Geometric functions and utilities. More...

Functions
vec3	barycenter (const vec3 &p1, const vec3 &p2)
	Computes the barycenter of two points in 3d. More...

vec3	barycenter (const vec3 &p1, const vec3 &p2, const vec3 &p3)
	Computes the barycenter of three points in 3d. More...

double	cos_angle (const vec3 &a, const vec3 &b)
	Computes the cosine of the angle between two 3d vectors. More...

double	angle (const vec3 &a, const vec3 &b)
	Computes the angle between two 3d vectors. More...

double	det (const vec2 &a, const vec2 &b)
	Computes the determinant of two vectors. More...

vec3	triangle_normal (const vec3 &p1, const vec3 &p2, const vec3 &p3)
	Computes the normal of a 3d triangle. More...

double	triangle_area_3d (const double p1, const double p2, const double *p3)
	Computes the area of a 3d triangle. More...

double	triangle_area (const vec3 &p1, const vec3 &p2, const vec3 &p3)
	Computes the area of a 3d triangle. More...

double	triangle_signed_area_2d (const double p1, const double p2, const double *p3)
	Computes the area of a 2d triangle. More...

double	triangle_signed_area (const vec2 &p1, const vec2 &p2, const vec2 &p3)
	Computes the area of a 2d triangle. More...

double	triangle_area_2d (const double p1, const double p2, const double *p3)
	Computes the area of a 2d triangle. More...

vec2	triangle_circumcenter (const vec2 &p1, const vec2 &p2, const vec2 &p3)
	Computes the center of the circumscribed circle of a 2d triangle. More...

bool	has_nan (const vec3 &v)
	Tests whether a 3d vector has a NaN (not a number) coordinate. More...

vec3	perpendicular (const vec3 &V)
	Computes a 3d vector orthogonal to another one. More...

double	tetra_signed_volume (const vec3 &p1, const vec3 &p2, const vec3 &p3, const vec3 &p4)
	Computes the signed volume of a 3d tetrahedron. More...

double	tetra_signed_volume (const double p1, const double p2, const double p3, const double p4)
	Computes the signed volume of a 3d tetrahedron. More...

double	tetra_volume (const vec3 &p1, const vec3 &p2, const vec3 &p3, const vec3 &p4)
	Computes the volume of a 3d tetrahedron. More...

vec3	tetra_circum_center (const vec3 &p1, const vec3 &p2, const vec3 &p3, const vec3 &p4)
	Computes the center of the circumscribed sphere of 3d tetrahedron. More...

void	triangle_centroid (const vec3 &p, const vec3 &q, const vec3 &r, double a, double b, double c, vec3 &Vg, double &V)
	Computes the centroid of a 3d triangle with weighted points. More...

double	triangle_mass (const vec3 &p, const vec3 &q, const vec3 &r, double a, double b, double c)
	Computes the mass of a 3d triangle with weighted points. More...

vec3	random_point_in_triangle (const vec3 &p1, const vec3 &p2, const vec3 &p3)
	Generates a random point in a 3d triangle. More...

template<class COORD_T >
double	distance2 (const COORD_T p1, const COORD_T p2, coord_index_t dim)
	Computes the squared distance between two nd points. More...

template<class COORD_T >
double	distance (const COORD_T p1, const COORD_T p2, coord_index_t dim)
	Computes the distance between two nd points. More...

template<class VEC >
double	distance2 (const VEC &p1, const VEC &p2)
	Computes the squared distance between two nd points. More...

template<class VEC >
double	distance (const VEC &p1, const VEC &p2)
	Computes the distance between two nd points. More...

template<class COORD_T >
double	triangle_area (const COORD_T p1, const COORD_T p2, const COORD_T *p3, coord_index_t dim)
	Computes the area of a nd triangle. More...

template<class COORD_T >
void	triangle_centroid (const COORD_T p, const COORD_T q, const COORD_T r, COORD_T a, COORD_T b, COORD_T c, double Vg, double &V, coord_index_t dim)
	Computes the centroid of a 3d triangle with weighted points. More...

template<class VEC >
double	triangle_area (const VEC &p1, const VEC &p2, const VEC &p3)
	Computes the area of a nd triangle. More...

template<class VEC >
double	triangle_mass (const VEC &p, const VEC &q, const VEC &r, double a, double b, double c)
	Computes the mass of a nd triangle with weighted points. More...

template<class POINT >
POINT	triangle_circumcenter (const POINT &Q1, const POINT &Q2, const POINT &Q3, double *denom=nullptr)
	Computes the center of the circumscribed circle of a nd triangle. More...

template<class VEC >
void	triangle_centroid (const VEC &p, const VEC &q, const VEC &r, double a, double b, double c, VEC &Vg, double &V)
	Computes the centroid of a nd triangle with weighted points. More...

template<class VEC >
VEC	random_point_in_triangle (const VEC &p1, const VEC &p2, const VEC &p3)
	Generates a random point in a nd triangle. More...

template<class VEC >
VEC	random_point_in_tetra (const VEC &p1, const VEC &p2, const VEC &p3, const VEC &p4)
	Generates a random point in a nd tetrahedron. More...

template<class VEC >
double	point_segment_squared_distance (const VEC &point, const VEC &V0, const VEC &V1, VEC &closest_point, double &lambda0, double &lambda1)
	Computes the point closest to a given point in a nd segment. More...

template<class VEC >
double	point_segment_squared_distance (const VEC &point, const VEC &V0, const VEC &V1)
	Computes the point closest to a given point in a nd segment. More...

template<class VEC >
double	point_triangle_squared_distance (const VEC &point, const VEC &V0, const VEC &V1, const VEC &V2, VEC &closest_point, double &lambda0, double &lambda1, double &lambda2)
	Computes the point closest to a given point in a nd triangle. More...

template<class VEC >
double	point_triangle_squared_distance (const VEC &p, const VEC &q1, const VEC &q2, const VEC &q3)
	Computes the squared distance between a point and a nd triangle. More...

