#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 \( ax +by +c = 0\). More...

	Line_2 (const Point_2< Kernel > &p, const Point_2< Kernel > &q)
	introduces a line `l` passing through the points `p` and `q`. More...

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

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

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

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

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. More...

bool	operator!= (const Line_2< Kernel > &h) const
	Test for inequality. More...

Kernel::RT	a () const
	returns the first coefficient of `l`. More...

Kernel::RT	b () const
	returns the second coefficient of `l`. More...

Kernel::RT	c () const
	returns the third coefficient of `l`. More...

Point_2< Kernel >	point (const Kernel::FT i) const
	returns an arbitrary point on `l`. More...

Point_2< Kernel >	projection (const Point_2< Kernel > &p) const
	returns the orthogonal projection of `p` onto `l`. More...

Kernel::FT	x_at_y (const Kernel::FT &y) const
	returns the \( x\)-coordinate of the point at `l` with given \( y\)-coordinate. More...

Kernel::FT	y_at_x (const Kernel::FT &x) const
	returns the \( y\)-coordinate of the point at `l` with given \( x\)-coordinate. More...

Predicates
bool	is_degenerate () const
	line `l` is degenerate, if the coefficients `a` and `b` of the line equation are zero. More...

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`. More...

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`. More...

Direction_2< Kernel >	direction () const
	returns the direction of `l`. More...

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

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. More...

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`. More...

Constructor & Destructor Documentation

◆ Line_2() [1/6]

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 \( ax +by +c = 0\).

◆ Line_2() [2/6]

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.

◆ Line_2() [3/6]

template<typename Kernel >

CGAL::Line_2< Kernel >::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() [4/6]

template<typename Kernel >

CGAL::Line_2< Kernel >::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() [5/6]

template<typename Kernel >

CGAL::Line_2< Kernel >::Line_2 ( const Segment_2< Kernel > & s )

introduces a line l supporting the segment s, oriented from source to target.

◆ Line_2() [6/6]

template<typename Kernel >

CGAL::Line_2< Kernel >::Line_2 ( const Ray_2< Kernel > & r )

introduces a line l supporting the ray r, with same orientation.

Member Function Documentation

◆ a()

template<typename Kernel >

Kernel::RT CGAL::Line_2< Kernel >::a ( ) const

returns the first coefficient of l.

◆ b()

template<typename Kernel >

Kernel::RT CGAL::Line_2< Kernel >::b ( ) const

returns the second coefficient of l.

◆ c()

template<typename Kernel >

Kernel::RT CGAL::Line_2< Kernel >::c ( ) const

returns the third coefficient of l.

◆ direction()

template<typename Kernel >

Direction_2<Kernel> CGAL::Line_2< Kernel >::direction ( ) const

returns the direction of l.

◆ has_on()

template<typename Kernel >

bool CGAL::Line_2< Kernel >::has_on ( const Point_2< Kernel > & p ) const

◆ has_on_negative_side()

template<typename Kernel >

bool CGAL::Line_2< Kernel >::has_on_negative_side ( const Point_2< Kernel > & p ) const

◆ has_on_positive_side()

template<typename Kernel >

bool CGAL::Line_2< Kernel >::has_on_positive_side ( const Point_2< Kernel > & p ) const

◆ is_degenerate()

template<typename Kernel >

bool CGAL::Line_2< Kernel >::is_degenerate ( ) const

line l is degenerate, if the coefficients a and b of the line equation are zero.

◆ is_horizontal()

template<typename Kernel >

bool CGAL::Line_2< Kernel >::is_horizontal ( ) const

◆ is_vertical()

template<typename Kernel >

bool CGAL::Line_2< Kernel >::is_vertical ( ) const

◆ operator!=()

template<typename Kernel >

bool CGAL::Line_2< Kernel >::operator!= ( const Line_2< Kernel > & h ) const

Test for inequality.

◆ operator==()

template<typename Kernel >

bool CGAL::Line_2< Kernel >::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.

◆ opposite()

template<typename Kernel >

Line_2<Kernel> CGAL::Line_2< Kernel >::opposite ( ) const

returns the line with opposite direction.

◆ oriented_side()

template<typename Kernel >

Oriented_side CGAL::Line_2< Kernel >::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.

◆ perpendicular()

template<typename Kernel >

Line_2<Kernel> CGAL::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.

◆ 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.

◆ projection()

template<typename Kernel >

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

returns the orthogonal projection of p onto l.

◆ to_vector()

template<typename Kernel >

Vector_2<Kernel> CGAL::Line_2< Kernel >::to_vector ( ) const

returns a vector having the direction of l.

◆ transform()

template<typename Kernel >

Line_2<Kernel> CGAL::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.

◆ 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.

Definition

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

Creation

Operations

Predicates

Convenience Boolean Functions

Miscellaneous

Constructor & Destructor Documentation

◆ Line_2() [1/6]

◆ Line_2() [2/6]

◆ Line_2() [3/6]

◆ Line_2() [4/6]

◆ Line_2() [5/6]

◆ Line_2() [6/6]

Member Function Documentation

◆ a()

◆ b()

◆ c()

◆ direction()

◆ has_on()

◆ has_on_negative_side()

◆ has_on_positive_side()

◆ is_degenerate()

◆ is_horizontal()

◆ is_vertical()

◆ operator!=()

◆ operator==()

◆ opposite()

◆ oriented_side()

◆ perpendicular()

◆ point()

◆ projection()

◆ to_vector()

◆ transform()

◆ x_at_y()

◆ y_at_x()

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