 ## 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 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 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 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 h ( Point_3 p, Point_3 q, Point_3 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 h ( Point_3 p, Vector_3 v); introduces a plane h that passes through point p and that is orthogonal to v. Plane_3 h ( Point_3 p, Direction_3 d); introduces a plane h that passes through point p and that has as an orthogonal direction equal to d. Plane_3 h ( Line_3 l, Point_3 p); introduces a plane h that is defined through the three points l.point(0), l.point(1) and p. Plane_3 h ( Ray_3 r, Point_3 p); introduces a plane h that is defined through the three points r.point(0), r.point(1) and p. Plane_3 h ( Segment_3 s, Point_3 p); introduces a plane h that is defined through the three points s.source(), s.target() and p. Plane_3 h ( Circle_3 c); introduces a plane h that is defined as the plane containing the circle.

### Operations

 bool h.operator== ( h2) Test for equality: two planes are equal, iff they have a non empty intersection and the same orientation. bool h.operator!= ( h2) Test for inequality. Kernel::RT h.a () returns the first coefficient of h. Kernel::RT h.b () returns the second coefficient of h. Kernel::RT h.c () returns the third coefficient of h. Kernel::RT h.d () returns the fourth coefficient of h. Line_3 h.perpendicular_line ( Point_3 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. Point_3 h.projection ( Point_3 p) returns the orthogonal projection of p on h. Plane_3 h.opposite () returns the plane with opposite orientation. Point_3 h.point () returns an arbitrary point on h. Vector_3 h.orthogonal_vector () returns a vector that is orthogonal to h and that is directed to the positive side of h. Direction_3 h.orthogonal_direction () returns the direction that is orthogonal to h and that is directed to the positive side of h. Vector_3 h.base1 () returns a vector orthogonal to orthogonal_vector(). Vector_3 h.base2 () 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 h.to_2d ( Point_3 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. Point_3 h.to_3d ( Point_2 p) returns a point q, such that to_2d( to_3d( p )) is equal to p.

### Predicates

 Oriented_side h.oriented_side ( Point_3 p) 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 p) bool h.has_on_positive_side ( Point_3 p) bool h.has_on_negative_side ( Point_3 p) bool h.has_on ( Line_3 l) bool h.has_on ( Circle_3 l) bool h.is_degenerate () Plane h is degenerate, if the coefficients a, b, and c of the plane equation are zero.

### Miscellaneous

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