Class

CGAL::Plane_3<Kernel>

Definition

An object h of the data type Plane_3<Kernel> 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.

Creation

Plane_3<Kernel> h ( Kernel::RT a, Kernel::RT b, Kernel::RT c, 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<Kernel> h ( Point_3<Kernel> p, Point_3<Kernel> q, 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<Kernel> h ( Point_3<Kernel> p, Vector_3<Kernel> v);
introduces a plane h that passes through point p and that is orthogonal to v.


Plane_3<Kernel> h ( Point_3<Kernel> p, 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<Kernel> h ( Line_3<Kernel> l, 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<Kernel> h ( Ray_3<Kernel> r, 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<Kernel> h ( Segment_3<Kernel> s, Point_3<Kernel> p);
introduces a plane h that is defined through the three points s.source(), s.target() and p.


Plane_3<Kernel> h ( Circle_3<Kernel> c);
introduces a plane h that is defined as the plane containing the circle.

Operations

bool h.operator== ( h2) const Test for equality: two planes are equal, iff they have a non empty intersection and the same orientation.

bool h.operator!= ( h2) const Test for inequality.

Kernel::RT h.a () const returns the first coefficient of h.
Kernel::RT h.b () const returns the second coefficient of h.
Kernel::RT h.c () const returns the third coefficient of h.
Kernel::RT h.d () const returns the fourth coefficient of h.

Line_3<Kernel> h.perpendicular_line ( 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.

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

Plane_3<Kernel> h.opposite () const returns the plane with opposite orientation.

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

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

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

Vector_3<Kernel> h.base1 () const returns a vector orthogonal to orthogonal_vector().

Vector_3<Kernel> h.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> h.to_2d ( 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> h.to_3d ( Point_2<Kernel> p) const
returns a point q, such that to_2d( to_3d( p )) is equal to p.

Predicates

Oriented_side h.oriented_side ( 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.

For convenience we provide the following Boolean functions:

bool h.has_on ( Point_3<Kernel> p) const
bool h.has_on_positive_side ( Point_3<Kernel> p) const
bool h.has_on_negative_side ( Point_3<Kernel> p) const

bool h.has_on ( Line_3<Kernel> l) const
bool h.has_on ( Circle_3<Kernel> l) const

bool h.is_degenerate () const Plane h is degenerate, if the coefficients a, b, and c of the plane equation are zero.

Miscellaneous

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

See Also

Kernel::Plane_3