double	tetra_volume_from_edge_lengths (double u, double U, double v, double V, double w, double W)
	Computes the volume of a tetrahedron from edge lengths. More...

template<class VEC >
double	tetra_volume (const VEC &p1, const VEC &p2, const VEC &p3, const VEC &p4)
	Computes the volume of a nd tetrahedron. More...

template<int DIM>
double	tetra_volume (const double p1, const double p2, const double p3, const double p4)
	Computes the volume of a nd tetrahedron. More...

template<>
double	tetra_volume< 3 > (const double p1, const double p2, const double p3, const double p4)
	Computes the volume of a 3d tetrahedron. More...

const vec3 &	mesh_vertex (const Mesh &M, index_t v)
	Gets a mesh vertex by its index. More...

const vec3 &	mesh_vertex_ref (const Mesh &M, index_t v)
	Gets a mesh vertex by its index. More...

vec3 &	mesh_vertex_ref (Mesh &M, index_t v)
	Gets a mesh vertex by its index. More...

const vec3 &	mesh_corner_vertex (const Mesh &M, index_t c)
	Gets a mesh vertex by an incident corner index. More...

vec3 &	mesh_corner_vertex_ref (Mesh &M, index_t c)
	Gets a mesh vertex by an incident corner index. More...

const vec3 &	mesh_vertex_normal (const Mesh &M, index_t v)
	Gets a mesh vertex normal by vertex index. More...

vec3 &	mesh_vertex_normal_ref (Mesh &M, index_t v)
	Gets a mesh vertex normal by vertex index. More...

const vec3 &	mesh_vertex_normal_ref (const Mesh &M, index_t v)
	Gets a mesh vertex normal by vertex index. More...

double	mesh_facet_area (const Mesh &M, index_t f, index_t dim=0)
	Computes the area of a facet. More...

vec3	mesh_facet_normal (const Mesh &M, index_t f)
	Computes the normal to a mesh facet. More...

vec3	mesh_facet_center (const Mesh &M, index_t f)
	Gets the centroid of the vertices of a facet in a mesh. More...

vec3	mesh_cell_center (const Mesh &M, index_t c)
	Gets the centroid of the vertices of a cell in a mesh. More...

vec3	mesh_tet_center (const Mesh &M, index_t t)
	Gets the centroid of a tetrahedron in a mesh. More...

vec3	mesh_corner_vector (const Mesh &M, index_t c1)
	Gets a vector by a mesh corner. More...

double	mesh_normal_angle (const Mesh &M, index_t c)
	Computes the angle between the normal vectors of two mesh facets sharing an edge. More...

double	mesh_unsigned_normal_angle (const Mesh &M, index_t f1, index_t f2)
	Computes the angle between the normal vectors of two mesh facets sharing an edge. More...

double	mesh_area (const Mesh &M, index_t dim)
	Computes the total surface area of a mesh in arbitrary dimension. More...

double	mesh_area (const Mesh &M)
	Computes the total surface area of a mesh. More...

double	mesh_enclosed_volume (const Mesh &M)
	Computes the volume enclosed by a surfacic mesh. More...

const vec3 &	halfedge_vertex_from (const Mesh &M, const MeshHalfedges::Halfedge &H)
	Gets the origin point of a Halfedge. More...

const vec3 &	halfedge_vertex_to (const Mesh &M, const MeshHalfedges::Halfedge &H)
	Gets the arrow extremity point of a Halfedge. More...

vec3	halfedge_vector (const Mesh &M, const MeshHalfedges::Halfedge &H)
	Gets a 3d vector that connects the origin with the arrow extremity of a Halfedge. More...

double	edge_length (const Mesh &M, const MeshHalfedges::Halfedge &H)
	Gets the length of a Halfedge. More...

index_t	colocate (const double *points, coord_index_t dim, index_t nb_points, vector< index_t > &old2new, double tolerance=0.0, index_t stride=0, const std::string &nn_algo="default")
	Finds sets of identical points in a point set. More...

index_t	colocate_by_lexico_sort (const double *points, coord_index_t dim, index_t nb_points, vector< index_t > &old2new, index_t stride)
	Finds sets of identical points in a point set. More...

Detailed Description

Geometric functions and utilities.

Function Documentation

◆ angle()

double GEO::Geom::angle	(	const vec3 &	a,
		const vec3 &	b
	)

inline

Computes the angle between two 3d vectors.

Computes the angle between two 2d vectors.

Parameters

[in]	a	first vector
[in]	b	second vector

Returns: the angle between a and b in radians, in the interval \( [ 0 \ldots \pi ] \).

Parameters

[in]	a	first vector
[in]	b	second vector

Returns: the angle between and in radians, in the interval \( [-\pi \ldots \pi] \)

Definition at line 266 of file geometry.h.

◆ barycenter() [1/2]

vec3 GEO::Geom::barycenter	(	const vec3 &	p1,
		const vec3 &	p2
	)

inline

Computes the barycenter of two points in 3d.

Computes the barycenter of two points in 2d.

Parameters

[in]	p1	first point
[in]	p2	second point

Returns: the barycenter of p1 and p2

Definition at line 189 of file geometry.h.

◆ barycenter() [2/2]

vec3 GEO::Geom::barycenter	(	const vec3 &	p1,
		const vec3 &	p2,
		const vec3 &	p3
	)

inline

Computes the barycenter of three points in 3d.

Computes the barycenter of three points in 2d.

Parameters

[in]	p1	first point
[in]	p2	second point
[in]	p3	third point

Returns: the barycenter of p1, p2 and p3

Definition at line 217 of file geometry.h.

◆ colocate()

index_t GEO::Geom::colocate	(	const double *	points,
		coord_index_t	dim,
		index_t	nb_points,
		vector< index_t > &	old2new,
		double	tolerance = `0.0`,
		index_t	stride = `0`,
		const std::string &	nn_algo = `"default"`
	)

Finds sets of identical points in a point set.

