Line coordinates

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.

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 is y = mx + b. Here m is the slope and b is the y-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 is lx + my + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of l and m are the negative reciprocals of the x and y-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 in which the equation, lx + my + n = 0. Here l and m may not both be 0. In this equation, only the ratios between l, m and n 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 of homogeneous coordinates for the line.

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

Just as f(x, y) = 0 can represent a curve 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, the dual of the original plane. The equation φ(l, m) = 0 then represents a curve in the dual plane.

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