Line Coordinates - Tangential Equations

Tangential Equations

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.

For a curve f(x, y) = 0 in the plane, the tangents to the curve form a curve in the dual space called the dual curve. If φ(l, m) = 0 is the equation of the dual curve, then it is called the tangential equation, for the original curve. A given equation φ(l, m) = 0 represents a curve in the original plane determined as the envelope of the lines that satisfy this equation. Similarly, if φ(l, m, n) is a homogeneous 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.