Ingeometry,line coordinates are used to specify the position of aline just as point coordinates (or simplycoordinates) are used to specify the position of a point.

There are several possible ways to specify the position of a line in the plane. A simple way is by the pair(m,b) where the equation of the line isy =mx + b. Herem is theslope andb is they-intercept. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates(l,m) where the equation of the line islx + my + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations ofl andm are the negative reciprocals of thex andy-intercept respectively.

The exclusion of lines passing through the origin can be resolved by using a system of three coordinates(l,m,n) to specify the line with the equationlx + my + n = 0. Herel andm may not both be 0. In this equation, only the ratios betweenl,m andn are significant, in other words if the coordinates are multiplied by a non-zero scalar then line represented remains the same. So(l,m,n) is a system ofhomogeneous coordinates for the line.

If points in thereal projective plane are represented by homogeneous coordinates(x,y,z), the equation of the line islx + my + nz = 0, provided(l,m,n) ≠ (0,0,0) . In particular, line coordinate(0, 0, 1) represents the linez = 0, which is theline at infinity in theprojective plane. Line coordinates(0, 1, 0) and(1, 0, 0) represent thex andy-axes respectively.

Just asf(x, y) = 0 can represent acurve as a subset of the points in the plane, the equation φ(l, m) = 0 represents a subset of the lines on the plane. The set of lines on the plane may, in an abstract sense, be thought of as the set of points in a projective plane, thedual of the original plane. The equation φ(l, m) = 0 then represents a curve in the dual plane.

For a curvef(x, y) = 0 in the plane, thetangents to the curve form a curve in the dual space called thedual curve. If φ(l, m) = 0 is the equation of the dual curve, then it is called thetangential equation, for the original curve. A given equation φ(l, m) = 0 represents a curve in the original plane determined as theenvelope of the lines that satisfy this equation. Similarly, if φ(l, m, n) is ahomogeneous function then φ(l, m, n) = 0 represents a curve in the dual space given in homogeneous coordinates, and may be called the homogeneous tangential equation of the enveloped curve.

Tangential equations are useful in the study of curves defined as envelopes, just as Cartesian equations are useful in the study of curves defined as loci.

A linear equation in line coordinates has the formal + bm + c = 0, wherea,b andc are constants. Suppose (l, m) is a line that satisfies this equation. Ifc is not 0 thenlx + my + 1 = 0, wherex = a/c andy = b/c, so every line satisfying the original equation passes through the point (x, y). Conversely, any line through (x, y) satisfies the original equation, soal + bm + c = 0 is the equation of set of lines through (x, y). For a given point (x, y), the equation of the set of lines though it islx + my + 1 = 0, so this may be defined as the tangential equation of the point. Similarly, for a point (x, y, z) given in homogeneous coordinates, the equation of the point in homogeneous tangential coordinates islx + my + nz = 0.

The intersection of the lines (l₁, m₁) and (l₂, m₂) is the solution to the linear equations

${\displaystyle l_{1}x+m_{1}y+1=0}$

${\displaystyle l_{2}x+m_{2}y+1=0.}$

ByCramer's rule, the solution is

${\displaystyle x=\frac{m_1-m_2}{l_1{m}_2-l_2{m}_1},y=-\frac{l_1-l_2}{l_1{m}_2-l_2{m}_1}.}$

The lines (l₁, m₁), (l₂, m₂), and (l₃, m₃) areconcurrent when thedeterminant

For homogeneous coordinates, the intersection of the lines (l₁, m₁, n₁) and (l₂, m₂, n₂) is thecross product:

${\displaystyle (m_1{n}_2-m_2{n}_1,l_2{n}_1-l_1{n}_2,l_1{m}_2-l_2{m}_1).}$

The lines (l₁, m₁, n₁), (l₂, m₂, n₂) and (l₃, m₃, n₃) areconcurrent when thedeterminant

Dually, the coordinates of the line containing (x₁, y₁, z₁) and (x₂, y₂, z₂) can be obtained via the cross product:

${\displaystyle (y_1{z}_2-y_2{z}_1,x_2{z}_1-x_1{z}_2,x_1{y}_2-x_2{y}_1).}$

For two given points in thereal projective plane, (x₁, y₁, z₁) and (x₂, y₂, z₂), the three determinants

${\displaystyle y_1{z}_2-y_2{z}_1,x_2{z}_1-x_1{z}_2,x_1{y}_2-x_2{y}_1}$

determine theprojective line containing them.

Similarly, for two points inRP³, (x₁, y₁, z₁, w₁) and (x₂, y₂, z₂, w₂), the line containing them is determined by the six determinants

${\displaystyle x_1{y}_2-x_2{y}_1,x_1{z}_2-x_2{z}_1,y_1{z}_2-y_2{z}_1,x_1{w}_2-x_2{w}_1,y_1{w}_2-y_2{w}_1,z_1{w}_2-z_2{w}_1.}$

This is the basis for a system of homogeneous line coordinates in three-dimensional space calledPlücker coordinates. Six numbers in a set of coordinates only represent a line when they satisfy an additional equation. This system maps the space of lines in three-dimensional space toprojective spaceRP⁵, but with the additional requirement the space of lines corresponds to theKlein quadric, which is amanifold of dimension four.

More generally, the lines inn-dimensional projective space are determined by a system ofn(n − 1)/2 homogeneous coordinates that satisfy a set of (n − 2)(n − 3)/2 conditions, resulting in a manifold of dimension 2n− 2.

Isaak Yaglom has shown^[1] howdual numbers provide coordinates for oriented lines in the Euclidean plane, andsplit-complex numbers form line coordinates for thehyperbolic plane. The coordinates depend on the presence of an origin and reference line on it. Then, given an arbitrary line its coordinates are found from the intersection with the reference line. The distances from the origin to the intersection and the angle θ of inclination between the two lines are used:

• ${\displaystyle z=\left(\tan \frac{\theta }{2}\right)(1+sϵ)}$ is the dual number^[1]: 81 for a Euclidean line, and
• ${\displaystyle z=\left(\tan \frac{\theta }{2}\right)(\cosh s+j\sinh s)}$ is the split-complex number^[1]: 118 for a line in the Lobachevski plane.

Since there are lines ultraparallel to the reference line in the Lobachevski plane, they need coordinates too: There is a uniquecommon perpendicular, says is the distance from the origin to this perpendicular, andd is the length of the segment between reference and the given line.

• ${\displaystyle z=\left(\tanh \frac{d}{2}\right)(\sinh s+j\cosh s)}$ denotes the ultraparallel line.^[1]: 118

The motions of the line geometry are described withlinear fractional transformations on the appropriate complex planes.^{[1]: 87, 123}

• Robotics conventions

• Baker, Henry Frederick (1923),Principles of geometry. Volume 3. Solid geometry. Quadrics, cubic curves in space, cubic surfaces., Cambridge Library Collection,Cambridge University Press, p. 56,ISBN 978-1-108-01779-4,MR 2857520{{citation}}:ISBN / Date incompatibility (help). Reprinted 2010.
• Jones, Alfred Clement (1912).An Introduction to Algebraical Geometry. Clarendon. p. 390.

• Coordinate systems
• Analytic geometry

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