Line Class Reference

A line defined by two points. More...

#include <segment.h>

Inheritance diagram for Line:

Public Member Functions
	Line ()
	Empty.
	Line (const Vector &, const Vector &)
	Creates a line.
Vector	Vertex (int) const
	Return one of the vertices of the line.
Vector &	Vertex (int)
	Return one of the vertices of the line.
double	R (const Vector &) const
	Compute the squared distance between a point and the line.
double	R (const Line &) const
	Compute the squared distance between two lines.

Additional Inherited Members
Protected Member Functions inherited from Segment
	Segment ()
	Empty.
	Segment (const Vector &, const Vector &)
	Creates a segment given its end vertices.
	Segment (const Segment2 &, const double &, const double &)
	Create a segment from a segment in the plane.
	~Segment ()
	Empty.
void	Rotate (const Matrix &)
	Rotates a segment.
void	Translate (const Vector &)
	Translates a segment.
void	Scale (const double &)
	Uniformly scales a segment.
Segment	Translated (const Vector &) const
	Translates a segment.
Segment	Scaled (const Vector &) const
	Scales a segment.
Segment	Scaled (const double &) const
	Uniformly scales a segment.
Segment	Rotated (const Matrix &) const
	Rotates a segment.
Segment	InverseTransformed (const Frame &) const
	Inverse transformation.
Quadric	Equation (const Ray &) const
	Computes the polynomial equation of the Euclidean distance between a ray and the line corresponding to the segment.
Vector	Vertex (int) const
	Return one of the end vertex of the axis.
Vector &	Vertex (int)
	Return one of the end vertex of the axis.
Vector	VertexAt (const double &) const
	Compute a point on the segment.
Vector	Center () const
	Compute the center of the segment.
bool	Intersect (const Box &) const
	Check if a segment intersects a box.
Vector	GetAxis () const
	Returns the normalized axis vector.
Box	GetBox () const
	Compute the bounding box of the segment.
double	Length () const
	Return axis length.
double	R (const Vector &) const
	Compute the squared distance to the segment.
double	R (const Vector &, double &) const
	Compute the squared distance to the segment.
double	R (const Segment &) const
	Compute the squared distance between two segments.
Vector	Normal (const Vector &) const
	Compute the normal vector between a point and its projection onto the segment.
bool	Equal (const Segment &, const double &) const
	Test if two segments are almost equal.
Static Protected Member Functions inherited from Segment
static Quadric	EdgeEquation (const Ray &, const Vector &, const Vector &, const Vector &)
	Compute the polynomial equation of the distance function along the ray.
static Vector	Intersect (const Vector &, const Vector &, const double, const double, double=0.0)
	Compute the intersection between the line f(x)=y and a line such that f(a)=va and f(b)=vb, on the segment ab.
Protected Attributes inherited from Segment
Vector	b = Vector::Z
	End vertices of the segment.

Detailed Description

A line defined by two points.

The line goes through the points Line::a and Line::b, its direction is the (not normalized) vector \( b-a \). The two points should be distinct.

Constructor & Destructor Documentation

Line()

Line::Line ( const Vector & a, const Vector & b )

inlineexplicit

Creates a line.

Parameters

a,b	Two points on the line.

Member Function Documentation

R() [1/2]

double Line::R

(

const Line &

line

)

const

Compute the squared distance between two lines.

Let \( u=b-a \) and \( v \) be the direction vectors of the two lines. The closest points are found by solving a 2x2 linear system whose determinant is \( e=\|u\|^2\|v\|^2-(u\cdot v)^2=\|u\|^2\|v\|^2\sin^2\theta \), where \( \theta \) is the angle between the lines.

The lines are considered parallel when \( \sin^2\theta<10^{-12} \), i.e. \( \theta<10^{-6} \). This test is relative to the lengths of the direction vectors, so that it does not depend on the scale of the points defining the lines. In that case the system is ill-conditioned, and the result is the squared distance between the first point of this line and the other line. It is exact for parallel lines, and differs from the squared distance between two almost parallel lines only by a term of the order of \( \theta \) times the offset of the closest points along the lines.

Parameters

line

Line.

Returns

The squared Euclidean distance, which is positive or null (null if the lines intersect or are identical).

See also

Line::R(const Vector&) const

R() [2/2]

double Line::R

(

const Vector &

p

)

const

Compute the squared distance between a point and the line.

The direction vector is not normalized, the squared distance is computed as \( \|p-a\|^2 - \frac{((p-a)\cdot(b-a))^2}{\|b-a\|^2} \).

Parameters

p

Point.

Returns

The squared Euclidean distance, which is positive or null.

See also

Line::R(const Line&) const