Elementary Mathematics from an Advanced Standpoint by Klein Felix;

Author:Klein, Felix;
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2004-04-28T04:00:00+00:00

2. Contact Transformations

These transformations, so named by Sophus Lie, are obtained if, instead of the bilinear equation (4a), we start with an arbitrary higher equation in the four point coordinates of the two planes:

(1) Ω(x,y; x′, y′) = 0

We shall assume that this equation satisfies the requisite conditions of continuity. It is called, after Plücker, the aequatio directrix or directrix equation. For plane geometry, all the relevant developments are found in Plücker's work mentioned above.1 To begin with, we keep x and y fixed, i.e., we consider a definite point P(x, y) in E. (See Fig. 79.) Then the equation Ω = 0 represents, in the running coordinates x′ and y′, a definite curve C′ in the plane E′, and we make this curve correspond, as a new element of the plane E′, to the point P, as we did earlier with the straight line. If, however, we now take a fixed point P′(x′, y′) in E′, say on the curve C′, then the same equation Ω = 0, in which we now think of x′ and y′ as fixed and of x and y as running coordinates, represents a definite curve C in E. Of course, the curve C must pass through the first point P. In this way, we have established a correspondence between the points P in E and the ∞2 curves C′ in E′, and between the points P′ in E′ and the ∞2 curves C in E, just as we established earlier a correspondence between points and straight lines.

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