CGAL::Plane_3< Kernel > Class Template Reference

#include <CGAL/Plane_3.h>

Definition

An object h of the data type Plane_3 is an oriented plane in the three-dimensional Euclidean space \( \E^3\).

It is defined by the set of points with Cartesian coordinates \( (x,y,z)\) that satisfy the plane equation

\[h :\; a\, x +b\, y +c\, z + d = 0.\]

The plane splits \( \E^3\) in a positive and a negative side. A point p with Cartesian coordinates \( (px, py, pz)\) is on the positive side of h, iff \( a\, px +b\, py +c\, pz + d > 0\). It is on the negative side, iff \( a\, px +b\, py\, +c\, pz + d < 0\).

See Also

Kernel::Plane_3

Creation
	Plane_3 (const Kernel::RT &a, const Kernel::RT &b, const Kernel::RT &c, const Kernel::RT &d)
	creates a plane h defined by the equation \( a\, px +b\, py +c\, pz + d = 0\). More...
	Plane_3 (const Point_3< Kernel > &p, const Point_3< Kernel > &q, const Point_3< Kernel > &r)
	creates a plane h passing through the points p, q and r. More...
	Plane_3 (const Point_3< Kernel > &p, const Vector_3< Kernel > &v)
	introduces a plane h that passes through point p and that is orthogonal to v.
	Plane_3 (const Point_3< Kernel > &p, const Direction_3< Kernel > &d)
	introduces a plane h that passes through point p and that has as an orthogonal direction equal to d.
	Plane_3 (const Line_3< Kernel > &l, const Point_3< Kernel > &p)
	introduces a plane h that is defined through the three points l.point(0), l.point(1) and p.
	Plane_3 (const Ray_3< Kernel > &r, const Point_3< Kernel > &p)
	introduces a plane h that is defined through the three points r.point(0), r.point(1) and p.
	Plane_3 (const Segment_3< Kernel > &s, const Point_3< Kernel > &p)
	introduces a plane h that is defined through the three points s.source(), s.target() and p.
	Plane_3 (const Circle_3< Kernel > &c)
	introduces a plane h that is defined as the plane containing the circle.

Operations
bool	operator== (const Plane_3< Kernel > &h2) const
	Test for equality: two planes are equal, iff they have a non empty intersection and the same orientation.
bool	operator!= (const Plane_3< Kernel > &h2) const
	Test for inequality.
Kernel::RT	a () const
	returns the first coefficient of h.
Kernel::RT	b () const
	returns the second coefficient of h.
Kernel::RT	c () const
	returns the third coefficient of h.
Kernel::RT	d () const
	returns the fourth coefficient of h.
Line_3< Kernel >	perpendicular_line (const Point_3< Kernel > &p) const
	returns the line that is perpendicular to h and that passes through point p. More...
Point_3< Kernel >	projection (const Point_3< Kernel > &p) const
	returns the orthogonal projection of p on h.
Plane_3< Kernel >	opposite () const
	returns the plane with opposite orientation.
Point_3< Kernel >	point () const
	returns an arbitrary point on h.
Vector_3< Kernel >	orthogonal_vector () const
	returns a vector that is orthogonal to h and that is directed to the positive side of h.
Direction_3< Kernel >	orthogonal_direction () const
	returns the direction that is orthogonal to h and that is directed to the positive side of h.
Vector_3< Kernel >	base1 () const
	returns a vector orthogonal to orthogonal_vector().
Vector_3< Kernel >	base2 () const
	returns a vector that is both orthogonal to base1(), and to orthogonal_vector(), and such that the result of orientation( point(), point() + base1(), point()+base2(), point() + orthogonal_vector() ) is positive.

2D Conversion
The following functions provide conversion between a plane and CGAL's two-dimensional space. The transformation is affine, but not necessarily an isometry. This means, the transformation preserves combinatorics, but not distances.
Point_2< Kernel >	to_2d (const Point_3< Kernel > &p) const
	returns the image point of the projection of p under an affine transformation, which maps h onto the \( xy\)-plane, with the \( z\)-coordinate removed.
Point_3< Kernel >	to_3d (const Point_2< Kernel > &p) const
	returns a point q, such that to_2d( to_3d( p )) is equal to p.

Predicates
Oriented_side	oriented_side (const Point_3< Kernel > &p) const
	returns either ON_ORIENTED_BOUNDARY, or the constant ON_POSITIVE_SIDE, or the constant ON_NEGATIVE_SIDE, determined by the position of p relative to the oriented plane h.

Convenience Boolean Functions
bool	has_on (const Point_3< Kernel > &p) const
bool	has_on_positive_side (const Point_3< Kernel > &p) const
bool	has_on_negative_side (const Point_3< Kernel > &p) const
bool	has_on (const Line_3< Kernel > &l) const
bool	has_on (const Circle_3< Kernel > &l) const
bool	is_degenerate () const
	Plane h is degenerate, if the coefficients a, b, and c of the plane equation are zero.

Miscellaneous
Plane_3< Kernel >	transform (const Aff_transformation_3< Kernel > &t) const
	returns the plane obtained by applying t on a point of h and the orthogonal direction of h.

Constructor & Destructor Documentation

template<typename Kernel >

CGAL::Plane_3< Kernel >::Plane_3	(	const Kernel::RT &	a,
		const Kernel::RT &	b,
		const Kernel::RT &	c,
		const Kernel::RT &	d
	)

creates a plane h defined by the equation \( a\, px +b\, py +c\, pz + d = 0\).

Notice that h is degenerate if \( a = b = c = 0\).

template<typename Kernel >

CGAL::Plane_3< Kernel >::Plane_3	(	const Point_3< Kernel > &	p,
		const Point_3< Kernel > &	q,
		const Point_3< Kernel > &	r
	)

creates a plane h passing through the points p, q and r.

The plane is oriented such that p, q and r are oriented in a positive sense (that is counterclockwise) when seen from the positive side of h. Notice that h is degenerate if the points are collinear.

Member Function Documentation

template<typename Kernel >

Line_3<Kernel> CGAL::Plane_3< Kernel >::perpendicular_line

(

const Point_3< Kernel > &

p)

const

returns the line that is perpendicular to h and that passes through point p.

The line is oriented from the negative to the positive side of h.