An object l of the data type Line_2<Kernel> is a directed straight line in the two-dimensional Euclidean plane 2. It is defined by the set of points with Cartesian coordinates (x,y) that satisfy the equation l : ax + by + c = 0

The line splits 2 in a positive and a negative side. 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. The positive side is to the left of l.


Line_2<Kernel> l ( Kernel::RT a, Kernel::RT b, Kernel::RT c);
introduces a line l with the line equation in Cartesian coordinates ax +by +c = 0.

Line_2<Kernel> l ( Point_2<Kernel> p, Point_2<Kernel> q);
introduces a line l passing through the points p and q. Line l is directed from p to q.

Line_2<Kernel> l ( Point_2<Kernel> p, Direction_2<Kernel> d);
introduces a line l passing through point p with direction d.

Line_2<Kernel> l ( Point_2<Kernel> p, Vector_2<Kernel> v);
introduces a line l passing through point p and oriented by v.

Line_2<Kernel> l ( Segment_2<Kernel> s);
introduces a line l supporting the segment s, oriented from source to target.

Line_2<Kernel> l ( Ray_2<Kernel> r);
introduces a line l supporting the ray r, with same orientation.


bool l.operator== ( h)
Test for equality: two lines are equal, iff they have a non empty intersection and the same direction.

bool l.operator!= ( h)
Test for inequality.

Kernel::RT l.a () returns the first coefficient of l.
Kernel::RT l.b () returns the second coefficient of l.
Kernel::RT l.c () returns the third coefficient of l.

Point_2<Kernel> 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.

Point_2<Kernel> l.projection ( Point_2<Kernel> p)
returns the orthogonal projection of p onto l.

Kernel::FT l.x_at_y ( Kernel::FT y)
returns the x-coordinate of the point at l with given y-coordinate.
Precondition: l is not horizontal.

Kernel::FT l.y_at_x ( Kernel::FT x)
returns the y-coordinate of the point at l with given x-coordinate.
Precondition: l is not vertical.


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 ()

Oriented_side l.oriented_side ( Point_2<Kernel> p)
returns ON_ORIENTED_BOUNDARY, ON_NEGATIVE_SIDE, or the constant 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 ( Point_2<Kernel> p)
bool l.has_on_positive_side ( Point_2<Kernel> p)
bool l.has_on_negative_side ( Point_2<Kernel> p)


Vector_2<Kernel> l.to_vector () returns a vector having the direction of l.

l.direction () returns the direction of l.

Line_2<Kernel> l.opposite () returns the line with opposite direction.

Line_2<Kernel> l.perpendicular ( Point_2<Kernel> p)
returns the line perpendicular to l and passing through p, where the direction is the direction of l rotated counterclockwise by 90 degrees.

Line_2<Kernel> l.transform ( Aff_transformation_2<Kernel> t)
returns the line obtained by applying t on a point on l and the direction of l.


Let us first define two Cartesian two-dimensional points in the Euclidean plane 2. Their dimension and the fact that they are Cartesian is expressed by the suffix _2 and the representation type Cartesian.

  Point_2< Cartesian<double> >  p(1.0,1.0), q(4.0,7.0);

To define a line l we write:

  Line_2< Cartesian<double> > l(p,q);

See Also