CGAL Namespace Reference

Classes
class	Aff_transformation_2
	The class Aff_transformation_2 represents two-dimensional affine transformations. More...
class	Aff_transformation_3
	The class Aff_transformation_3 represents three-dimensional affine transformations. More...
struct	Ambient_dimension
	The class Ambient_dimension allows to retrieve the dimension of the ambient space of a type T in a kernel K. More...
class	Bbox_2
	An object b of the class Bbox_2 is a bounding box in the two-dimensional Euclidean plane \( \E^2\). More...
class	Bbox_3
	An object b of the class Bbox_3 is a bounding box in the three-dimensional Euclidean space \( \E^3\). More...
struct	Cartesian
	A model for Kernel that uses Cartesian coordinates to represent the geometric objects. More...
class	Cartesian_converter
	Cartesian_converter converts objects from the kernel traits K1 to the kernel traits K2 using NTConverter to do the conversion. More...
class	Circle_2
	An object c of type Circle_2 is a circle in the two-dimensional Euclidean plane \( \E^2\). More...
class	Circle_3
	An object c of type Circle_3 is a circle in the three-dimensional Euclidean space \( \E^3\). More...
struct	Dimension_tag
	An object of the class Dimension_tag is an empty object which can be used for dispatching functions based on the dimension of an object, as provided by the dim parameter. More...
class	Direction_2
	An object d of the class Direction_2 is a vector in the two-dimensional vector space \( \mathbb{R}^2\) where we forget about its length. More...
class	Direction_3
	An object of the class Direction_3 is a vector in the three-dimensional vector space \( \mathbb{R}^3\) where we forget about their length. More...
struct	Dynamic_dimension_tag
	An object of the class Dynamic_dimension_tag is an empty object which can be used for dispatching functions based on the dimension of an object. More...
class	Exact_predicates_exact_constructions_kernel
	A typedef to a kernel which has the following properties: More...
class	Exact_predicates_exact_constructions_kernel_with_kth_root
	A typedef to a kernel which has the following properties: More...
class	Exact_predicates_exact_constructions_kernel_with_root_of
	A typedef to a kernel which has the following properties: More...
class	Exact_predicates_exact_constructions_kernel_with_sqrt
	A typedef to a kernel which has the following properties: More...
class	Exact_predicates_inexact_constructions_kernel
	A typedef to a kernel that has the following properties: More...
struct	Feature_dimension
	The class Feature_dimension allows to retrieve the geometric dimension of a type T in a kernel K. More...
struct	Filtered_kernel
	Filtered_kernel is a kernel that uses a filtering technique based on interval arithmetic form to achieve exact and efficient predicates. More...
struct	Filtered_kernel_adaptor
	Filtered_kernel_adaptor is a kernel that uses a filtering technique to obtain a kernel with exact and efficient predicate functors. More...
class	Filtered_predicate
	Filtered_predicate is an adaptor for predicate function objects that allows one to produce efficient and exact predicates. More...
struct	Homogeneous
	A model for a Kernel using homogeneous coordinates to represent the geometric objects. More...
class	Homogeneous_converter
	Homogeneous_converter converts objects from the kernel traits K1 to the kernel traits K2. More...
class	Identity_transformation
	Tag class for affine transformations. More...
class	Iso_cuboid_3
	An object c of the data type Iso_cuboid_3 is a cuboid in the Euclidean space \( \E^3\) with edges parallel to the \( x\), \( y\) and \( z\) axis of the coordinate system. More...
class	Iso_rectangle_2
	An object r of the data type Iso_rectangle_2 is a rectangle in the Euclidean plane \( \E^2\) with sides parallel to the \( x\) and \( y\) axis of the coordinate system. More...
struct	Kernel_traits
	The class Kernel_traits provides access to the kernel model to which the argument type T belongs. More...
class	Line_2
	An object l of the data type Line_2 is a directed straight line in the two-dimensional Euclidean plane \( \E^2\). More...
class	Line_3
	An object l of the data type Line_3 is a directed straight line in the three-dimensional Euclidean space \( \E^3\). More...
class	Null_vector
	CGAL defines a symbolic constant NULL_VECTOR to construct zero length vectors. More...
class	Origin
	CGAL defines a symbolic constant ORIGIN which denotes the point at the origin. More...
class	Plane_3
	An object h of the data type Plane_3 is an oriented plane in the three-dimensional Euclidean space \( \E^3\). More...
class	Point_2
	An object p of the class Point_2 is a point in the two-dimensional Euclidean plane \( \E^2\). More...
class	Point_3
	An object of the class Point_3 is a point in the three-dimensional Euclidean space \( \E^3\). More...
class	Projection_traits_3
	The class Projection_traits_3 works similarly to the Projection_traits_xy_3, Projection_traits_xz_3, and Projection_traits_yz_3 traits classes, enabling the use of 2D algorithms on the projections of 3D data onto an arbitrary plane. More...
class	Projection_traits_xy_3
	The class Projection_traits_xy_3 is an adapter to apply 2D algorithms to the projections of 3D data on the xy-plane. More...
class	Projection_traits_xz_3
	The class Projection_traits_xz_3 is an adapter to apply 2D algorithms to the projections of 3D data on the xz-plane. More...
class	Projection_traits_yz_3
	The class Projection_traits_yz_3 is an adapter to apply 2D algorithms to the projections of 3D data on the yz-plane. More...
class	Ray_2
	An object r of the data type Ray_2 is a directed straight ray in the two-dimensional Euclidean plane \( \E^2\). More...
class	Ray_3
	An object r of the data type Ray_3 is a directed straight ray in the three-dimensional Euclidean space \( \E^3\). More...
class	Reflection
	Tag class for affine transformations. More...
class	Rotation
	Tag class for affine transformations. More...
class	Scaling
	Tag class for affine transformations. More...
class	Segment_2
	An object s of the data type Segment_2 is a directed straight line segment in the two-dimensional Euclidean plane \( \E^2\), i.e. a straight line segment \( [p,q]\) connecting two points \( p,q \in \mathbb{R}^2\). More...
class	Segment_3
	An object s of the data type Segment_3 is a directed straight line segment in the three-dimensional Euclidean space \( \E^3\), that is a straight line segment \( [p,q]\) connecting two points \( p,q \in \R^3\). More...
struct	Simple_cartesian
	A model for a Kernel using Cartesian coordinates to represent the geometric objects. More...
struct	Simple_homogeneous
	A model for a Kernel using homogeneous coordinates to represent the geometric objects. More...
class	Sphere_3
	An object of type Sphere_3 is a sphere in the three-dimensional Euclidean space \( \E^3\). More...
class	Tetrahedron_3
	An object t of the class Tetrahedron_3 is an oriented tetrahedron in the three-dimensional Euclidean space \( \E^3\). More...
class	Translation
	Tag class for affine transformations. More...
class	Triangle_2
	An object t of the class Triangle_2 is a triangle in the two-dimensional Euclidean plane \( \E^2\). More...
class	Triangle_3
	An object t of the class Triangle_3 is a triangle in the three-dimensional Euclidean space \( \E^3\). More...
class	Vector_2
	An object v of the class Vector_2 is a vector in the two-dimensional vector space \( \mathbb{R}^2\). More...
class	Vector_3
	An object of the class Vector_3 is a vector in the three-dimensional vector space \( \mathbb{R}^3\). More...
class	Weighted_point_2
	An object of the class Weighted_point_2 is a tuple of a two-dimensional point and a scalar weight. More...
class	Weighted_point_3
	An object of the class Weighted_point_3 is a tuple of a three-dimensional point and a scalar weight. More...