Parameters

[in]	points	the point array
[in]	dim	dimension of the points
[in]	nb_points	number of points
[out]	old2new	an array of size nb_points.
[in]	tolerance	threshold for colocating points.
[in]	stride	number of doubles between two consecutive points (set to dim if unspecified).
[in]	nn_algo	factory name for nearest neighbor search.

Returns: the number of unique points

◆ colocate_by_lexico_sort()

index_t GEO::Geom::colocate_by_lexico_sort	(	const double *	points,
		coord_index_t	dim,
		index_t	nb_points,
		vector< index_t > &	old2new,
		index_t	stride
	)

Finds sets of identical points in a point set.

This version uses a lexicographic sort. It does not have a 'tolerance' parameter (only points with exactly the same coordinates can be colocated).

Parameters

[in]	points	the point array
[in]	dim	dimension of the points
[in]	nb_points	number of points
[out]	old2new	an array of size nb_points.
[in]	stride	number of doubles between two consecutive points (set to dim if unspecified).

Returns: the number of unique points

◆ cos_angle()

double GEO::Geom::cos_angle	(	const vec3 &	a,
		const vec3 &	b
	)

inline

Computes the cosine of the angle between two 3d vectors.

Computes the cosine of the angle between two 2d vectors.

Parameters

[in]	a	first vector
[in]	b	second vector

Returns: the cosine of the angle between a and b

Definition at line 249 of file geometry.h.

◆ det()

double GEO::Geom::det	(	const vec2 &	a,
		const vec2 &	b
	)

inline

Computes the determinant of two vectors.

Parameters

[in]	a	first vector
[in]	b	second vector

Returns: the determinant of a and b

Definition at line 292 of file geometry.h.

◆ distance() [1/2]

template<class COORD_T >

double GEO::Geom::distance	(	const COORD_T *	p1,
		const COORD_T *	p2,
		coord_index_t	dim
	)

inline

Computes the distance between two nd points.

Parameters

[in]	p1	a pointer to the coordinates of the first point
[in]	p2	a pointer to the coordinates of the second point
[in]	dim	dimension (number of coordinates of the points)

Returns: the distance between p1 and p2

Template Parameters

COORD_T the numeric type of the point coordinates

Definition at line 84 of file geometry_nd.h.

◆ distance() [2/2]

template<class VEC >

double GEO::Geom::distance	(	const VEC &	p1,
		const VEC &	p2
	)

inline

Computes the distance between two nd points.

Parameters

[in]	p1	first point
[in]	p2	second point

Template Parameters

VEC	the class that represents the points. VEC needs to implement data(), that returns a pointer to the coordinates of the point.

Returns: the distance between p1 and p2

Definition at line 120 of file geometry_nd.h.

◆ distance2() [1/2]

template<class COORD_T >

double GEO::Geom::distance2	(	const COORD_T *	p1,
		const COORD_T *	p2,
		coord_index_t	dim
	)

inline

Computes the squared distance between two nd points.

Parameters

[in]	p1	a pointer to the coordinates of the first point
[in]	p2	a pointer to the coordinates of the second point
[in]	dim	dimension (number of coordinates of the points)

Returns: the squared distance between p1 and p2

Template Parameters

COORD_T the numeric type of the point coordinates

Definition at line 65 of file geometry_nd.h.

◆ distance2() [2/2]

template<class VEC >

double GEO::Geom::distance2	(	const VEC &	p1,
		const VEC &	p2
	)

inline

Computes the squared distance between two nd points.

Parameters

[in]	p1	first point
[in]	p2	second point

Template Parameters

VEC	the class that represents the points. VEC needs to implement data(), that returns a pointer to the coordinates of the point.

Returns: the squared distance between p1 and p2

Definition at line 100 of file geometry_nd.h.

◆ edge_length()

double GEO::Geom::edge_length	(	const Mesh &	M,
		const MeshHalfedges::Halfedge &	H
	)

inline

Gets the length of a Halfedge.

Parameters

[in]	M	the Mesh
[in]	H	the Halfedge

Returns: the 3d length of H

Definition at line 358 of file mesh_halfedges.h.

◆ halfedge_vector()

vec3 GEO::Geom::halfedge_vector	(	const Mesh &	M,
		const MeshHalfedges::Halfedge &	H
	)

inline

Gets a 3d vector that connects the origin with the arrow extremity of a Halfedge.

Parameters

[in]	M	the Mesh
[in]	H	the Halfedge

Returns: a 3d vector that connects the origin with the arrow extremity of H

Definition at line 346 of file mesh_halfedges.h.

◆ halfedge_vertex_from()

const vec3& GEO::Geom::halfedge_vertex_from	(	const Mesh &	M,
		const MeshHalfedges::Halfedge &	H
	)

inline

Gets the origin point of a Halfedge.

Parameters

[in]	M	the mesh
[in]	H	the Halfedge

Returns: a const reference to the origin of H

Definition at line 319 of file mesh_halfedges.h.

◆ halfedge_vertex_to()

const vec3& GEO::Geom::halfedge_vertex_to	(	const Mesh &	M,
		const MeshHalfedges::Halfedge &	H
	)

inline

Gets the arrow extremity point of a Halfedge.

Parameters

[in]	M	the mesh
[in]	H	the Halfedge

Returns: a const reference to the arrow extremity of H

Definition at line 331 of file mesh_halfedges.h.

◆ has_nan()

bool GEO::Geom::has_nan ( const vec3 & v )

inline

Tests whether a 3d vector has a NaN (not a number) coordinate.

Tests whether a 2d vector has a NaN (not a number) coordinate.

Parameters

[in] v a 3d vector

Returns: true if one of the coordinates is a NaN, false otherwise

Parameters

[in] v a 2d vector

Returns: true if one of the coordinates is a NaN, false otherwise

Definition at line 429 of file geometry.h.

◆ mesh_area() [1/2]

double GEO::Geom::mesh_area ( const Mesh & M )

inline

Computes the total surface area of a mesh.

