Line coordinates

Template:Short description Template:Distinguish In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point. The idea of line coordinates is fundamental to line geometry, which is an approach to geometry where lines are taken as fundamental and indivisible objects, rather than points.

Lines in the plane

There are several possible ways to specify the position of a line in the plane. A simple way is by the pair Template:Math where the equation of the line is Template:Math. Here Template:Mvar is the slope and Template:Mvar is the [[y-intercept|Template:Mvar-intercept]]. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates Template:Math where the equation of the line is Template:Math. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of Template:Mvar and Template:Mvar are the negative reciprocals of the Template:Mvar and [[y-intercept|Template:Mvar-intercept]] respectively.

The exclusion of lines passing through the origin can be resolved by using a system of three coordinates Template:Math to specify the line with the equation Template:Math. Here Template:Mvar and Template:Mvar may not both be 0. In this equation, only the ratios between Template:Mvar, Template:Mvar and Template:Mvar are significant, in other words if the coordinates are multiplied by a non-zero scalar then line represented remains the same. So Template:Math is a system of homogeneous coordinates for the line.

If points in the real projective plane are represented by homogeneous coordinates Template:Math, the equation of the line is Template:Math, provided Template:Math. In particular, line coordinate Template:Math represents the line Template:Math, which is the line at infinity in the projective plane. Line coordinates Template:Math and Template:Math represent the Template:Mvar and Template:Mvar-axes respectively.

Tangential equations

Just as Template:Math can represent a curve as a subset of the points in the plane, the equation Template:Math 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, the dual of the original plane. The equation Template:Math then represents a curve in the dual plane.

For a curve Template:Math in the plane, the tangents to the curve form a curve in the dual space called the dual curve. If Template:Math is the equation of the dual curve, then it is called the tangential equation, for the original curve. A given equation Template:Math represents a curve in the original plane determined as the envelope of the lines that satisfy this equation. Similarly, if Template:Math is a homogeneous function then Template:Math 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.

Tangential equation of a point

A linear equation in line coordinates has the form Template:Math, where Template:Mvar, Template:Mvar and Template:Mvar are constants. Suppose Template:Math is a line that satisfies this equation. If Template:Mvar is not 0 then Template:Math, where Template:Math and Template:Math, so every line satisfying the original equation passes through the point Template:Math. Conversely, any line through Template:Math satisfies the original equation, so Template:Math is the equation of set of lines through Template:Math. For a given point Template:Math, the equation of the set of lines though it is Template:Math, so this may be defined as the tangential equation of the point. Similarly, for a point Template:Math given in homogeneous coordinates, the equation of the point in homogeneous tangential coordinates is Template:Math.

Formulas

The intersection of the lines Template:Math and Template:Math is the solution to the linear equations

$$\begin{array}{cc}{l}_{1}x+m_{1}y+1& =0,\\ l_{2}x+m_{2}y+1& =0.\end{array}$$

By Cramer's rule, the solution is

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

The lines Template:Math, Template:Math, and Template:Math are concurrent when the determinant

$$\left|\begin{array}{ccc}{l}_1& m_1& 1\\ l_2& m_2& 1\\ l_3& m_3& 1\end{array}\right|=0.$$

For homogeneous coordinates, the intersection of the lines Template:Math and Template:Math is the cross product:

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

The lines Template:Math, Template:Math and Template:Math are concurrent when the determinant

$$\left|\begin{array}{ccc}{l}_1& m_1& n_1\\ l_2& m_2& n_2\\ l_3& m_3& n_3\end{array}\right|=0.$$

Dually, the coordinates of the line containing Template:Math and Template:Math can be obtained via the cross product: $$(y_1{z}_2-y_2{z}_1,x_2{z}_1-x_1{z}_2,x_1{y}_2-x_2{y}_1).$$

Lines in three-dimensional space

Template:Main For two given points in the real projective plane, Template:Math and Template:Math, the three determinants

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

determine the projective line containing them.

Similarly, for two points in Template:Tmath Template:Math and Template:Math, the line containing them is determined by the six determinants

$$\begin{array}{cccc}& x_1{y}_2-x_2{y}_1,& & x_1{w}_2-x_2{w}_1,\\ & x_1{z}_2-x_2{z}_1,& & y_1{w}_2-y_2{w}_1,\\ & y_1{z}_2-y_2{z}_1,& & z_1{w}_2-z_2{w}_1.\end{array}$$

This is the basis for a system of homogeneous line coordinates in three-dimensional space called Plücker 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 to projective space Template:Tmath but with the additional requirement the space of lines corresponds to the Klein quadric, which is a manifold of dimension four.

More generally, the lines in Template:Mvar-dimensional projective space are determined by a system of Template:Math homogeneous coordinates that satisfy a set of Template:Math conditions, resulting in a manifold of dimension Template:Math.

With complex numbers

Template:Full article Isaak Yaglom has shown how dual numbers provide coordinates for oriented lines in the Euclidean plane, and split-complex numbers form line coordinates for the hyperbolic 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 distance Template:Mvar from the origin to the intersection and the angle Template:Mvar of inclination between the two lines are used:

• $z=\tan \left(\frac{\theta }{2}\right)(1+sϵ)$ is the dual numberTemplate:Rp for a Euclidean line, and
• $z=\tan \left(\frac{\theta }{2}\right)(\cosh s+j\sinh s)$ is the split-complex numberTemplate:Rp 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 unique common perpendicular, say Template:Mvar is the distance from the origin to this perpendicular, and Template:Mvar is the length of the segment between reference and the given line.

• $z=\tanh \left(\frac{d}{2}\right)(\sinh s+j\cosh s)$ denotes the ultraparallel line.Template:Rp

The motions of the line geometry are described with linear fractional transformations on the appropriate complex planes.Template:Rp

See also

• Robotics conventions
• Line complex

References

• Template:Citation. Reprinted 2010.
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