#include <CGAL/Plane_3.h>

Definition

template<typename Kernel>
class CGAL::Plane_3< Kernel >

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 \( a\, px +b\, py\, +c\, pz + d < 0\).

Is model of: 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 \( a\, px +b\, py +c\, pz + d = 0\).

	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`.

	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`.

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

◆ Plane_3() [1/3]

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 \( a = b = c = 0\).

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ Plane_3() [2/3]

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.

◆ Plane_3() [3/3]

template<typename Kernel >

CGAL::Plane_3< Kernel >::Plane_3 ( const Circle_3< Kernel > & c )

introduces a plane h that is defined as the plane containing the circle.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

Member Function Documentation

◆ a()

template<typename Kernel >

Kernel::RT CGAL::Plane_3< Kernel >::a ( ) const

returns the first coefficient of h.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ b()

template<typename Kernel >

Kernel::RT CGAL::Plane_3< Kernel >::b ( ) const

returns the second coefficient of h.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ c()

template<typename Kernel >

Kernel::RT CGAL::Plane_3< Kernel >::c ( ) const

returns the third coefficient of h.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ d()

template<typename Kernel >

Kernel::RT CGAL::Plane_3< Kernel >::d ( ) const

returns the fourth coefficient of h.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ opposite()

template<typename Kernel >

Plane_3< Kernel > CGAL::Plane_3< Kernel >::opposite ( ) const

returns the plane with opposite orientation.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ orthogonal_direction()

template<typename Kernel >

Direction_3< Kernel > CGAL::Plane_3< Kernel >::orthogonal_direction ( ) const

returns the direction that is orthogonal to h and that is directed to the positive side of h.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ orthogonal_vector()

template<typename Kernel >

Vector_3< Kernel > CGAL::Plane_3< Kernel >::orthogonal_vector ( ) const

returns a vector that is orthogonal to h and that is directed to the positive side of h.

Exactness: This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

◆ perpendicular_line()

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.

Definition

Creation

Operations

2D Conversion

Predicates

Convenience Boolean Functions

Miscellaneous

Constructor & Destructor Documentation

◆ Plane_3() [1/3]

◆ Plane_3() [2/3]

◆ Plane_3() [3/3]

Member Function Documentation

◆ a()

◆ b()

◆ c()

◆ d()

◆ opposite()

◆ orthogonal_direction()

◆ orthogonal_vector()

◆ perpendicular_line()