3D Plane (CGAL_Plane_3)

Definition

h : a x + b y + c z + d = 0

The plane splits $$E³ in a positive and a negative side.[^1] 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.

#include <CGAL/Plane_3.h>

Creation

CGAL_Plane_3<R> h ( R::RT a, R::RT b, R::RT c, R::RT d);
	introduces a plane h defined by the equation a px + b py + c pz + d = 0.
CGAL_Plane_3<R> h ( CGAL_Point_3<R> p, CGAL_Point_3<R> q, CGAL_Point_3<R> r);
	introduces 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.
CGAL_Plane_3<R> h ( CGAL_Point_3<R> p, CGAL_Direction_3<R> d);
	introduces a plane h that passes through point p and that has as an orthogonal direction equal to d.
CGAL_Plane_3<R> h ( CGAL_Line_3<R> l, CGAL_Point_3<R> p);
	introduces a plane h that is defined through the three points l.point(0), l.point(1) and p.
CGAL_Plane_3<R> h ( CGAL_Ray_3<R> r, CGAL_Point_3<R> p);
	introduces a plane h that is defined through the three points r.point(0), r.point(1) and p.
CGAL_Plane_3<R> h ( CGAL_Segment_3<R> s, CGAL_Point_3<R> p);
	introduces a plane h that is defined through the three points s.source(), s.target() and p.

Operations

bool	h == h2	Test for equality: two planes are equal, iff they have a non empty intersection and the same orientation.
bool	h != h2	Test for inequality.
R::RT	h.a ()	returns the first coefficient of H.
R::RT	h.b ()	returns the second coefficient of H.
R::RT	h.c ()	returns the third coefficient of H.
R::RT	h.d ()	returns the fourth coefficient of H.
CGAL_Line_3<R>	h.perpendicular_line ( CGAL_Point_3<R> p)
		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.
CGAL_Plane_3<R>	h.opposite ()	returns the plane with opposite orientation.
CGAL_Point_3<R>	h.projection ( CGAL_Point_3<R> p)
		returns the orthogonal projection of $p$ on h.
CGAL_Point_3<R>	h.point ()	returns an arbitrary point on h.
CGAL_Vector_3<R>	h.orthogonal_vector ()
		returns a vector that is orthogonal to h and that is directed to the positive side of h.
CGAL_Direction_3<R>
	h.orthogonal_direction ()
		returns the direction that is orthogonal to h and that is directed to the positive side of h.
CGAL_Vector_3<R>	h.base1 ()	returns a vector orthogonal to orthogonal_vector().
CGAL_Vector_3<R>	h.base2 ()	returns a vector that is both orthogonal to base1(), and to orthogonal_vector(), and such that the result of CGAL_orientation( point(), point() + base1(), point()+base2(), point() + orthogonal_vector() ) is positive.

Projection

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.

CGAL_Point_2<R>	h.to_2d ( CGAL_Point_3<R> p)
		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.
CGAL_Point_3<R>	h.to_3d ( CGAL_Point_2<R> p)
		returns a point $q$, such that to_2d( to_3d( p )) is equal to p.
CGAL_Oriented_side	h.oriented_side ( CGAL_Point_3<R> p)
		returns either CGAL_ON_ORIENTED_BOUNDARY, or the constant CGAL_ON_POSITIVE_SIDE, or the con\-stant CGAL_ON_NEGATIVE_SIDE, determined by the position of $p$ relative to the oriented plane h.

Predicates

bool	h.has_on ( CGAL_Point_3<R> p)
bool	h.has_on_boundary ( CGAL_Point_3<R> p)
bool	h.has_on_positive_side ( CGAL_Point_3<R> p)
bool	h.has_on_negative_side ( CGAL_Point_3<R> p)
bool	h.has_on ( CGAL_Line_3<R> l)
bool	h.has_on_boundary ( CGAL_Line_3<R> l)
bool	h.is_degenerate ()	Plane h is degenerate, if the coefficients a, b, and c of the plane equation are zero.

Miscellaneous

CGAL_Plane_3<R>	h.transform ( CGAL_Aff_transformation_3<R> t)
		returns the plane obtained by applying $t$ on a point of h and the orthogonal direction of h.

Footnotes

1. See Chapter [ref:Utilities] for the definition of CGAL_Oriented_side.