Typedefs
typedef Sign	Orientation

Enumerations
enum	Angle { OBTUSE , RIGHT , ACUTE }
enum	Bounded_side { ON_UNBOUNDED_SIDE , ON_BOUNDARY , ON_BOUNDED_SIDE }
enum	Comparison_result { SMALLER , EQUAL , LARGER }
enum	Sign { NEGATIVE , ZERO , POSITIVE }
enum	Oriented_side { ON_NEGATIVE_SIDE , ON_ORIENTED_BOUNDARY , ON_POSITIVE_SIDE }
enum	Box_parameter_space_2 {   LEFT_BOUNDARY = 0 , RIGHT_BOUNDARY , BOTTOM_BOUNDARY , TOP_BOUNDARY ,   INTERIOR , EXTERIOR }

Functions
template<typename T , typename U >
T	enum_cast (const U &u)
	converts between the various enums provided by the CGAL kernel.
Oriented_side	opposite (const Oriented_side &o)
	returns the opposite side (for example CGAL::ON_POSITIVE_SIDE if o==CGAL::ON_NEGATIVE_SIDE), or CGAL::ON_ORIENTED_BOUNDARY if o==CGAL::ON_ORIENTED_BOUNDARY.
Bounded_side	opposite (const Bounded_side &o)
	returns the opposite side (for example CGAL::ON_BOUNDED_SIDE if o==CGAL::ON_UNBOUNDED_SIDE), or returns CGAL::ON_BOUNDARY if o==CGAL::ON_BOUNDARY.
bool	do_intersect (Type1< Kernel > obj1, Type2< Kernel > obj2)
	checks whether obj1 and obj2 intersect.
bool	do_intersect (Plane_3< Kernel > obj1, Plane_3< Kernel > obj2, Plane_3< Kernel > obj3)
	checks whether obj1, obj2 and obj3 intersect.
template<typename Kernel >
decltype(auto)	intersection (Type1< Kernel > obj1, Type2< Kernel > obj2)
	Two objects obj1 and obj2 intersect if there is a point p that is part of both obj1 and obj2.
template<typename Kernel >
decltype(auto)	intersection (const Plane_3< Kernel > &pl1, const Plane_3< Kernel > &pl2, const Plane_3< Kernel > &pl3)
	returns the intersection of 3 planes, which can be a point, a line, a plane, or empty.
template<typename Kernel >
Angle	angle (const CGAL::Vector_2< Kernel > &u, const CGAL::Vector_2< Kernel > &v)
	returns CGAL::OBTUSE, CGAL::RIGHT or CGAL::ACUTE depending on the angle formed by the two vectors u and v.
template<typename Kernel >
Angle	angle (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns CGAL::OBTUSE, CGAL::RIGHT or CGAL::ACUTE depending on the angle formed by the three points p, q, r (q being the vertex of the angle).
template<typename Kernel >
Angle	angle (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r, const CGAL::Point_2< Kernel > &s)
	returns CGAL::OBTUSE, CGAL::RIGHT or CGAL::ACUTE depending on the angle formed by the two vectors pq, rs.
template<typename Kernel >
Angle	angle (const CGAL::Vector_3< Kernel > &u, const CGAL::Vector_3< Kernel > &v)
	returns CGAL::OBTUSE, CGAL::RIGHT or CGAL::ACUTE depending on the angle formed by the two vectors u and v.
template<typename Kernel >
Angle	angle (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns CGAL::OBTUSE, CGAL::RIGHT or CGAL::ACUTE depending on the angle formed by the three points p, q, r (q being the vertex of the angle).
template<typename Kernel >
Angle	angle (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	returns CGAL::OBTUSE, CGAL::RIGHT or CGAL::ACUTE depending on the angle formed by the two vectors pq, rs.
template<typename Kernel >
Angle	angle (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Vector_3< Kernel > &v)
	returns CGAL::OBTUSE, CGAL::RIGHT or CGAL::ACUTE depending on the angle formed by the normal of the triangle pqr and the vector v.
template<typename Kernel >
Kernel::FT	approximate_angle (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns an approximation of the angle between p-q and r-q.
template<typename Kernel >
Kernel::FT	approximate_angle (const CGAL::Vector_3< Kernel > &u, const CGAL::Vector_3< Kernel > &v)
	returns an approximation of the angle between u and v.
template<typename Kernel >
Kernel::FT	approximate_dihedral_angle (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	returns an approximation of the signed dihedral angle in the tetrahedron pqrs of edge pq.
template<typename Kernel >
Kernel::FT	area (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns the signed area of the triangle defined by the points p, q and r.
template<typename Kernel >
bool	are_ordered_along_line (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true, iff the three points are collinear and q lies between p and r.
template<typename Kernel >
bool	are_ordered_along_line (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns true, iff the three points are collinear and q lies between p and r.
template<typename Kernel >
bool	are_strictly_ordered_along_line (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true, iff the three points are collinear and q lies strictly between p and r.
template<typename Kernel >
bool	are_strictly_ordered_along_line (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns true, iff the three points are collinear and q lies strictly between p and r.
template<typename Kernel >
CGAL::Point_2< Kernel >	barycenter (const CGAL::Point_2< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_2< Kernel > &p2)
	compute the barycenter of the points p1 and p2 with corresponding weights w1 and 1-w1.
template<typename Kernel >
CGAL::Point_2< Kernel >	barycenter (const CGAL::Point_2< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_2< Kernel > &p2, const Kernel::FT &w2)
	compute the barycenter of the points p1 and p2 with corresponding weights w1 and w2.
template<typename Kernel >
CGAL::Point_2< Kernel >	barycenter (const CGAL::Point_2< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_2< Kernel > &p2, const Kernel::FT &w2, const CGAL::Point_2< Kernel > &p3)
	compute the barycenter of the points p1, p2 and p3 with corresponding weights w1, w2 and 1-w1-w2.
template<typename Kernel >
CGAL::Point_2< Kernel >	barycenter (const CGAL::Point_2< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_2< Kernel > &p2, const Kernel::FT &w2, const CGAL::Point_2< Kernel > &p3, const Kernel::FT &w3)
	compute the barycenter of the points p1, p2 and p3 with corresponding weights w1, w2 and w3.
template<typename Kernel >
CGAL::Point_2< Kernel >	barycenter (const CGAL::Point_2< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_2< Kernel > &p2, const Kernel::FT &w2, const CGAL::Point_2< Kernel > &p3, const Kernel::FT &w3, const CGAL::Point_2< Kernel > &p4)
	compute the barycenter of the points p1, p2, p3 and p4 with corresponding weights w1, w2, w3 and 1-w1-w2-w3.
