2D Line (CGAL_Line_2)

Definition

l : ax + by + c = 0

The line splits $$E² in a positive and a negative side.[^1] A point $p$ with Cartesian coordinates $(px,\; py)$ is on the positive side of l, iff a px + b py + c > 0, it is on the negative side of l, iff a px + b py + c < 0.

#include <CGAL/Line_2.h>

Creation

CGAL_Line_2<R> l ( R::RT a, R::RT b, R::RT c);
	introduces a line l with the line equation in Cartesian coordinates $ax\; +by\; +c\; =\; 0$.
CGAL_Line_2<R> l ( CGAL_Point_2<R> p, CGAL_Point_2<R> q);
	introduces a line l passing through the points $p$ and $q$. Line l is directed from $p$ to $q$.
CGAL_Line_2<R> l ( CGAL_Point_2<R> p, CGAL_Direction_2<R> d);
	introduces a line l passing through point $p$ with direction $d$.
CGAL_Line_2<R> l ( CGAL_Segment_2<R> s);
	introduces a line l supporting the segment $s$, oriented from source to target.
CGAL_Line_2<R> l ( CGAL_Ray_2<R> r);
	introduces a line l supporting the ray $r$, with same orientation.

Operations

bool	l == h	Test for equality: two lines are equal, iff they have a non empty intersection and the same direction.
bool	l != h	Test for inequality.
R::RT	l.a ()	returns the first coefficient of $$L.
R::RT	l.b ()	returns the second coefficient of $$L.
R::RT	l.c ()	returns the third coefficient of $$L.
CGAL_Point_2<R>	l.point ( int i)	returns an arbitrary point on l. It holds point(i) == point(j), iff i==j. Furthermore, l is directed from point(i) to point(j), for all i j.
CGAL_Point_2<R>	l.projection ( CGAL_Point_2<R> p)
		returns the orthogonal projection of $p$ onto l.
R::FT	l.x_at_y ( R::FT y)
		returns the $x$-coordinate of the point at l with given $y$-coordinate. Precondition: l is not horizontal.
R::FT	l.y_at_x ( R::FT x)
		returns the $y$-coordinate of the point at l with given $x$-coordinate. Precondition: l is not vertical.

Predicates

bool	l.is_degenerate ()	line l is degenerate, if the coefficients a and b of the line equation are zero.
bool	l.is_horizontal ()
bool	l.is_vertical ()
CGAL_Oriented_side	l.oriented_side ( CGAL_Point_2<R> p)
		returns CGAL_ON_ORIENTED_BOUNDARY, CGAL_ON_NEGATIVE_SIDE, or the constant CGAL_ON_POSITIVE_SIDE, depending on the position of $p$ relative to the oriented line l.

For convenience we provide the following boolean functions:

bool	l.has_on ( CGAL_Point_2<R> p)
bool	l.has_on_boundary ( CGAL_Point_2<R> p)
		returns has_on().
bool	l.has_on_positive_side ( CGAL_Point_2<R> p)
bool	l.has_on_negative_side ( CGAL_Point_2<R> p)

Miscellaneous

CGAL_Direction_2<R>
	l.direction ()	returns the direction of l.
CGAL_Line_2<R>	l.opposite ()	returns the line with opposite direction.
CGAL_Line_2<R>	l.perpendicular ( CGAL_Point_2<R> p)
		returns the line perpendicular to l and passing through $p$, where the direction is the direction of l rotated counterclockwise by 90 degrees.
CGAL_Line_2<R>	l.transform ( CGAL_Aff_transformation_2<R> t)
		returns the line obtained by applying $t$ on a point on l and the direction of l.

Implementation

Example

  CGAL_Point_2< CGAL_Cartesian<double> >  p(1.0,1.0), q(4.0,7.0);

To define a line $l$ we write:

  CGAL_Line_2< CGAL_Cartesian<double> > l(p,q);

Footnotes

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