Parameters

[in] M the mesh

Returns: the area of the mesh computed in M.vertices.dimension() dim.

Definition at line 301 of file mesh_geometry.h.

◆ mesh_area() [2/2]

double GEO::Geom::mesh_area	(	const Mesh &	M,
		index_t	dim
	)

Computes the total surface area of a mesh in arbitrary dimension.

Parameters

[in]	M	the mesh
[in]	dim	the dimension to be used for the computation

Returns: the area of the mesh M computed in dim d.

Precondition: dim <= M.vertices.dimension()

◆ mesh_cell_center()

vec3 GEO::Geom::mesh_cell_center	(	const Mesh &	M,
		index_t	c
	)

inline

Gets the centroid of the vertices of a cell in a mesh.

Parameters

[in]	M	the mesh
[in]	c	the index of the facet

Returns: the 3d centroid of facet f in M

Definition at line 219 of file mesh_geometry.h.

◆ mesh_corner_vector()

vec3 GEO::Geom::mesh_corner_vector	(	const Mesh &	M,
		index_t	c1
	)

inline

Gets a vector by a mesh corner.

Parameters

[in]	M	a const reference to the mesh
[in]	c1	a corner index in `M`

Returns: a vector originating at c1 and pointing at the next corner around the facet incident to c1

Precondition: M.facets.are_simplices()

Definition at line 256 of file mesh_geometry.h.

◆ mesh_corner_vertex()

const vec3& GEO::Geom::mesh_corner_vertex	(	const Mesh &	M,
		index_t	c
	)

inline

Gets a mesh vertex by an incident corner index.

Parameters

[in]	M	the mesh
[in]	c	the index of a corner incident to the vertex

Returns: a reference to the vth vertex of a mesh

Precondition: M.vertices.dimension() >= 3

Definition at line 100 of file mesh_geometry.h.

◆ mesh_corner_vertex_ref()

vec3& GEO::Geom::mesh_corner_vertex_ref	(	Mesh &	M,
		index_t	c
	)

inline

Gets a mesh vertex by an incident corner index.

Parameters

[in]	M	the mesh
[in]	c	the index of a corner incident to the vertex

Returns: a const reference to the vth vertex of a mesh

Precondition: M.vertices.dimension() >= 3

Definition at line 111 of file mesh_geometry.h.

◆ mesh_enclosed_volume()

double GEO::Geom::mesh_enclosed_volume ( const Mesh & M )

Computes the volume enclosed by a surfacic mesh.

Parameters

[in] M a closed surfacic mesh.

Returns: the volume enclosed by M.

◆ mesh_facet_area()

double GEO::Geom::mesh_facet_area	(	const Mesh &	M,
		index_t	f,
		index_t	dim = `0`
	)

inline

Computes the area of a facet.

Parameters

[in]	M	a const reference to the mesh
[in]	f	index of the facet
[in]	dim	dimension that will be used to compute the area

Returns: the area of the facet, obtained by considering the dim first coordinates of the vertices only

Definition at line 159 of file mesh_geometry.h.

◆ mesh_facet_center()

vec3 GEO::Geom::mesh_facet_center	(	const Mesh &	M,
		index_t	f
	)

inline

Gets the centroid of the vertices of a facet in a mesh.

Parameters

[in]	M	the mesh
[in]	f	the index of the facet

Returns: the 3d centroid of facet f in M

Definition at line 202 of file mesh_geometry.h.

◆ mesh_facet_normal()

vec3 GEO::Geom::mesh_facet_normal	(	const Mesh &	M,
		index_t	f
	)

Computes the normal to a mesh facet.

Parameters

[in]	M	the mesh
[in]	f	the facet index in `M`

Returns: the normal to facet f

Precondition: dimension >= 3

Note: the computed vector is not normalized.

◆ mesh_normal_angle()

double GEO::Geom::mesh_normal_angle	(	const Mesh &	M,
		index_t	c
	)

Computes the angle between the normal vectors of two mesh facets sharing an edge.

Parameters

[in]	M	a const reference to the mesh
[in]	c	a corner index in `M`

Returns: the angle between the facet that contains c and the facet adjacent to c

Precondition: M.facets.are_simplices() && M.corner_adjacent_facet(c) != -1

◆ mesh_tet_center()

vec3 GEO::Geom::mesh_tet_center	(	const Mesh &	M,
		index_t	t
	)

inline

Gets the centroid of a tetrahedron in a mesh.

Parameters

[in]	M	the mesh
[in]	t	the index of the tetrahedron

Returns: the 3d centroid of tetrahedron t in M

Definition at line 235 of file mesh_geometry.h.

◆ mesh_unsigned_normal_angle()

double GEO::Geom::mesh_unsigned_normal_angle	(	const Mesh &	M,
		index_t	f1,
		index_t	f2
	)

Computes the angle between the normal vectors of two mesh facets sharing an edge.

Parameters

[in]	M	a const reference to the mesh
[in]	f1,f2	two facets of the mesh

Returns: the angle between f1 and f2 in radians

◆ mesh_vertex()

const vec3& GEO::Geom::mesh_vertex	(	const Mesh &	M,
		index_t	v
	)

inline

Gets a mesh vertex by its index.

Parameters

[in]	M	the mesh
[in]	v	the index of the vertex

Returns: a const reference to the vth vertex of a mesh

Precondition: M.vertices.dimension() >= 3

Definition at line 64 of file mesh_geometry.h.

◆ mesh_vertex_normal()

const vec3& GEO::Geom::mesh_vertex_normal	(	const Mesh &	M,
		index_t	v
	)

inline

Gets a mesh vertex normal by vertex index.

Parameters

[in]	M	the mesh
[in]	v	the index of the vertex

Returns: a const reference to the stored normal of vertex v

Precondition: M.vertices.dimension() >= 6

Definition at line 122 of file mesh_geometry.h.