template<typename Kernel >
CGAL::Point_2< Kernel >	barycenter (const CGAL::Point_2< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_2< Kernel > &p2, const Kernel::FT &w2, const CGAL::Point_2< Kernel > &p3, const Kernel::FT &w3, const CGAL::Point_2< Kernel > &p4, const Kernel::FT &w4)
	compute the barycenter of the points p1, p2, p3 and p4 with corresponding weights w1, w2, w3 and w4.
template<typename Kernel >
CGAL::Point_3< Kernel >	barycenter (const CGAL::Point_3< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_3< Kernel > &p2)
	compute the barycenter of the points p1 and p2 with corresponding weights w1 and 1-w1.
template<typename Kernel >
CGAL::Point_3< Kernel >	barycenter (const CGAL::Point_3< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_3< Kernel > &p2, const Kernel::FT &w2)
	compute the barycenter of the points p1 and p2 with corresponding weights w1 and w2.
template<typename Kernel >
CGAL::Point_3< Kernel >	barycenter (const CGAL::Point_3< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_3< Kernel > &p2, const Kernel::FT &w2, const CGAL::Point_3< Kernel > &p3)
	compute the barycenter of the points p1, p2 and p3 with corresponding weights w1, w2 and 1-w1-w2.
template<typename Kernel >
CGAL::Point_3< Kernel >	barycenter (const CGAL::Point_3< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_3< Kernel > &p2, const Kernel::FT &w2, const CGAL::Point_3< Kernel > &p3, const Kernel::FT &w3)
	compute the barycenter of the points p1, p2 and p3 with corresponding weights w1, w2 and w3.
template<typename Kernel >
CGAL::Point_3< Kernel >	barycenter (const CGAL::Point_3< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_3< Kernel > &p2, const Kernel::FT &w2, const CGAL::Point_3< Kernel > &p3, const Kernel::FT &w3, const CGAL::Point_3< Kernel > &p4)
	compute the barycenter of the points p1, p2, p3 and p4 with corresponding weights w1, w2, w3 and 1-w1-w2-w3.
template<typename Kernel >
CGAL::Point_3< Kernel >	barycenter (const CGAL::Point_3< Kernel > &p1, const Kernel::FT &w1, const CGAL::Point_3< Kernel > &p2, const Kernel::FT &w2, const CGAL::Point_3< Kernel > &p3, const Kernel::FT &w3, const CGAL::Point_3< Kernel > &p4, const Kernel::FT &w4)
	compute the barycenter of the points p1, p2, p3 and p4 with corresponding weights w1, w2, w3 and w4.
template<typename Kernel >
CGAL::Line_2< Kernel >	bisector (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	constructs the bisector line of the two points p and q.
template<typename Kernel >
CGAL::Line_2< Kernel >	bisector (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2)
	constructs the bisector of the two lines l1 and l2.
template<typename Kernel >
CGAL::Plane_3< Kernel >	bisector (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	constructs the bisector plane of the two points p and q.
template<typename Kernel >
CGAL::Plane_3< Kernel >	bisector (const CGAL::Plane_3< Kernel > &h1, const CGAL::Plane_3< Kernel > &h2)
	constructs the bisector of the two planes h1 and h2.
template<typename Kernel >
CGAL::Point_2< Kernel >	centroid (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	compute the centroid of the points p, q, and r.
template<typename Kernel >
CGAL::Point_2< Kernel >	centroid (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r, const CGAL::Point_2< Kernel > &s)
	compute the centroid of the points p, q, r, and s.
template<typename Kernel >
CGAL::Point_2< Kernel >	centroid (const CGAL::Triangle_2< Kernel > &t)
	compute the centroid of the triangle t.
template<typename Kernel >
CGAL::Point_3< Kernel >	centroid (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	compute the centroid of the points p, q, and r.
template<typename Kernel >
CGAL::Point_3< Kernel >	centroid (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	compute the centroid of the points p, q, r, and s.
template<typename Kernel >
CGAL::Point_3< Kernel >	centroid (const CGAL::Triangle_3< Kernel > &t)
	compute the centroid of the triangle t.
template<typename Kernel >
CGAL::Point_3< Kernel >	centroid (const CGAL::Tetrahedron_3< Kernel > &t)
	compute the centroid of the tetrahedron t.
template<typename Kernel >
CGAL::Point_2< Kernel >	circumcenter (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	compute the center of the smallest circle passing through the points p and q.
template<typename Kernel >
CGAL::Point_2< Kernel >	circumcenter (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	compute the center of the circle passing through the points p, q, and r.
template<typename Kernel >
CGAL::Point_2< Kernel >	circumcenter (const CGAL::Triangle_2< Kernel > &t)
	compute the center of the circle passing through the vertices of t.
template<typename Kernel >
CGAL::Point_3< Kernel >	circumcenter (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	compute the center of the smallest sphere passing through the points p and q.
template<typename Kernel >
CGAL::Point_3< Kernel >	circumcenter (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	compute the center of the circle passing through the points p, q, and r.
template<typename Kernel >
CGAL::Point_3< Kernel >	circumcenter (const CGAL::Triangle_3< Kernel > &t)
	compute the center of the circle passing through the vertices of t.
template<typename Kernel >
CGAL::Point_3< Kernel >	circumcenter (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	compute the center of the sphere passing through the points p, q, r, and s.
template<typename Kernel >
CGAL::Point_3< Kernel >	circumcenter (const CGAL::Tetrahedron_3< Kernel > &t)
	compute the center of the sphere passing through the vertices of t.
template<typename Kernel >
bool	collinear_are_ordered_along_line (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true, iff q lies between p and r.
template<typename Kernel >
bool	collinear_are_ordered_along_line (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns true, iff q lies between p and r.
template<typename Kernel >
bool	collinear_are_strictly_ordered_along_line (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true, iff q lies strictly between p and r.
template<typename Kernel >
bool	collinear_are_strictly_ordered_along_line (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns true, iff q lies strictly between p and r.
template<typename Kernel >
bool	collinear (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true, iff p, q, and r are collinear.
template<typename Kernel >
bool	collinear (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns true, iff p, q, and r are collinear.
template<typename Kernel >
Comparison_result	compare_angle (const CGAL::Point_3< Kernel > &a, const CGAL::Point_3< Kernel > &b, const CGAL::Point_3< Kernel > &c, const Kernel::FT &cosine)
	compares the angles \( \theta_1\) and \( \theta_2\), where \( \theta_1\) is the angle in \( [0, \pi]\) of the triangle \( (a, b, c)\) at the vertex b, and \( \theta_2\) is the angle in \( [0, \pi]\) such that \( cos(\theta_2) = cosine\).
template<typename Kernel >
Comparison_result	compare_dihedral_angle (const CGAL::Point_3< Kernel > &a1, const CGAL::Point_3< Kernel > &b1, const CGAL::Point_3< Kernel > &c1, const CGAL::Point_3< Kernel > &d1, const Kernel::FT &cosine)
	compares the dihedral angles \( \theta_1\) and \( \theta_2\), where \( \theta_1\) is the dihedral angle, in \( [0, \pi]\), of the tetrahedron (a1, b1, c1, d1) at the edge (a1, b1), and \( \theta_2\) is the angle in \( [0, \pi]\) such that \( cos(\theta_2) = cosine\).
