Triangle Class Reference

Base minimum storage triangle class. More...

#include <triangle.h>

Public Member Functions
	Triangle ()
	Empty.
	Triangle (const Vector &, const Vector &, const Vector &)
	Create a triangle. More...
	~Triangle ()
	Empty.
Vector	operator[] (int) const
	Return the i-th vertex. More...
Vector	Vertex (const double &, const double &) const
	Compute a point in the triangle, given uv-coordinates. More...
bool	Intersect (const Ray &, double &) const
	Compute the intersection between a ray and a triangle. More...
bool	Intersect (const Ray &, double &, double &, double &) const
	Compute the intersection between a ray and a triangle. More...
bool	Intersect (const Sphere &) const
	Check if a sphere intersects or contains a triangle. More...
void	Translate (const Vector &)
	Translates a triangle by a given vector. More...
void	Rotate (const Matrix &)
	Rotates a triangle. More...
void	Scale (const Vector &)
	Scale a triangle. More...
void	Transform (const Frame &)
	Transforms a triangle given a frame transformation. More...
Triangle	Transformed (const FrameScaled &) const
	Transforms a triangle given a frame transformation. More...
void	Shrink (const double &)
	Shrinks a triangle. More...
Vector	Normal () const
	Compute the normal vector of the triangle. More...
Vector	AreaNormal () const
	Compute the normal vector of the triangle, and scale the normal using its area. More...
Vector	Center () const
	Compute the barycenter of the triangle.
double	R (const Vector &) const
	Computes the squared distance between a point and a triangle. More...
Vector	Normal (const Vector &) const
	Computes the normal vector to a triangle. More...
Sphere	Inscribed () const
	Compute the sphere inscribed in the triangle. More...
Sphere	Circumscribed () const
	Compute the sphere circumscribing the triangle. More...
double	Area () const
	Compute the area of the triangle.
double	Aspect () const
	Computes the aspect ratio of the triangle. More...
Box	GetBox () const
	Computes the axis aligned box enclosing the triangle.
int	Intersect (const Triangle &, int &, Vector &, Vector &) const
	Check if two triangles intersect. More...
int	Intersect (const Plane &, Segment &) const
	Compute the intersection between a triangle and a plane. More...
int	Intersect (const double &, Segment &) const
	Compute the intersection between a triangle and a horizontal plane. More...
Vector	BarycentricCoordinates (const Vector &) const
	Compute the barycentric coordinates of an argument vector with respect to the triangle. More...
void	Subdivide (int, QVector< Vector > &, QVector< int > &) const
	Create an n-adic subdivision of a triangle. More...
double	InscribedRadius () const
	Compute the radius of the circle inscribed in the triangle. More...
double	CircumscribedRadius () const
	Compute the radius of the circumscribed circle of the triangle.
Vector	RandomInside (Random &=Random::R239) const
	Generate a random vector inside the triangle. More...

Protected Attributes
Vector	p [3]
	Array of vertices.

Static Protected Attributes
static constexpr const double	epsilon = 1.0e-7
	Internal epsilon constant.

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

Detailed Description

Base minimum storage triangle class.

The data-structure of the triangle does not include the normalized normal vector.

Constructor & Destructor Documentation

Triangle()

Triangle::Triangle ( const Vector &  a, const Vector &  b, const Vector &  c  )

inlineexplicit

Create a triangle.

Parameters

a,b,c

Vertices of the triangle.

Member Function Documentation

AreaNormal()

Vector Triangle::AreaNormal

(

)

const

Compute the normal vector of the triangle, and scale the normal using its area.

This function is less expensive than Triangle::Normal() as it does not require normalizing the cross product of the edge vectors.

IIt is useful for computing the vertex normals of a triangle mesh by averaging the face normals of the triangles and weighting the influence of the different triangles with their area.

See also

Triangle::Normal()

Aspect()

double Triangle::Aspect

(

)

const

Computes the aspect ratio of the triangle.

It is defined as the radius of the inscribed circle divided by the radius of the circumscribing circle. This function is faster than:

Triangle t;
double a=2.0*t.InscribedRadius()/t.CircumscribedRadius();

BarycentricCoordinates()

Vector Triangle::BarycentricCoordinates

(

const Vector &

p

)

const

Compute the barycentric coordinates of an argument vector with respect to the triangle.

If the point lies within the triangle, the coordinates are positive and form a partition of the unity. This no longer holds if the point is outside of the triangle.

Note that this function is not the most efficient, as it relies on the evaluation of the area formed by the triangles pab, pbc, and pca.

Parameters

p

Point.

Circumscribed()

Sphere Triangle::Circumscribed

(

)

const

Compute the sphere circumscribing the triangle.

This function directly relies on the constructor of the class Sphere.

See also

Sphere::Sphere(const Vector&,const Vector&,const Vector&);

Inscribed()

Sphere Triangle::Inscribed

(

)

const

Compute the sphere inscribed in the triangle.

The center of the sphere is the center of (A,a), (B,b) and (C,c) where A, B and C are the vertices of the triangle and coefficients a, b and c represent the length of their facing edge.

InscribedRadius()

double Triangle::InscribedRadius

(

)

const

Compute the radius of the circle inscribed in the triangle.

Some algebra will show that it is half the ratio of the half perimeter and the surface of the triangle.

Intersect() [1/6]

int Triangle::Intersect	(	const double &	z,
		Segment &	segment
	)		const

Compute the intersection between a triangle and a horizontal plane.

