CGAL::Point_d<Kernel>

Definition

An instance of data type Point_d<Kernel> is a point of Euclidean space in dimension $$d. A point $$p = (p₀,...,p _{d - 1} ) in $$d-dimensional space can be represented by homogeneous coordinates $$(h₀,h₁,...,h_d) of number type RT such that $$p_i = h_i/h_d, which is of type FT. The homogenizing coordinate $$h_d is positive.

We call $$p_i, $$0 i < d the $$i-th Cartesian coordinate and $$h_i, $$0 i d, the $$i-th homogeneous coordinate. We call $$d the dimension of the point.

Types

Point_d<Kernel>::RT
	the ring type.
Point_d<Kernel>::FT
	the field type.
Point_d<Kernel>::LA
	the linear algebra layer.
Point_d<Kernel>::Cartesian_const_iterator
	a read-only iterator for the Cartesian coordinates.
Point_d<Kernel>::Homogeneous_const_iterator
	a read-only iterator for the homogeneous coordinates.

Creation

Point_d<Kernel> p;
	introduces a variable p of type Point_d<Kernel>.
Point_d<Kernel> p ( int d, Origin);
	introduces a variable p of type Point_d<Kernel> in $$d-dimensional space, initialized to the origin.
template <class InputIterator>
Point_d<Kernel> p ( int d, InputIterator first, InputIterator last);
	introduces a variable p of type Point_d<Kernel> in dimension d. If size [first,last) == d this creates a point with Cartesian coordinates set [first,last). If size [first,last) == p+1 the range specifies the homogeneous coordinates $$H = set [first,last) = ( ± h0, ± h1, ..., ± hd) where the sign chosen is the sign of $$hd. Precondition: d is nonnegative, [first,last) has d or d+1 elements where the last has to be non-zero. Requirement: The value type of InputIterator is RT.
template <class InputIterator>
Point_d<Kernel> p ( int d, InputIterator first, InputIterator last, RT D);
	introduces a variable p of type Point_d<Kernel> in dimension d initialized to the point with homogeneous coordinates as defined by H = set [first,last) and D: $$( ± H[0], ± H[1], ..., ± H[d-1], ± D). The sign chosen is the sign of $$D. Precondition: D is non-zero, the iterator range defines a $$d-tuple of RT. Requirement: The value type of InputIterator is RT.
Point_d<Kernel> p ( RT x, RT y, RT w = 1);
	introduces a variable p of type Point_d<Kernel> in $$2-dimensional space. Precondition: $$w 0.
Point_d<Kernel> p ( RT x, RT y, RT z, RT w);
	introduces a variable p of type Point_d<Kernel> in $$3-dimensional space. Precondition: $$w 0.

Operations

int	p.dimension ()	returns the dimension of p.
FT	p.cartesian ( int i)
		returns the $$i-th Cartesian coordinate of p. Precondition: $$0 i < d.
FT	p [ int i]	returns the $$i-th Cartesian coordinate of p. Precondition: $$0 i < d.
RT	p.homogeneous ( int i)
		returns the $$i-th homogeneous coordinate of p. Precondition: $$0 i d.
Cartesian_const_iterator
	p.cartesian_begin ()
		returns an iterator pointing to the zeroth Cartesian coordinate $$p0 of p.
Cartesian_const_iterator
	p.cartesian_end ()	returns an iterator pointing beyond the last Cartesian coordinate of p.
Homogeneous_const_iterator
	p.homogeneous_begin ()
		returns an iterator pointing to the zeroth homogeneous coordinate $$h0 of p.
Homogeneous_const_iterator
	p.homogeneous_end ()
		returns an iterator pointing beyond the last homogeneous coordinate of p.
Point_d<Kernel>	p.transform ( Aff_transformation_d<Kernel> t)
		returns $$t(p).

Arithmetic Operators, Tests and IO

Vector_d<Kernel>	p - Origin o	returns the vector $$p-O.
Vector_d<Kernel>	p - q	returns $$p - q. Precondition: p.dimension() == q.dimension().
Point_d<Kernel>	p + Vector_d<Kernel> v
		returns $$p + v. Precondition: p.dimension() == v.dimension().
Point_d<Kernel>	p - Vector_d<Kernel> v
		returns $$p - v. Precondition: p.dimension() == v.dimension().
Point_d<Kernel>&	p += Vector_d<Kernel> v
		adds v to p. Precondition: p.dimension() == v.dimension().
Point_d<Kernel>&	p -= Vector_d<Kernel> v
		subtracts v from p. Precondition: p.dimension() == v.dimension().
bool	p == Origin	returns true if p is the origin.

Downward compatibility

Implementation

Points are implemented by arrays of RT items. All operations like creation, initialization, tests, point - vector arithmetic, input and output on a point $$p take time $$O(p.dimension()). dimension(), coordinate access and conversions take constant time. The space requirement for points is $$O(p.dimension()).