template<typename Kernel >
Comparison_result	compare_dihedral_angle (const CGAL::Point_3< Kernel > &a1, const CGAL::Point_3< Kernel > &b1, const CGAL::Point_3< Kernel > &c1, const CGAL::Point_3< Kernel > &d1, const CGAL::Point_3< Kernel > &a2, const CGAL::Point_3< Kernel > &b2, const CGAL::Point_3< Kernel > &c2, const CGAL::Point_3< Kernel > &d2)
	compares the dihedral angles \( \theta_1\) and \( \theta_2\), where \( \theta_i\) is the dihedral angle in the tetrahedron (a_i, b_i, c_i, d_i) at the edge (a_i, b_i).
template<typename Kernel >
Comparison_result	compare_dihedral_angle (const CGAL::Vector_3< Kernel > &u1, const CGAL::Vector_3< Kernel > &v1, const CGAL::Vector_3< Kernel > &w1, const Kernel::FT &cosine)
	compares the dihedral angles \( \theta_1\) and \( \theta_2\), where \( \theta_1\) is the dihedral angle, in \( [0, \pi]\), between the vectorial planes defined by (u_1, v_1) and (u_1, w_1), and \( \theta_2\) is the angle in \( [0, \pi]\) such that \( cos(\theta_2) = cosine\).
template<typename Kernel >
Comparison_result	compare_dihedral_angle (const CGAL::Vector_3< Kernel > &u1, const CGAL::Vector_3< Kernel > &v1, const CGAL::Vector_3< Kernel > &w1, const CGAL::Vector_3< Kernel > &u2, const CGAL::Vector_3< Kernel > &v2, const CGAL::Vector_3< Kernel > &w2)
	compares the dihedral angles \( \theta_1\) and \( \theta_2\), where \( \theta_i\) is the dihedral angle between the vectorial planes defined by (u_i, v_i) and (u_i, w_i).
template<typename Kernel >
Comparison_result	compare_distance_to_point (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	compares the distances of points q and r to point p.
template<typename Kernel >
Comparison_result	compare_distance_to_point (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	compares the distances of points q and r to point p.
template<typename Kernel >
Comparison_result	compare_lexicographically (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	Compares the Cartesian coordinates of points p and q lexicographically in \( xy\) order: first \( x\)-coordinates are compared, if they are equal, \( y\)-coordinates are compared.
template<typename Kernel >
Comparison_result	compare_lexicographically (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	Compares the Cartesian coordinates of points p and q lexicographically in \( xyz\) order: first \( x\)-coordinates are compared, if they are equal, \( y\)-coordinates are compared, and if both \( x\)- and \( y\)- coordinate are equal, \( z\)-coordinates are compared.
template<typename Kernel >
Comparison_result	compare_signed_distance_to_line (const CGAL::Line_2< Kernel > &l, const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns CGAL::LARGER iff the signed distance of p and l is larger than the signed distance of q and l, CGAL::SMALLER, iff it is smaller, and CGAL::EQUAL iff both are equal.
template<typename Kernel >
Comparison_result	compare_signed_distance_to_line (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r, const CGAL::Point_2< Kernel > &s)
	returns CGAL::LARGER iff the signed distance of r and l is larger than the signed distance of s and l, CGAL::SMALLER, iff it is smaller, and CGAL::EQUAL iff both are equal, where l is the directed line through p and q.
template<typename Kernel >
Comparison_result	compare_signed_distance_to_plane (const CGAL::Plane_3< Kernel > &h, const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns CGAL::LARGER iff the signed distance of p and h is larger than the signed distance of q and h, CGAL::SMALLER, iff it is smaller, and CGAL::EQUAL iff both are equal.
template<typename Kernel >
Comparison_result	compare_signed_distance_to_plane (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s, const CGAL::Point_3< Kernel > &t)
	returns CGAL::LARGER iff the signed distance of s and h is larger than the signed distance of t and h, CGAL::SMALLER, iff it is smaller, and CGAL::EQUAL iff both are equal, where h is the oriented plane through p, q and r.
template<typename Kernel >
Comparison_result	compare_slope (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2)
	compares the slopes of the lines l1 and l2
template<typename Kernel >
Comparison_result	compare_slope (const CGAL::Segment_2< Kernel > &s1, const CGAL::Segment_2< Kernel > &s2)
	compares the slopes of the segments s1 and s2, where the slope is the variation of the y-coordinate from the left to the right endpoint of the segments.
template<typename Kernel >
Comparison_result	compare_slope (const CGAL::Point_2< Kernel > &s1s, const CGAL::Point_2< Kernel > &s1t, const CGAL::Point_2< Kernel > &s2s, const CGAL::Point_2< Kernel > &s2t)
	compares the slopes of the segments (s1s,s1t) and (s2s,s2t), where the slope is the variation of the y-coordinate from the left to the right endpoint of the segments.
template<typename Kernel >
Comparison_result	compare_slope (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	compares the slopes of the segments (p,q) and (r,s), where the slope is the variation of the z-coordinate from the first to the second point of the segment divided by the length of the segment.
template<typename Kernel >
Comparison_result	compare_squared_distance (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const typename Kernel::FT &d2)
	compares the squared distance of points p and q to d2.
template<typename Kernel >
Comparison_result	compare_squared_distance (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const typename Kernel::FT &d2)
	compares the squared distance of points p and q to d2.
template<typename Kernel >
Comparison_result	compare_squared_radius (const CGAL::Point_3< Kernel > &p, const typename Kernel::FT &sr)
	compares the squared radius of the sphere of radius 0 centered at p to sr.
template<typename Kernel >
Comparison_result	compare_squared_radius (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const typename Kernel::FT &sr)
	compares the squared radius of the sphere defined by the points p and q to sr.
template<typename Kernel >
Comparison_result	compare_squared_radius (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const typename Kernel::FT &sr)
	compares the squared radius of the sphere defined by the points p, q, and r to sr.
template<typename Kernel >
Comparison_result	compare_squared_radius (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s, const typename Kernel::FT &sr)
	compares the squared radius of the sphere defined by the points p, q, r, and r to sr.
template<typename Kernel >
Comparison_result	compare_x (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	compares the \( x\)-coordinates of p and q.
template<typename Kernel >
Comparison_result	compare_x (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	compares the \( x\)-coordinates of p and q.
template<typename Kernel >
Comparison_result	compare_x (const CGAL::Point_2< Kernel > &p, const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2)
	compares the \( x\)-coordinates of p and the intersection of lines l1 and l2.
template<typename Kernel >
Comparison_result	compare_x (const CGAL::Line_2< Kernel > &l, const CGAL::Line_2< Kernel > &h1, const CGAL::Line_2< Kernel > &h2)
	compares the \( x\)-coordinates of the intersection of line l with line h1 and with line h2.