Parameters

z	Height of horizontal plane.
segment	Returned segment if intersection occurs.

Returns

The number of intersections, denoted as n. If n=1 then only one vertex of the triangle is one the plane, whereas if n=3 then the triangle is on the plane.

See also

int Triangle::Intersect(const double&,Segment&)

Intersect() [2/6]

int Triangle::Intersect	(	const Plane &	plane,
		Segment &	segment
	)		const

Compute the intersection between a triangle and a plane.

Parameters

plane	The plane.
segment	Returned segment if intersection occurs.

Returns

The number of intersections, denoted as n. If n=1 then only one vertex of the triangle is one the plane, whereas if n=3 then the triangle is on the plane.

See also

int Triangle::Intersect(const double&,Segment&)

Intersect() [3/6]

bool Triangle::Intersect	(	const Ray &	ray,
		double &	t
	)		const

Compute the intersection between a ray and a triangle.

After Tomas Moller and Ben Trumbore, Fast, minimum storage ray-triangle intersection, Journal of graphics tools, 2(1):21-28, 1997.

Parameters

ray	The ray (direction should be of unit length).
t	Intersection depth.

Intersect() [4/6]

bool Triangle::Intersect	(	const Ray &	ray,
		double &	t,
		double &	u,
		double &	v
	)		const

Compute the intersection between a ray and a triangle.

Return the barycentric coordinates of the intersection if within the triangle.

After Tomas Moller and Ben Trumbore, Fast, minimum storage ray-triangle intersection, Journal of graphics tools, 2(1):21-28, 1997.

Parameters

ray	The ray (direction should be of unit length).
t	Intersection depth.
u,v	Parametric coordinates of the intersection depth in the triangle.

Intersect() [5/6]

bool Triangle::Intersect

(

const Sphere &

sphere

)

const

Check if a sphere intersects or contains a triangle.

It does not attempt to compute the set of intersection.

If a triangle vertex lies exactly on the sphere, then no intersection occurs. Apart from this limit case, the algorithm proceeds as follows.
1. All three triangle vertices are contained in the sphere. The sphere completely contains the triangle.
2. At least one vertex is inside the sphere and at least one vertex is outside the sphere. The sphere and triangle intersect.
3. All three vertices are outside the sphere. The sphere and triangle intersect when the distance from sphere center to triangle is less than the sphere radius.

Parameters

sphere

The sphere.

Intersect() [6/6]

int Triangle::Intersect	(	const Triangle &	U,
		int &	coplanar,
		Vector &	isectpt1,
		Vector &	isectpt2
	)		const

Check if two triangles intersect.

This function analyses all cases : edge intersecting coplanar triangles, fully inside or outside triangles, face intersecting triangles in three dimensionnal space.

Parameters

U	Triangle.
isectpt1,isectpt2	end vertices of segment if intersection occurs.
coplanar	Coplanar flag.

After T. Moller, A Fast Triangle-Triangle Intersection Test, Journal of graphics tools, 2(2):25-30, 1997.

Normal() [1/2]

Vector Triangle::Normal

(

)

const

Compute the normal vector of the triangle.

This function is expensive as it requires normalizing the cross product of the edge vectors.

See also

TriangleEdge

Normal() [2/2]

Vector Triangle::Normal

(

const Vector &

q

)

const

Computes the normal vector to a triangle.

Parameters

q

Point.

operator[]()

Vector Triangle::operator[] ( int  i ) const

inline

Return the i-th vertex.

Parameters

i

Index.

R()

double Triangle::R

(

const Vector &

q

)

const

Computes the squared distance between a point and a triangle.

Parameters

q

Point.

RandomInside()

Vector Triangle::RandomInside

(

Random &

random = Random::R239

)

const

Generate a random vector inside the triangle.

After R. Osada, T. Funkhouser, B. Chazelle, D. Dobkin. Shape Distributions, ACM Transactions on Graphics, 21(4), 807-832, 2002.

Parameters

random

Random number generator.

Rotate()

void Triangle::Rotate

(

const Matrix &

r

)

Rotates a triangle.

Parameters

r	Rotation matrix.

Scale()

void Triangle::Scale

(

const Vector &

u

)

Scale a triangle.

Parameters

u	Scaling vector.

Shrink()

void Triangle::Shrink

(

const double &

e

)

Shrinks a triangle.

This function erodes a triangle by moving its edges by a given distance.

Parameters

e	Erosion radius.

Subdivide()

void Triangle::Subdivide	(	int	n,
		QVector< Vector > &	vertex,
		QVector< int > &	index
	)		const

Create an n-adic subdivision of a triangle.

Parameters

n	Subdivision level.
vertex	Array of points.
index	Array of indexes defining the triangles.

Transform()

void Triangle::Transform

(

const Frame &

t

)

Transforms a triangle given a frame transformation.

Parameters

t	Transformation.

Transformed()

Triangle Triangle::Transformed

(

const FrameScaled &

t

)

const

Transforms a triangle given a frame transformation.

Parameters

t	Transformation.

Translate()

void Triangle::Translate

(

const Vector &

u

)

Translates a triangle by a given vector.

Parameters

u	Translation vector.

Vertex()

Vector Triangle::Vertex	(	const double &	u,
		const double &	v
	)		const

Compute a point in the triangle, given uv-coordinates.

Parameters

u,v	Coordinates.

Friends And Related Function Documentation

operator<<

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

friend

Overloaded.

Parameters

t	Triangle.
s	Stream.