◆ mesh_vertex_normal_ref() [1/2]

const vec3& GEO::Geom::mesh_vertex_normal_ref	(	const Mesh &	M,
		index_t	v
	)

inline

Gets a mesh vertex normal by vertex index.

Parameters

[in]	M	the mesh
[in]	v	the index of the vertex

Returns: a const reference to the stored normal of vertex v

Precondition: M.vertices.dimension() >= 6

Definition at line 146 of file mesh_geometry.h.

◆ mesh_vertex_normal_ref() [2/2]

vec3& GEO::Geom::mesh_vertex_normal_ref	(	Mesh &	M,
		index_t	v
	)

inline

Gets a mesh vertex normal by vertex index.

Parameters

[in]	M	the mesh
[in]	v	the index of the vertex

Returns: a reference to the stored normal of vertex v

Precondition: M.vertices.dimension() >= 6

Definition at line 134 of file mesh_geometry.h.

◆ mesh_vertex_ref() [1/2]

const vec3& GEO::Geom::mesh_vertex_ref	(	const Mesh &	M,
		index_t	v
	)

inline

Gets a mesh vertex by its index.

Parameters

[in]	M	the mesh
[in]	v	the index of the vertex

Returns: a const reference to the vth vertex of a mesh

Precondition: M.vertices.dimension() >= 3

Definition at line 76 of file mesh_geometry.h.

◆ mesh_vertex_ref() [2/2]

vec3& GEO::Geom::mesh_vertex_ref	(	Mesh &	M,
		index_t	v
	)

inline

Gets a mesh vertex by its index.

Parameters

[in]	M	the mesh
[in]	v	the index of the vertex

Returns: a reference to the vth vertex of a mesh

Precondition: M.vertices.dimension() >= 3

Definition at line 88 of file mesh_geometry.h.

◆ perpendicular()

vec3 GEO::Geom::perpendicular ( const vec3 & V )

Computes a 3d vector orthogonal to another one.

Parameters

[in] V a 3d vector

Returns: a 3d vector orthogonal to V

◆ point_segment_squared_distance() [1/2]

template<class VEC >

double GEO::Geom::point_segment_squared_distance	(	const VEC &	point,
		const VEC &	V0,
		const VEC &	V1
	)

inline

Computes the point closest to a given point in a nd segment.

Parameters

[in]	point	the query point
[in]	V0	first extremity of the segment
[in]	V1	second extremity of the segment

Template Parameters

VEC	the class that represents the points.

Returns: the squared distance between the point and the segment [V0, V1]

Definition at line 441 of file geometry_nd.h.

◆ point_segment_squared_distance() [2/2]

template<class VEC >

double GEO::Geom::point_segment_squared_distance	(	const VEC &	point,
		const VEC &	V0,
		const VEC &	V1,
		VEC &	closest_point,
		double &	lambda0,
		double &	lambda1
	)

inline

Computes the point closest to a given point in a nd segment.

Parameters

[in]	point	the query point
[in]	V0	first extremity of the segment
[in]	V1	second extremity of the segment
[out]	closest_point	the point closest to `point` in the segment [`V0`, `V1`]
[out]	lambda0	barycentric coordinate of the closest point relative to `V0`
[out]	lambda1	barycentric coordinate of the closest point relative to `V1`

Template Parameters

VEC	the class that represents the points.

Returns: the squared distance between the point and the segment [V0, V1]

Definition at line 403 of file geometry_nd.h.

◆ point_triangle_squared_distance() [1/2]

template<class VEC >

double GEO::Geom::point_triangle_squared_distance	(	const VEC &	p,
		const VEC &	q1,
		const VEC &	q2,
		const VEC &	q3
	)

inline

Computes the squared distance between a point and a nd triangle.

See http://www.geometrictools.com/LibMathematics/Distance/Distance.html

Parameters

[in]	p	the query point
[in]	q1	first vertex of the triangle
[in]	q2	second vertex of the triangle
[in]	q3	third vertex of the triangle

Template Parameters

VEC	the class that represents the points.

Returns: the squared distance between the point and the triangle (V0, V1, V2)

Definition at line 696 of file geometry_nd.h.

◆ point_triangle_squared_distance() [2/2]

template<class VEC >

double GEO::Geom::point_triangle_squared_distance	(	const VEC &	point,
		const VEC &	V0,
		const VEC &	V1,
		const VEC &	V2,
		VEC &	closest_point,
		double &	lambda0,
		double &	lambda1,
		double &	lambda2
	)

inline

Computes the point closest to a given point in a nd triangle.

See http://www.geometrictools.com/LibMathematics/Distance/Distance.html

Parameters

[in]	point	the query point
[in]	V0	first vertex of the triangle
[in]	V1	second vertex of the triangle
[in]	V2	third vertex of the triangle
[out]	closest_point	the point closest to `point` in the triangle (`V0`, `V1`, `V2`)
[out]	lambda0	barycentric coordinate of the closest point relative to `V0`
[out]	lambda1	barycentric coordinate of the closest point relative to `V1`
[out]	lambda2	barycentric coordinate of the closest point relative to `V2`

Template Parameters

VEC	the class that represents the points.

Returns: the squared distance between the point and the triangle (V0, V1, V2)

Definition at line 475 of file geometry_nd.h.

◆ random_point_in_tetra()

template<class VEC >

VEC GEO::Geom::random_point_in_tetra	(	const VEC &	p1,
		const VEC &	p2,
		const VEC &	p3,
		const VEC &	p4
	)

inline

Generates a random point in a nd tetrahedron.

Uses Greg Turk's second method (see article in Graphic Gems).

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle
[in]	p4	fourth vertex of the triangle

Returns: a random point in tetrahedron ( p1, p2, p3, p4)

Template Parameters

VEC	the class used to represent the vertices of the triangle

Definition at line 364 of file geometry_nd.h.

◆ random_point_in_triangle() [1/2]

template<class VEC >

VEC GEO::Geom::random_point_in_triangle	(	const VEC &	p1,
		const VEC &	p2,
		const VEC &	p3
	)

inline

Generates a random point in a nd triangle.

