CGAL::Line_2< Kernel > Class Template Reference

#include <CGAL/Line_2.h>

Definition

template<typename Kernel>
class CGAL::Line_2< Kernel >

An object l of the data type Line_2 is a directed straight line in the two-dimensional Euclidean plane \( \E^2\).

It is defined by the set of points with Cartesian coordinates \( (x,y)\) that satisfy the equation

\[ l:\; a\, x +b\, y +c = 0. \]

The line splits \( \E^2\) 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.

Example

Let us first define two Cartesian two-dimensional points in the Euclidean plane \( \E^2\). 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);

Is model of

Kernel::Line_2

Creation
	Line_2 (const Kernel::RT &a, const Kernel::RT &b, const Kernel::RT &c)
	introduces a line l with the line equation in Cartesian coordinates \( ax +by +c = 0\).
	Line_2 (const Point_2< Kernel > &p, const Point_2< Kernel > &q)
	introduces a line l passing through the points p and q.
	Line_2 (const Point_2< Kernel > &p, const Direction_2< Kernel > &d)
	introduces a line l passing through point p with direction d.
	Line_2 (const Point_2< Kernel > &p, const Vector_2< Kernel > &v)
	introduces a line l passing through point p and oriented by v.
	Line_2 (const Segment_2< Kernel > &s)
	introduces a line l supporting the segment s, oriented from source to target.
	Line_2 (const Ray_2< Kernel > &r)
	introduces a line l supporting the ray r, with same orientation.

Operations
bool	operator== (const Line_2< Kernel > &h) const
	Test for equality: two lines are equal, iff they have a non empty intersection and the same direction.
bool	operator!= (const Line_2< Kernel > &h) const
	Test for inequality.
Kernel::RT	a () const
	returns the first coefficient of l.
Kernel::RT	b () const
	returns the second coefficient of l.
Kernel::RT	c () const
	returns the third coefficient of l.
Point_2< Kernel >	point (const Kernel::FT i) const
	returns an arbitrary point on l.
Point_2< Kernel >	projection (const Point_2< Kernel > &p) const
	returns the orthogonal projection of p onto l.
Kernel::FT	x_at_y (const Kernel::FT &y) const
	returns the \( x\)-coordinate of the point at l with given \( y\)-coordinate.
Kernel::FT	y_at_x (const Kernel::FT &x) const
	returns the \( y\)-coordinate of the point at l with given \( x\)-coordinate.

Predicates
bool	is_degenerate () const
	line l is degenerate, if the coefficients a and b of the line equation are zero.
bool	is_horizontal () const
bool	is_vertical () const
Oriented_side	oriented_side (const Point_2< Kernel > &p) const
	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.

Convenience Boolean Functions
bool	has_on (const Point_2< Kernel > &p) const
bool	has_on_positive_side (const Point_2< Kernel > &p) const
bool	has_on_negative_side (const Point_2< Kernel > &p) const

Miscellaneous
Vector_2< Kernel >	to_vector () const
	returns a vector having the direction of l.
Direction_2< Kernel >	direction () const
	returns the direction of l.
Line_2< Kernel >	opposite () const
	returns the line with opposite direction.
Line_2< Kernel >	perpendicular (const Point_2< Kernel > &p) const
	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 >	transform (const Aff_transformation_2< Kernel > &t) const
	returns the line obtained by applying t on a point on l and the direction of l.

Constructor & Destructor Documentation

Line_2() [1/2]

template<typename Kernel >

CGAL::Line_2< Kernel >::Line_2	(	const Kernel::RT &	a,
		const Kernel::RT &	b,
		const Kernel::RT &	c
	)

introduces a line l with the line equation in Cartesian coordinates \( ax +by +c = 0\).

Exactness

This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

Line_2() [2/2]

template<typename Kernel >

CGAL::Line_2< Kernel >::Line_2	(	const Point_2< Kernel > &	p,
		const Point_2< Kernel > &	q
	)

introduces a line l passing through the points p and q.

Line l is directed from p to q.

Member Function Documentation

a()

template<typename Kernel >

Kernel::RT CGAL::Line_2< Kernel >::a

(

)

const

returns the first coefficient of l.

Exactness

This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

b()

template<typename Kernel >

Kernel::RT CGAL::Line_2< Kernel >::b

(

)

const

returns the second coefficient of l.

Exactness

This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

c()

template<typename Kernel >

Kernel::RT CGAL::Line_2< Kernel >::c

(

)

const

returns the third coefficient of l.

Exactness

This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

opposite()

template<typename Kernel >

Line_2< Kernel > CGAL::Line_2< Kernel >::opposite

(

)

const

returns the line with opposite direction.

Exactness

This construction is trivial and therefore always exact in Exact_predicates_inexact_constructions_kernel.

point()

template<typename Kernel >

Point_2< Kernel > CGAL::Line_2< Kernel >::point

(

const Kernel::FT

i

)

const

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.

x_at_y()

template<typename Kernel >

Kernel::FT CGAL::Line_2< Kernel >::x_at_y

(

const Kernel::FT &

y

)

const

returns the \( x\)-coordinate of the point at l with given \( y\)-coordinate.

Precondition

l is not horizontal.

y_at_x()

template<typename Kernel >

Kernel::FT CGAL::Line_2< Kernel >::y_at_x

(

const Kernel::FT &

x

)

const

returns the \( y\)-coordinate of the point at l with given \( x\)-coordinate.

Precondition

l is not vertical.