## 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<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.

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.

### 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<Kernel> | h.perpendicular_line ( Point_3<Kernel> 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<Kernel> | h.projection ( Point_3<Kernel> p)
|  | returns the orthogonal projection of p on h.
Plane_3<Kernel> | h.opposite () | returns the plane with opposite orientation.
Point_3<Kernel> | h.point () | returns an arbitrary point on h.
Vector_3<Kernel> | h.orthogonal_vector ()
|  | returns a vector that is orthogonal to h and that is directed to the positive side of h.
Direction_3<Kernel>
| h.orthogonal_direction ()
|  | returns the direction that is orthogonal to h and that is directed to the positive side of h.
Vector_3<Kernel> | h.base1 () | returns a vector orthogonal to orthogonal_vector().
Vector_3<Kernel> | 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<Kernel> | h.to_2d ( Point_3<Kernel> 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<Kernel> | h.to_3d ( Point_2<Kernel> p)
|  | 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)
|  | 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)
bool | h.has_on_positive_side ( Point_3<Kernel> p)
bool | h.has_on_negative_side ( Point_3<Kernel> p)
bool | h.has_on ( Line_3<Kernel> 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<Kernel> | h.transform ( Aff_transformation_3<Kernel> t)
|  | returns the plane obtained by applying t on a point of h and the orthogonal direction of h.

### See Also

Kernel::Plane_3