Uses Greg Turk's second method (see article in Graphic Gems).

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle

Returns: a random point in triangle ( p1, p2, p3 )

Template Parameters

VEC	the class used to represent the vertices of the triangle

Definition at line 338 of file geometry_nd.h.

◆ random_point_in_triangle() [2/2]

vec3 GEO::Geom::random_point_in_triangle	(	const vec3 &	p1,
		const vec3 &	p2,
		const vec3 &	p3
	)

inline

Generates a random point in a 3d triangle.

Uses Greg Turk's second method. Reference: Greg Turk, Generating Random Points in Triangles, Graphics Gems, p. 24-28, code: p. 649-650.

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle

Returns: a random point in triangle ( p1, p2, p3 )

Definition at line 582 of file geometry.h.

◆ tetra_circum_center()

vec3 GEO::Geom::tetra_circum_center	(	const vec3 &	p1,
		const vec3 &	p2,
		const vec3 &	p3,
		const vec3 &	p4
	)

Computes the center of the circumscribed sphere of 3d tetrahedron.

Parameters

[in]	p1	first vertex of the tetrahedron
[in]	p2	second vertex of the tetrahedron
[in]	p3	third vertex of the tetrahedron
[in]	p4	fourth vertex of the tetrahedron

Returns: the circumcenter of the tetrahedron (p1, p2, p3, p4)

◆ tetra_signed_volume() [1/2]

double GEO::Geom::tetra_signed_volume	(	const double *	p1,
		const double *	p2,
		const double *	p3,
		const double *	p4
	)

inline

Computes the signed volume of a 3d tetrahedron.

Parameters

[in]	p1	first vertex of the tetrahedron
[in]	p2	second vertex of the tetrahedron
[in]	p3	third vertex of the tetrahedron
[in]	p4	fourth vertex of the tetrahedron

Returns: the signed volume of the tetrahedron (p1, p2, p3, p4)

Definition at line 479 of file geometry.h.

◆ tetra_signed_volume() [2/2]

double GEO::Geom::tetra_signed_volume	(	const vec3 &	p1,
		const vec3 &	p2,
		const vec3 &	p3,
		const vec3 &	p4
	)

inline

Computes the signed volume of a 3d tetrahedron.

Parameters

[in]	p1	first vertex of the tetrahedron
[in]	p2	second vertex of the tetrahedron
[in]	p3	third vertex of the tetrahedron
[in]	p4	fourth vertex of the tetrahedron

Returns: the signed volume of the tetrahedron (p1, p2, p3, p4)

Definition at line 463 of file geometry.h.

◆ tetra_volume() [1/3]

template<int DIM>

double GEO::Geom::tetra_volume	(	const double *	p1,
		const double *	p2,
		const double *	p3,
		const double *	p4
	)

inline

Computes the volume of a nd tetrahedron.

Uses a form of generalized Heron formula: W. Kahan, "What has the Volume of a Tetrahedron to do with Computer Programming Languages?"

Parameters

[in]	p1	first vertex of the tetrahedron
[in]	p2	second vertex of the tetrahedron
[in]	p3	third vertex of the tetrahedron
[in]	p4	fourth vertex of the tetrahedron

Template Parameters

DIM	dimension of the points

Returns: the volume of the tetrahedron (p1, p2, p3, p4)

Definition at line 783 of file geometry_nd.h.

◆ tetra_volume() [2/3]

template<class VEC >

double GEO::Geom::tetra_volume	(	const VEC &	p1,
		const VEC &	p2,
		const VEC &	p3,
		const VEC &	p4
	)

inline

Computes the volume of a nd tetrahedron.

Uses a form of generalized Heron formula: W. Kahan, "What has the Volume of a Tetrahedron to do with Computer Programming Languages?"

Parameters

[in]	p1	first vertex of the tetrahedron
[in]	p2	second vertex of the tetrahedron
[in]	p3	third vertex of the tetrahedron
[in]	p4	fourth vertex of the tetrahedron

Template Parameters

VEC	the class that represents the points

Returns: the volume of the tetrahedron (p1, p2, p3, p4)

Definition at line 757 of file geometry_nd.h.

◆ tetra_volume() [3/3]

double GEO::Geom::tetra_volume	(	const vec3 &	p1,
		const vec3 &	p2,
		const vec3 &	p3,
		const vec3 &	p4
	)

inline

Computes the volume of a 3d tetrahedron.

Parameters

[in]	p1	first vertex of the tetrahedron
[in]	p2	second vertex of the tetrahedron
[in]	p3	third vertex of the tetrahedron
[in]	p4	fourth vertex of the tetrahedron

Returns: the volume of the tetrahedron (p1, p2, p3, p4)

Definition at line 500 of file geometry.h.

◆ tetra_volume< 3 >()

template<>

double GEO::Geom::tetra_volume< 3 >	(	const double *	p1,
		const double *	p2,
		const double *	p3,
		const double *	p4
	)

inline

Computes the volume of a 3d tetrahedron.

Partial specialization of tetra_volume() for DIM == 3. It uses the standard formula, much simpler than the N dimension version.

Parameters

[in]	p1	first vertex of the tetrahedron
[in]	p2	second vertex of the tetrahedron
[in]	p3	third vertex of the tetrahedron
[in]	p4	fourth vertex of the tetrahedron

Returns: the volume of the tetrahedron (p1, p2, p3, p4)

Definition at line 809 of file geometry_nd.h.

◆ tetra_volume_from_edge_lengths()

double GEO::Geom::tetra_volume_from_edge_lengths	(	double	u,
		double	U,
		double	v,
		double	V,
		double	w,
		double	W
	)

inline

Computes the volume of a tetrahedron from edge lengths.

Uses a form of generalized Heron formula: W. Kahan, "What has the Volume of a Tetrahedron to do with Computer Programming Languages?"