template<typename Kernel >
Comparison_result	compare_x (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2, const CGAL::Line_2< Kernel > &h1, const CGAL::Line_2< Kernel > &h2)
	compares the \( x\)-coordinates of the intersection of lines l1 and l2 and the intersection of lines h1 and h2.
template<typename CircularKernel >
Comparison_result	compare_x (const CGAL::Circular_arc_point_2< CircularKernel > &p, const CGAL::Circular_arc_point_2< CircularKernel > &q)
	compares the \( x\)-coordinates of p and q.
template<typename CircularKernel >
Comparison_result	compare_x (const CGAL::Circular_arc_point_2< CircularKernel > &p, const CGAL::Point_2< CircularKernel > &q)
	compares the \( x\)-coordinates of p and q.
template<typename SphericalKernel >
Comparison_result	compare_x (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Circular_arc_point_3< SphericalKernel > &q)
	compares the \( x\)-coordinates of p and q.
template<typename SphericalKernel >
Comparison_result	compare_x (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Point_3< SphericalKernel > &q)
	compares the \( x\)-coordinates of p and q.
template<typename Kernel >
Comparison_result	compare_xy (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	Compares the Cartesian coordinates of points p and q lexicographically in \( xy\) order: first \( x\)-coordinates are compared, if they are equal, \( y\)-coordinates are compared.
template<typename Kernel >
Comparison_result	compare_xy (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	Compares the Cartesian coordinates of points p and q lexicographically in \( xy\) order: first \( x\)-coordinates are compared, if they are equal, \( y\)-coordinates are compared.
template<typename CircularKernel >
Comparison_result	compare_xy (const CGAL::Circular_arc_point_2< CircularKernel > &p, const CGAL::Circular_arc_point_2< CircularKernel > &q)
	Compares the \( x\) and \( y\) Cartesian coordinates of points p and q lexicographically.
template<typename CircularKernel >
Comparison_result	compare_xy (const CGAL::Circular_arc_point_2< CircularKernel > &p, const CGAL::Point_2< CircularKernel > &q)
	Compares the \( x\) and \( y\) Cartesian coordinates of points p and q lexicographically.
template<typename SphericalKernel >
Comparison_result	compare_xy (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Circular_arc_point_3< SphericalKernel > &q)
	Compares the \( x\) and \( y\) Cartesian coordinates of points p and q lexicographically.
template<typename SphericalKernel >
Comparison_result	compare_xy (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Point_3< SphericalKernel > &q)
	Compares the \( x\) and \( y\) Cartesian coordinates of points p and q lexicographically.
template<typename Kernel >
Comparison_result	compare_x_at_y (const CGAL::Point_2< Kernel > &p, const CGAL::Line_2< Kernel > &h)
	compares the \( x\)-coordinates of p and the horizontal projection of p on h.
template<typename Kernel >
Comparison_result	compare_x_at_y (const CGAL::Point_2< Kernel > &p, const CGAL::Line_2< Kernel > &h1, const CGAL::Line_2< Kernel > &h2)
	This function compares the \( x\)-coordinates of the horizontal projection of p on h1 and on h2.
template<typename Kernel >
Comparison_result	compare_x_at_y (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2, const CGAL::Line_2< Kernel > &h)
	Let p be the intersection of lines l1 and l2.
template<typename Kernel >
Comparison_result	compare_x_at_y (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2, const CGAL::Line_2< Kernel > &h1, const CGAL::Line_2< Kernel > &h2)
	Let p be the intersection of lines l1 and l2.
template<typename Kernel >
Comparison_result	compare_y_at_x (const CGAL::Point_2< Kernel > &p, const CGAL::Line_2< Kernel > &h)
	compares the \( y\)-coordinates of p and the vertical projection of p on h.
template<typename Kernel >
Comparison_result	compare_y_at_x (const CGAL::Point_2< Kernel > &p, const CGAL::Line_2< Kernel > &h1, const CGAL::Line_2< Kernel > &h2)
	compares the \( y\)-coordinates of the vertical projection of p on h1 and on h2.
template<typename Kernel >
Comparison_result	compare_y_at_x (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2, const CGAL::Line_2< Kernel > &h)
	Let p be the intersection of lines l1 and l2.
template<typename Kernel >
Comparison_result	compare_y_at_x (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2, const CGAL::Line_2< Kernel > &h1, const CGAL::Line_2< Kernel > &h2)
	Let p be the intersection of lines l1 and l2.
template<typename Kernel >
Comparison_result	compare_y_at_x (const CGAL::Point_2< Kernel > &p, const CGAL::Segment_2< Kernel > &s)
	compares the \( y\)-coordinates of p and the vertical projection of p on s.
template<typename Kernel >
Comparison_result	compare_y_at_x (const CGAL::Point_2< Kernel > &p, const CGAL::Segment_2< Kernel > &s1, const CGAL::Segment_2< Kernel > &s2)
	compares the \( y\)-coordinates of the vertical projection of p on s1 and on s2.
template<typename Kernel >
Comparison_result	compare_y (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	compares Cartesian \( y\)-coordinates of p and q.
template<typename Kernel >
Comparison_result	compare_y (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	compares Cartesian \( y\)-coordinates of p and q.
template<typename Kernel >
Comparison_result	compare_y (const CGAL::Point_2< Kernel > &p, const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2)
	compares the \( y\)-coordinates of p and the intersection of lines l1 and l2.
template<typename Kernel >
Comparison_result	compare_y (const CGAL::Line_2< Kernel > &l, const CGAL::Line_2< Kernel > &h1, const CGAL::Line_2< Kernel > &h2)
	compares the \( y\)-coordinates of the intersection of line l with line h1 and with line h2.
template<typename Kernel >
Comparison_result	compare_y (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2, const CGAL::Line_2< Kernel > &h1, const CGAL::Line_2< Kernel > &h2)
	compares the \( y\)-coordinates of the intersection of lines l1 and l2 and the intersection of lines h1 and h2.
template<typename CircularKernel >
Comparison_result	compare_y (const CGAL::Circular_arc_point_2< CircularKernel > &p, const CGAL::Circular_arc_point_2< CircularKernel > &q)
	compares the \( y\)-coordinates of p and q.
template<typename CircularKernel >
Comparison_result	compare_y (const CGAL::Circular_arc_point_2< CircularKernel > &p, const CGAL::Point_2< CircularKernel > &q)
	compares the \( y\)-coordinates of p and q.
template<typename SphericalKernel >
Comparison_result	compare_y (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Circular_arc_point_3< SphericalKernel > &q)
	compares the \( y\)-coordinates of p and q.
template<typename SphericalKernel >
Comparison_result	compare_y (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Point_3< SphericalKernel > &q)
	compares the \( y\)-coordinates of p and q.
template<typename Kernel >
Comparison_result	compare_xyz (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	Compares the Cartesian coordinates of points p and q lexicographically in \( xyz\) order: first \( x\)-coordinates are compared, if they are equal, \( y\)-coordinates are compared, and if both \( x\)- and \( y\)- coordinate are equal, \( z\)-coordinates are compared.
template<typename SphericalKernel >
Comparison_result	compare_xyz (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Circular_arc_point_3< SphericalKernel > &q)
	Compares the Cartesian coordinates of points p and q lexicographically.
