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CGAL 5.0.1 - 2D and 3D Linear Geometry Kernel
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CGAL 5.0.1 - 2D and 3D Linear Geometry Kernel
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#include <CGAL/Plane_3.h>
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 \( a\, px +b\, py\, +c\, pz + d < 0\).
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< Kernel > | perpendicular_line (const Point_3< Kernel > &p) const |
returns the line that is perpendicular to h and that passes through point p. More... | |
| Point_3< Kernel > | projection (const Point_3< Kernel > &p) const |
returns the orthogonal projection of p on h. | |
| Plane_3< Kernel > | opposite () const |
| returns the plane with opposite orientation. | |
| Point_3< Kernel > | point () const |
returns an arbitrary point on h. | |
| Vector_3< Kernel > | orthogonal_vector () const |
returns a vector that is orthogonal to h and that is directed to the positive side of h. | |
| Direction_3< Kernel > | orthogonal_direction () const |
returns the direction that is orthogonal to h and that is directed to the positive side of h. | |
| Vector_3< Kernel > | base1 () const |
returns a vector orthogonal to orthogonal_vector(). | |
| Vector_3< Kernel > | 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 > | to_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< Kernel > | to_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< Kernel > | transform (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. | |
| 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 \( a = b = c = 0\).
| 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.
1.8.13