Parameters

[in]	u	distance between p1 and p4
[in]	U	distance between p2 and p3
[in]	v	distance between p2 and p4
[in]	V	distance between p3 and p1
[in]	w	distance between p3 and p4
[in]	W	distance between p1 and p2

Returns: the volume of the tetrahedron

Definition at line 720 of file geometry_nd.h.

◆ triangle_area() [1/3]

template<class COORD_T >

double GEO::Geom::triangle_area	(	const COORD_T *	p1,
		const COORD_T *	p2,
		const COORD_T *	p3,
		coord_index_t	dim
	)

inline

Computes the area of a nd triangle.

Uses Heron formula (that computes the area from the lengths of the three edges).

Parameters

[in]	p1	a pointer to the coordinates of the first vertex of the triangle
[in]	p2	a pointer to the coordinates of the second vertex of the triangle
[in]	p3	a pointer to the coordinates of the third vertex of the triangle
[in]	dim	dimension of the points

Template Parameters

COORD_T the numeric type that represents the coordinates of the points

Returns: the area of triangle ( p1, p2, p3)

Definition at line 143 of file geometry_nd.h.

◆ triangle_area() [2/3]

template<class VEC >

double GEO::Geom::triangle_area	(	const VEC &	p1,
		const VEC &	p2,
		const VEC &	p3
	)

inline

Computes the area of a nd triangle.

Uses Heron formula (that compures the area from the lengths of the three edges).

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle

Template Parameters

VEC	the class used to represent the vertices of the triangle

Returns: the area of triangle (p1, p2, p3)

Definition at line 212 of file geometry_nd.h.

◆ triangle_area() [3/3]

double GEO::Geom::triangle_area	(	const vec3 &	p1,
		const vec3 &	p2,
		const vec3 &	p3
	)

inline

Computes the area of a 3d triangle.

Computes the area of a 2d triangle.

Parameters

[in] p1,p2,p3 the three vertices of the triangle

Returns: the area of the triangle (p1, p2, p3).

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle

Returns: the area of the triangle (p1, p2, p3).

Definition at line 348 of file geometry.h.

◆ triangle_area_2d()

double GEO::Geom::triangle_area_2d	(	const double *	p1,
		const double *	p2,
		const double *	p3
	)

inline

Computes the area of a 2d triangle.

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle

Returns: the area of the triangle (p1, p2, p3).

Definition at line 406 of file geometry.h.

◆ triangle_area_3d()

double GEO::Geom::triangle_area_3d	(	const double *	p1,
		const double *	p2,
		const double *	p3
	)

inline

Computes the area of a 3d triangle.

Parameters

[in] p1,p2,p3 the three vertices of the triangle

Returns: the area of the triangle (p1, p2, p3).

Definition at line 326 of file geometry.h.

◆ triangle_centroid() [1/3]

template<class COORD_T >

void GEO::Geom::triangle_centroid	(	const COORD_T *	p,
		const COORD_T *	q,
		const COORD_T *	r,
		COORD_T	a,
		COORD_T	b,
		COORD_T	c,
		double *	Vg,
		double &	V,
		coord_index_t	dim
	)

inline

Computes the centroid of a 3d triangle with weighted points.

The integrated weight varies linearly in the triangle.

Parameters

[in]	p	a pointer to the coordinates of the first vertex of the triangle
[in]	q	a pointer to the coordinates of the second vertex of the triangle
[in]	r	a pointer to the coordinates of the third vertex of the triangle
[in]	a	the weight associated with vertex `p`
[in]	b	the weight associated with vertex `q`
[in]	c	the weight associated with vertex `r`
[out]	Vg	the total weight times the centroid ( a pointer to a caller-allocated array of dim COORD_Ts)
[out]	V	the total weight
[in]	dim	the dimension of the vertices

Template Parameters

COORD_T the numeric type that represents the coordinates of the points

Definition at line 178 of file geometry_nd.h.

◆ triangle_centroid() [2/3]

template<class VEC >

void GEO::Geom::triangle_centroid	(	const VEC &	p,
		const VEC &	q,
		const VEC &	r,
		double	a,
		double	b,
		double	c,
		VEC &	Vg,
		double &	V
	)

inline

Computes the centroid of a nd triangle with weighted points.

The integrated weight varies linearly in the triangle.

Parameters

[in]	p	first vertex of the triangle
[in]	q	second vertex of the triangle
[in]	r	third vertex of the triangle
[in]	a	the weight associated with vertex `p`
[in]	b	the weight associated with vertex `q`
[in]	c	the weight associated with vertex `r`
[out]	Vg	the total weight times the centroid
[out]	V	the total weight

Template Parameters

VEC	the class used to represent the vertices of the triangle

Definition at line 311 of file geometry_nd.h.

◆ triangle_centroid() [3/3]

void GEO::Geom::triangle_centroid	(	const vec3 &	p,
		const vec3 &	q,
		const vec3 &	r,
		double	a,
		double	b,
		double	c,
		vec3 &	Vg,
		double &	V
	)

inline

Computes the centroid of a 3d triangle with weighted points.

The integrated weight varies linearly in the triangle.

Parameters

[in]	p	first vertex of the triangle
[in]	q	second vertex of the triangle
[in]	r	third vertex of the triangle
[in]	a	the weight associated with vertex `p`
[in]	b	the weight associated with vertex `q`
[in]	c	the weight associated with vertex `r`
[out]	Vg	the total weight times the centroid
[out]	V	the total weight

Definition at line 534 of file geometry.h.

◆ triangle_circumcenter() [1/2]

template<class POINT >

POINT GEO::Geom::triangle_circumcenter	(	const POINT &	Q1,
		const POINT &	Q2,
		const POINT &	Q3,
		double *	denom = `nullptr`
	)

Computes the center of the circumscribed circle of a nd triangle.

Parameters

[in]	Q1	first vertex of the triangle
[in]	Q2	second vertex of the triangle
[in]	Q3	third vertex of the triangle
[out]	denom	if the parameter is non null, it is set to the denominator of the barycentric coordinates of the circumcenter.