template<typename SphericalKernel >
Comparison_result	compare_xyz (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Point_3< SphericalKernel > &q)
	Compares the Cartesian coordinates of points p and q lexicographically.
template<typename Kernel >
Comparison_result	compare_z (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	compares the \( z\)-coordinates of p and q.
template<typename SphericalKernel >
Comparison_result	compare_z (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Circular_arc_point_3< SphericalKernel > &q)
	compares the \( z\)-coordinates of p and q.
template<typename SphericalKernel >
Comparison_result	compare_z (const CGAL::Circular_arc_point_3< SphericalKernel > &p, const CGAL::Point_3< SphericalKernel > &q)
	compares the \( z\)-coordinates of p and q.
template<typename Kernel >
Comparison_result	compare_yx (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	Compares the Cartesian coordinates of points p and q lexicographically in \( yx\) order: first \( y\)-coordinates are compared, if they are equal, \( x\)-coordinates are compared.
template<typename Kernel >
bool	coplanar (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	returns true, if p, q, r, and s are coplanar.
template<typename Kernel >
Orientation	coplanar_orientation (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	Let P be the plane defined by the points p, q, and r.
template<typename Kernel >
Orientation	coplanar_orientation (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	If p,q,r are collinear, then CGAL::COLLINEAR is returned.
template<typename Kernel >
Bounded_side	coplanar_side_of_bounded_circle (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	returns the bounded side of the circle defined by p, q, and r on which s lies.
template<typename Kernel >
CGAL::Vector_3< Kernel >	cross_product (const CGAL::Vector_3< Kernel > &u, const CGAL::Vector_3< Kernel > &v)
	returns the cross product of u and v.
template<typename Kernel >
Kernel::FT	determinant (const CGAL::Vector_2< Kernel > &v, const CGAL::Vector_2< Kernel > &w)
	returns the determinant of v and w.
template<typename Kernel >
Kernel::FT	determinant (const CGAL::Vector_3< Kernel > &u, const CGAL::Vector_3< Kernel > &v, const CGAL::Vector_3< Kernel > &w)
	returns the determinant of u, v and w.
template<typename Kernel >
CGAL::Line_3< Kernel >	equidistant_line (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	constructs the line which is at the same distance from the three points p, q and r.
template<typename Kernel >
bool	has_larger_distance_to_point (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true iff the distance between q and p is larger than the distance between r and p.
template<typename Kernel >
bool	has_larger_distance_to_point (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns true iff the distance between q and p is larger than the distance between r and p.
template<typename Kernel >
bool	has_larger_signed_distance_to_line (const CGAL::Line_2< Kernel > &l, const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns true iff the signed distance of p and l is larger than the signed distance of q and l.
template<typename Kernel >
bool	has_larger_signed_distance_to_line (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r, const CGAL::Point_2< Kernel > &s)
	returns true iff the signed distance of r and l is larger than the signed distance of s and l, where l is the directed line through points p and q.
template<typename Kernel >
bool	has_larger_signed_distance_to_plane (const CGAL::Plane_3< Kernel > &h, const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns true iff the signed distance of p and h is larger than the signed distance of q and h.
template<typename Kernel >
bool	has_larger_signed_distance_to_plane (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s, const CGAL::Point_3< Kernel > &t)
	returns true iff the signed distance of s and h is larger than the signed distance of t and h, where h is the oriented plane through p, q and r.
template<typename Kernel >
bool	has_smaller_distance_to_point (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true iff the distance between q and p is smaller than the distance between r and p.
template<typename Kernel >
bool	has_smaller_distance_to_point (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns true iff the distance between q and p is smaller than the distance between r and p.
template<typename Kernel >
bool	has_smaller_signed_distance_to_line (const CGAL::Line_2< Kernel > &l, const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns true iff the signed distance of p and l is smaller than the signed distance of q and l.
template<typename Kernel >
bool	has_smaller_signed_distance_to_line (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r, const CGAL::Point_2< Kernel > &s)
	returns true iff the signed distance of r and l is smaller than the signed distance of s and l, where l is the oriented line through p and q.
template<typename Kernel >
bool	has_smaller_signed_distance_to_plane (const CGAL::Plane_3< Kernel > &h, const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns true iff the signed distance of p and h is smaller than the signed distance of q and h.
template<typename Kernel >
bool	has_smaller_signed_distance_to_plane (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s, const CGAL::Point_3< Kernel > &t)
	returns true iff the signed distance of p and h is smaller than the signed distance of q and h, where h is the oriented plane through p, q and r.
template<typename Kernel >
Kernel::FT	l_infinity_distance (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns the distance between p and q in the L-infinity metric.
template<typename Kernel >
Kernel::FT	l_infinity_distance (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns the distance between p and q in the L-infinity metric.
template<typename Kernel >
bool	left_turn (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true iff p, q, and r form a left turn.
template<typename Kernel >
bool	lexicographically_xy_larger (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns true iff p is lexicographically larger than q with respect to \( xy\) order.
template<typename Kernel >
bool	lexicographically_xy_larger_or_equal (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns true iff p is lexicographically not smaller than q with respect to \( xy\) order.
template<typename Kernel >
bool	lexicographically_xy_smaller (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns true iff p is lexicographically smaller than q with respect to \( xy\) order.
template<typename Kernel >
bool	lexicographically_xy_smaller_or_equal (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns true iff p is lexicographically not larger than q with respect to \( xy\) order.
template<typename Kernel >
bool	lexicographically_xyz_smaller (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns true iff p is lexicographically smaller than q with respect to \( xyz\) order.
template<typename Kernel >
bool	lexicographically_xyz_smaller_or_equal (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns true iff p is lexicographically not larger than q with respect to \( xyz\) order.
template<typename Kernel >
CGAL::Point_2< Kernel >	max_vertex (const CGAL::Iso_rectangle_2< Kernel > &ir)
	computes the vertex with the lexicographically largest coordinates of the iso rectangle ir.
template<typename Kernel >
CGAL::Point_3< Kernel >	max_vertex (const CGAL::Iso_cuboid_3< Kernel > &ic)
	computes the vertex with the lexicographically largest coordinates of the iso cuboid ic.
template<typename Kernel >
CGAL::Point_2< Kernel >	midpoint (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	computes the midpoint of the segment pq.
template<typename Kernel >
CGAL::Point_2< Kernel >	midpoint (const CGAL::Segment_2< Kernel > &s)
	computes the midpoint of the segment s.
template<typename Kernel >
CGAL::Point_3< Kernel >	midpoint (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	computes the midpoint of the segment pq.
template<typename Kernel >
CGAL::Point_3< Kernel >	midpoint (const CGAL::Segment_3< Kernel > &s)
	computes the midpoint of the segment s.
template<typename Kernel >
CGAL::Point_2< Kernel >	min_vertex (const CGAL::Iso_rectangle_2< Kernel > &ir)
	computes the vertex with the lexicographically smallest coordinates of the iso rectangle ir.
