CGAL 5.4 - 2D and 3D Linear Geometry Kernel
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 $$a\, px +b\, py\, +c\, pz + d < 0$$.

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 $$a\, px +b\, py +c\, pz + d = 0$$. 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< Kernelperpendicular_line (const Point_3< Kernel > &p) const
returns the line that is perpendicular to h and that passes through point p. More...

Point_3< Kernelprojection (const Point_3< Kernel > &p) const
returns the orthogonal projection of p on h.

Plane_3< Kernelopposite () const
returns the plane with opposite orientation.

Point_3< Kernelpoint () const
returns an arbitrary point on h.

Vector_3< Kernelorthogonal_vector () const
returns a vector that is orthogonal to h and that is directed to the positive side of h.

Direction_3< Kernelorthogonal_direction () const
returns the direction that is orthogonal to h and that is directed to the positive side of h.

Vector_3< Kernelbase1 () const
returns a vector orthogonal to orthogonal_vector().

Vector_3< Kernelbase2 () 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< Kernelto_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< Kernelto_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< Kerneltransform (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.

## ◆ Plane_3() [1/2]

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 $$a = b = c = 0$$.

## ◆ Plane_3() [2/2]

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.

## ◆ perpendicular_line()

template<typename Kernel >
 Line_3 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.