Template Parameters

POINT the class used to represent the vertices of the triangle

Returns: the circumcenter of the triangle (p1, p2, p3).

Definition at line 267 of file geometry_nd.h.

◆ triangle_circumcenter() [2/2]

vec2 GEO::Geom::triangle_circumcenter	(	const vec2 &	p1,
		const vec2 &	p2,
		const vec2 &	p3
	)

Computes the center of the circumscribed circle of a 2d triangle.

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle

Returns: the circumcenter of the triangle (p1, p2, p3).

◆ triangle_mass() [1/2]

template<class VEC >

double GEO::Geom::triangle_mass	(	const VEC &	p,
		const VEC &	q,
		const VEC &	r,
		double	a,
		double	b,
		double	c
	)

inline

Computes the mass of a nd triangle with weighted points.

The integrated weight varies linearly in the triangle.

Parameters

[in]	p	first vertex of the triangle
[in]	q	second vertex of the triangle
[in]	r	third vertex of the triangle
[in]	a	the weight associated with vertex `p`
[in]	b	the weight associated with vertex `q`
[in]	c	the weight associated with vertex `r`

Template Parameters

VEC	the class used to represent the vertices of the triangle

Returns: the mass of the weighted triangle ( p, a), ( q, b), ( r, c)

Definition at line 237 of file geometry_nd.h.

◆ triangle_mass() [2/2]

double GEO::Geom::triangle_mass	(	const vec3 &	p,
		const vec3 &	q,
		const vec3 &	r,
		double	a,
		double	b,
		double	c
	)

inline

Computes the mass of a 3d triangle with weighted points.

The integrated weight varies linearly in the triangle.

Parameters

[in]	p	first vertex of the triangle
[in]	q	second vertex of the triangle
[in]	r	third vertex of the triangle
[in]	a	the weight associated with vertex `p`
[in]	b	the weight associated with vertex `q`
[in]	c	the weight associated with vertex `r`

Returns: the mass of the weighted triangle ( p, a), ( q, b), ( r, c)

Definition at line 563 of file geometry.h.

◆ triangle_normal()

vec3 GEO::Geom::triangle_normal	(	const vec3 &	p1,
		const vec3 &	p2,
		const vec3 &	p3
	)

inline

Computes the normal of a 3d triangle.

Parameters

[in] p1,p2,p3 the three vertices of the triangle

Returns: the normal of the triangle (p1, p2, p3).

Definition at line 315 of file geometry.h.

◆ triangle_signed_area()

double GEO::Geom::triangle_signed_area	(	const vec2 &	p1,
		const vec2 &	p2,
		const vec2 &	p3
	)

inline

Computes the area of a 2d triangle.

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle

Returns: the signed area of the triangle (p1, p2, p3), positive if the triangle is oriented clockwise, negative otherwise.

Definition at line 380 of file geometry.h.

◆ triangle_signed_area_2d()

double GEO::Geom::triangle_signed_area_2d	(	const double *	p1,
		const double *	p2,
		const double *	p3
	)

inline

Computes the area of a 2d triangle.

Parameters

[in]	p1	first vertex of the triangle
[in]	p2	second vertex of the triangle
[in]	p3	third vertex of the triangle

Returns: the signed area of the 2D triangle (p1, p2, p3), positive if the triangle is oriented clockwise, negative otherwise.

Definition at line 362 of file geometry.h.

Functions

Detailed Description

Function Documentation

◆ angle()

◆ barycenter() [1/2]

◆ barycenter() [2/2]

◆ colocate()

◆ colocate_by_lexico_sort()

◆ cos_angle()

◆ det()

◆ distance() [1/2]

◆ distance() [2/2]

◆ distance2() [1/2]

◆ distance2() [2/2]

◆ edge_length()

◆ halfedge_vector()

◆ halfedge_vertex_from()

◆ halfedge_vertex_to()

◆ has_nan()

◆ mesh_area() [1/2]

◆ mesh_area() [2/2]

◆ mesh_cell_center()

◆ mesh_corner_vector()

◆ mesh_corner_vertex()

◆ mesh_corner_vertex_ref()

◆ mesh_enclosed_volume()

◆ mesh_facet_area()

◆ mesh_facet_center()

◆ mesh_facet_normal()

◆ mesh_normal_angle()

◆ mesh_tet_center()

◆ mesh_unsigned_normal_angle()

◆ mesh_vertex()

◆ mesh_vertex_normal()

◆ mesh_vertex_normal_ref() [1/2]

◆ mesh_vertex_normal_ref() [2/2]

◆ mesh_vertex_ref() [1/2]

◆ mesh_vertex_ref() [2/2]

◆ perpendicular()

◆ point_segment_squared_distance() [1/2]

◆ point_segment_squared_distance() [2/2]

◆ point_triangle_squared_distance() [1/2]

◆ point_triangle_squared_distance() [2/2]

◆ random_point_in_tetra()

◆ random_point_in_triangle() [1/2]

◆ random_point_in_triangle() [2/2]

◆ tetra_circum_center()

◆ tetra_signed_volume() [1/2]

◆ tetra_signed_volume() [2/2]

◆ tetra_volume() [1/3]

◆ tetra_volume() [2/3]

◆ tetra_volume() [3/3]

◆ tetra_volume< 3 >()

◆ tetra_volume_from_edge_lengths()

◆ triangle_area() [1/3]

◆ triangle_area() [2/3]

◆ triangle_area() [3/3]

◆ triangle_area_2d()

◆ triangle_area_3d()

◆ triangle_centroid() [1/3]

◆ triangle_centroid() [2/3]

◆ triangle_centroid() [3/3]

◆ triangle_circumcenter() [1/2]

◆ triangle_circumcenter() [2/2]

◆ triangle_mass() [1/2]

◆ triangle_mass() [2/2]

◆ triangle_normal()

◆ triangle_signed_area()

◆ triangle_signed_area_2d()