template<typename Kernel >
CGAL::Point_3< Kernel >	min_vertex (const CGAL::Iso_cuboid_3< Kernel > &ic)
	computes the vertex with the lexicographically smallest coordinates of the iso cuboid ic.
template<typename Kernel >
CGAL::Vector_3< Kernel >	normal (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	computes the normal vector for the vectors q-p and r-p.
template<typename Kernel >
Orientation	orientation (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns CGAL::LEFT_TURN, if r lies to the left of the oriented line l defined by p and q, returns CGAL::RIGHT_TURN if r lies to the right of l, and returns CGAL::COLLINEAR if r lies on l.
template<typename Kernel >
Orientation	orientation (const CGAL::Vector_2< Kernel > &u, const CGAL::Vector_2< Kernel > &v)
	returns CGAL::LEFT_TURN if u and v form a left turn, returns CGAL::RIGHT_TURN if u and v form a right turn, and returns CGAL::COLLINEAR if u and v are collinear.
template<typename Kernel >
Orientation	orientation (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	returns CGAL::POSITIVE, if s lies on the positive side of the oriented plane h defined by p, q, and r, returns CGAL::NEGATIVE if s lies on the negative side of h, and returns CGAL::COPLANAR if s lies on h.
template<typename Kernel >
Orientation	orientation (const CGAL::Vector_3< Kernel > &u, const CGAL::Vector_3< Kernel > &v, const CGAL::Vector_3< Kernel > &w)
	returns CGAL::NEGATIVE if u, v and w are negatively oriented, CGAL::POSITIVE if u, v and w are positively oriented, and CGAL::COPLANAR if u, v and w are coplanar.
template<typename Kernel >
CGAL::Vector_3< Kernel >	orthogonal_vector (const CGAL::Plane_3< Kernel > &p)
	computes an orthogonal vector of the plane p, which is directed to the positive side of this plane.
template<typename Kernel >
CGAL::Vector_3< Kernel >	orthogonal_vector (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	computes an orthogonal vector of the plane defined by p, q and r, which is directed to the positive side of this plane.
template<typename Kernel >
bool	parallel (const CGAL::Line_2< Kernel > &l1, const CGAL::Line_2< Kernel > &l2)
	returns true, if l1 and l2 are parallel or if one of those (or both) is degenerate.
template<typename Kernel >
bool	parallel (const CGAL::Ray_2< Kernel > &r1, const CGAL::Ray_2< Kernel > &r2)
	returns true, if r1 and r2 are parallel or if one of those (or both) is degenerate.
template<typename Kernel >
bool	parallel (const CGAL::Segment_2< Kernel > &s1, const CGAL::Segment_2< Kernel > &s2)
	returns true, if s1 and s2 are parallel or if one of those (or both) is degenerate.
template<typename Kernel >
bool	parallel (const CGAL::Line_3< Kernel > &l1, const CGAL::Line_3< Kernel > &l2)
	returns true, if l1 and l2 are parallel or if one of those (or both) is degenerate.
template<typename Kernel >
bool	parallel (const CGAL::Plane_3< Kernel > &h1, const CGAL::Plane_3< Kernel > &h2)
	returns true, if h1 and h2 are parallel or if one of those (or both) is degenerate.
template<typename Kernel >
bool	parallel (const CGAL::Ray_3< Kernel > &r1, const CGAL::Ray_3< Kernel > &r2)
	returns true, if r1 and r2 are parallel or if one of those (or both) is degenerate.
template<typename Kernel >
bool	parallel (const CGAL::Segment_3< Kernel > &s1, const CGAL::Segment_3< Kernel > &s2)
	returns true, if s1 and s2 are parallel or if one of those (or both) is degenerate.
CGAL::Plane_3< Kernel >	radical_plane (const CGAL::Sphere_3< Kernel > &s1, const CGAL::Sphere_3< Kernel > &s2)
	returns the radical plane of the two spheres.
template<typename Kernel >
CGAL::Line_2< Kernel >	radical_line (const CGAL::Circle_2< Kernel > &c1, const CGAL::Circle_2< Kernel > &c2)
	returns the radical line of the two circles.
template<typename Kernel >
bool	right_turn (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	returns true iff p, q, and r form a right turn.
template<typename Kernel >
Kernel::FT	scalar_product (const CGAL::Vector_2< Kernel > &u, const CGAL::Vector_2< Kernel > &v)
	returns the scalar product of u and v.
template<typename Kernel >
Kernel::FT	scalar_product (const CGAL::Vector_3< Kernel > &u, const CGAL::Vector_3< Kernel > &v)
	returns the scalar product of u and v.
template<typename Kernel >
Bounded_side	side_of_bounded_circle (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r, const CGAL::Point_2< Kernel > &t)
	returns the relative position of point t to the circle defined by p, q and r.
template<typename Kernel >
Bounded_side	side_of_bounded_circle (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &t)
	returns the position of the point t relative to the circle that has pq as its diameter.
template<typename Kernel >
Bounded_side	side_of_bounded_sphere (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s, const CGAL::Point_3< Kernel > &t)
	returns the relative position of point t to the sphere defined by p, q, r, and s.
template<typename Kernel >
Bounded_side	side_of_bounded_sphere (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &t)
	returns the position of the point t relative to the sphere passing through p, q, and r and whose center is in the plane defined by these three points.
template<typename Kernel >
Bounded_side	side_of_bounded_sphere (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &t)
	returns the position of the point t relative to the sphere that has pq as its diameter.
template<typename Kernel >
Oriented_side	side_of_oriented_circle (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r, const CGAL::Point_2< Kernel > &test)
	returns the relative position of point test to the oriented circle defined by p, q and r.
template<typename Kernel >
Oriented_side	side_of_oriented_sphere (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s, const CGAL::Point_3< Kernel > &test)
	returns the relative position of point test to the oriented sphere defined by p, q, r and s.
template<typename Kernel >
Kernel::FT	squared_area (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	returns the squared area of the triangle defined by the points p, q and r.
template<typename Kernel >
FT	squared_length (const CGAL::Vector_2< Kernel > &v)
	compute the squared length of vector v.
template<typename Kernel >
FT	squared_length (const CGAL::Segment_2< Kernel > &s)
	compute the squared length of segment s.
template<typename Kernel >
FT	squared_length (const CGAL::Vector_3< Kernel > &v)
	compute the squared length of vector v.
template<typename Kernel >
FT	squared_length (const CGAL::Segment_3< Kernel > &s)
	compute the squared length of segment s.
template<typename Kernel >
FT	squared_radius (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q, const CGAL::Point_2< Kernel > &r)
	compute the squared radius of the circle passing through the points p, q, and r.
template<typename Kernel >
FT	squared_radius (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	compute the squared radius of the smallest circle passing through p, and q, i.e. one fourth of the squared distance between p and q.
template<typename Kernel >
FT	squared_radius (const CGAL::Point_2< Kernel > &p)
	compute the squared radius of the smallest circle passing through p, i.e. \( 0\).
template<typename Kernel >
FT	squared_radius (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r, const CGAL::Point_3< Kernel > &s)
	compute the squared radius of the sphere passing through the points p, q, r and s.
template<typename Kernel >
FT	squared_radius (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	compute the squared radius of the sphere passing through the points p, q, and r and whose center is in the same plane as those three points.
template<typename Kernel >
FT	squared_radius (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	compute the squared radius of the smallest circle passing through p, and q, i.e. one fourth of the squared distance between p and q.
template<typename Kernel >
FT	squared_radius (const CGAL::Point_3< Kernel > &p)
	compute the squared radius of the smallest circle passing through p, i.e. \( 0\).
