of a tetrahedron simply calls the right Sphere constructor.

Constructor & Destructor Documentation

◆ Tetrahedra() [1/2]

Tetrahedra::Tetrahedra	(	const Vector &	a,
		const Vector &	b,
		const Vector &	c,
		const Vector &	d
	)

explicit

Creates a tetrahedron.

Note that vertices should be ordered so that triangle normals should point outward.

Parameters

a,b,c,d Vertices.

◆ Tetrahedra() [2/2]

Tetrahedra::Tetrahedra	(	const Box &	box,
		int	k
	)

explicit

Returns the k-th tetrahedron of the box.

The box is divided into six tetrahedra.

Parameters

box	The box.
k	Index of the tetrahedron.

Member Function Documentation

◆ Area()

double Tetrahedra::Area ( ) const

Compute the surface area of a tetrahedron.

The surface is defined as the sum of the surface areas of its four triangle faces. Other techniques define the surface area as:
A=(u+v+w).(v-w+u).(w-u+v).(u-v+w)
Here u, v and w denote the three edges length.

◆ GetPlane()

Plane Tetrahedra::GetPlane ( int i ) const

Returns the i-th plane of the tetrahedron.

Parameters

i Index.

◆ GetSphere()

Sphere Tetrahedra::GetSphere ( ) const

Compute the sphere bounding the argument tetrahedron.

This is convenience function, simply calling the constructor of Sphere with four vertices.

See also: Sphere(const Vector&, const Vector&,const Vector&, const Vector&)

◆ GetTriangle()

Triangle Tetrahedra::GetTriangle ( int i ) const

inline

Returns the i-th triangle of the tetrahedron.

Parameters

i Index.

◆ Inside() [1/2]

bool Tetrahedra::Inside ( const Plane & plane ) const

Compute if the tetrahedra is inside a half plane.

Parameters

plane The plane.

◆ Inside() [2/2]

bool Tetrahedra::Inside ( const Vector & p ) const

Check if a point lies inside the tetrahedron.

Parameters

p Point.

◆ Intersect() [1/3]

bool Tetrahedra::Intersect	(	const Ray &	ray,
		double &	ta,
		double &	tb
	)		const

Compute the intersection between tetrahedra and a ray.

Sorted intersection depths are returned if intersection occurs.

Parameters

ray	The ray.
ta,tb	Intersection depths.

◆ Intersect() [2/3]

bool Tetrahedra::Intersect	(	const Ray &	ray,
		double &	ta,
		double &	tb,
		Vector &	na,
		Vector &	nb
	)		const

Compute the intersection between tetrahedra and a ray.

Sorted intersection depths are returned if intersection occurs.

Parameters

ray	The ray.
ta,tb	Returned intersection depths.
na,nb	Returned normals at intersection points.

◆ Intersect() [3/3]

bool Tetrahedra::Intersect ( const Tetrahedra & T ) const

Check if two tetrahedra overlap.

After F. Ganovelli, F. Ponchio, and C. Rocchini. Fast tetrahedron-tetrahedron overlap algorithm. Journal of Graphics Tools, 7(2):17-26, 2002.

Parameters

T	Argument tetrahedron.

◆ Normal() [1/2]

Vector Tetrahedra::Normal ( const Vector & x ) const

Computes the vector distance between the tetrahedron and a point.

Parameters

x Point.

◆ Normal() [2/2]

Vector Tetrahedra::Normal ( int i ) const

inline

Returns the normal of the i-th triangle of the tetrahedron.

Parameters

i Index.

◆ Normal2()

Vector Tetrahedra::Normal2 ( const Vector & x ) const

Computes the vector distance between the tetrahedron and a point.

Parameters

x Point.

◆ R()

double Tetrahedra::R ( const Vector & x ) const

Computes the squared distance between the tetrahedron and a point.

Parameters

x Point.

◆ RandomInside()

Vector Tetrahedra::RandomInside ( Random & r = Random::R239 ) const

Generate a random point inside the tetrahedron.

The method proposed here is a generalization of one of the techniques used by Turk to generate a random point inside a triangle. Generate a random point in a parallelogram and reflectit around the center of the parallelogram.

After C. Rocchini and P. Cignoni, Generating random points in a tetrahedron, Journal of graphics tools, 5(4):9-12, 2000.

Parameters

r	Random number generator.

◆ Rotated()

Tetrahedra Tetrahedra::Rotated ( const Matrix & r ) const

Rotate the tetrahedron.

Parameters

r	Rotation matrix.

◆ Scaled() [1/2]

Tetrahedra Tetrahedra::Scaled ( const double & s ) const

Uniform scaling.

Parameters

s	Scaling factor.

◆ Scaled() [2/2]

Tetrahedra Tetrahedra::Scaled ( const Vector & s ) const

Scale the tetrahedron.

Parameters

s	Scaling vector.

◆ Signed()

double Tetrahedra::Signed ( const Vector & x ) const

Computes the signed distance between the tetrahedron and a point.

This function does not generate an Euclidean distance.

Parameters

x Point.

◆ Translated()

Tetrahedra Tetrahedra::Translated ( const Vector & t ) const

Translate the tetrahedron.

Parameters

t	Translation vector.

◆ Vertex()

Vector Tetrahedra::Vertex ( int i ) const

inline

Returns the i-thvertex of the tetrahedron.

Parameters

i Index.

◆ VertexIndex()

constexpr int Tetrahedra::VertexIndex	(	int	i,
		int	j
	)		const

inlineconstexpr

Return the index of the j-th vertex of the i-th triangle.

Parameters

i	Triangle index.
j	Vertex index.

◆ Volume()

double Tetrahedra::Volume ( ) const

Compute the volume of a tetrahedron.

The volume may be defined as one-sixth the volume of a parallelepiped whose three main edges emanate from a single vertex.

Friends And Related Function Documentation

◆ operator<<

std::ostream & operator<<	(	std::ostream &	s,
		const Tetrahedra &	t
	)

friend

Overloaded.

Parameters

s	Stream.
t	The tetrahedron.

Public Member Functions

Static Public Attributes

Protected Attributes

Static Protected Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ Tetrahedra() [1/2]

◆ Tetrahedra() [2/2]

Member Function Documentation

◆ Area()

◆ GetPlane()

◆ GetSphere()

◆ GetTriangle()

◆ Inside() [1/2]

◆ Inside() [2/2]

◆ Intersect() [1/3]

◆ Intersect() [2/3]

◆ Intersect() [3/3]

◆ Normal() [1/2]

◆ Normal() [2/2]

◆ Normal2()

◆ R()

◆ RandomInside()

◆ Rotated()

◆ Scaled() [1/2]

◆ Scaled() [2/2]

◆ Signed()

◆ Translated()

◆ Vertex()

◆ VertexIndex()

◆ Volume()

Friends And Related Function Documentation

◆ operator<<