CGAL::Vector_3< Kernel >	unit_normal (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q, const CGAL::Point_3< Kernel > &r)
	computes the unit normal vector for the vectors q-p and r-p.
template<typename Kernel >
Kernel::FT	volume (const CGAL::Point_3< Kernel > &p0, const CGAL::Point_3< Kernel > &p1, const CGAL::Point_3< Kernel > &p2, const CGAL::Point_3< Kernel > &p3)
	Computes the signed volume of the tetrahedron defined by the four points p0, p1, p2 and p3.
template<typename Kernel >
bool	x_equal (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns true, iff p and q have the same x-coordinate.
template<typename Kernel >
bool	x_equal (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns true, iff p and q have the same x-coordinate.
template<typename Kernel >
bool	y_equal (const CGAL::Point_2< Kernel > &p, const CGAL::Point_2< Kernel > &q)
	returns true, iff p and q have the same y-coordinate.
template<typename Kernel >
bool	y_equal (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns true, iff p and q have the same y-coordinate.
template<typename Kernel >
bool	z_equal (const CGAL::Point_3< Kernel > &p, const CGAL::Point_3< Kernel > &q)
	returns true, iff p and q have the same z-coordinate.
template<RingNumberType >
void	rational_rotation_approximation (const RingNumberType &dirx, const RingNumberType &diry, RingNumberType &sin_num, RingNumberType &cos_num, RingNumberType &denom, const RingNumberType &eps_num, const RingNumberType &eps_den)
	computes integers sin_num, cos_num and denom, such that sin_num/denom approximates the sine of direction \( (\)dirx,diry \( )\).
template<typename Kernel >
Kernel::FT	squared_distance (Type1< Kernel > obj1, Type2< Kernel > obj2)
	computes the square of the Euclidean distance between two geometric objects.

Variables
const CGAL::Orientation	CLOCKWISE = NEGATIVE
const CGAL::Orientation	COUNTERCLOCKWISE = POSITIVE
const CGAL::Orientation	COLLINEAR = ZERO
const CGAL::Orientation	LEFT_TURN = POSITIVE
const CGAL::Orientation	RIGHT_TURN = NEGATIVE
const CGAL::Orientation	COPLANAR = ZERO
const CGAL::Orientation	DEGENERATE = ZERO
const CGAL::Null_vector	NULL_VECTOR
	A symbolic constant used to construct zero length vectors.
const CGAL::Origin	ORIGIN
	A symbolic constant which denotes the point at the origin.
template<typename RT >
Point_2< Homogeneous< RT > >	cartesian_to_homogeneous (const Point_2< Cartesian< RT > > &cp)
	Functions to convert between Cartesian and homogeneous kernels.
template<typename RT >
Point_3< Homogeneous< RT > >	cartesian_to_homogeneous (const Point_3< Cartesian< RT > > &cp)
	converts 3D point cp with Cartesian representation into a 3D point with homogeneous representation with the same number type.
template<typename FT >
Point_2< Cartesian< FT > >	homogeneous_to_cartesian (const Point_2< Homogeneous< FT > > &hp)
	converts 2D point hp with homogeneous representation into a 2D point with Cartesian representation with the same number type.
template<typename FT >
Point_3< Cartesian< FT > >	homogeneous_to_cartesian (const Point_3< Homogeneous< FT > > &hp)
	converts 3D point hp with homogeneous representation into a 3D point with Cartesian representation with the same number type.
template<typename RT >
Point_2< Cartesian< Quotient< RT > > >	homogeneous_to_quotient_cartesian (const Point_2< Homogeneous< RT > > &hp)
	converts the 2D point hp with homogeneous representation with number type RT into a 2D point with Cartesian representation with number type Quotient<RT>.
template<typename RT >
Point_3< Cartesian< Quotient< RT > > >	homogeneous_to_quotient_cartesian (const Point_3< Homogeneous< RT > > &hp)
	converts the 3D point hp with homogeneous representation with number type RT into a 3D point with Cartesian representation with number type Quotient<RT>.
template<typename RT >
Point_2< Homogeneous< RT > >	quotient_cartesian_to_homogeneous (const Point_2< Cartesian< Quotient< RT > > > &cp)
	converts 2D point cp with Cartesian representation with number type Quotient<RT> into a 2D point with homogeneous representation with number type RT.
template<typename RT >
Point_3< Homogeneous< RT > >	quotient_cartesian_to_homogeneous (const Point_3< Cartesian< Quotient< RT > > > &cp)
	converts 3D point cp with Cartesian representation with number type Quotient<RT> into a 3D point with homogeneous representation with number type RT.
bool	do_intersect (Type1< CircularKernel > obj1, Type2< CircularKernel > obj2)
	checks whether obj1 and obj2 intersect.
template<typename Type1 , typename Type2 , typename OutputIterator >
OutputIterator	intersection (const Type1 &obj1, const Type2 &obj2, OutputIterator intersections)
	Constructs the intersection elements between the two input objects and stores them in the OutputIterator in lexicographic order, where both, Type1 and Type2, can be either.

With the 2D Circular Kernel
See 2D Circular Geometry Kernel. #include <CGAL/global_functions_circular_kernel_2.h>
template<typename CircularKernel >
Comparison_result	compare_y_at_x (const CGAL::Circular_arc_point_2< CircularKernel > &p, const CGAL::Circular_arc_2< CircularKernel > &a)
	Same as above, for a point and a circular arc.
template<typename CircularKernel >
Comparison_result	compare_y_at_x (const CGAL::Circular_arc_point_2< CircularKernel > &p, const CGAL::Line_arc_2< CircularKernel > &a)
	Same as above, for a point and a line segment.
bool	do_intersect (Type1< SphericalKernel > obj1, Type2< SphericalKernel > obj2)
	checks whether obj1 and obj2 intersect.
bool	do_intersect (Type1< SphericalKernel > obj1, Type2< SphericalKernel > obj2, Type3< SphericalKernel > obj3)
	checks whether obj1, obj2 and obj3 intersect.
template<typename SphericalType1 , typename SphericalType1 , typename OutputIterator >
OutputIterator	intersection (const SphericalType1 &obj1, const SphericalType2 &obj2, OutputIterator intersections)
	Constructs the intersection elements between the two input objects and stores them in the OutputIterator in lexicographic order, where both, SphericalType1 and SphericalType2, can be either.
template<typename Type1 , typename Type2 , typename Type3 , typename OutputIterator >
OutputIterator	intersection (const Type1 &obj1, const Type2 &obj2, const Type3 &obj3, OutputIterator intersections)
	Copies in the output iterator the intersection elements between the three objects.