Inversion in a Circle

Abstract

A function may be thought of as a transformation, or “mapping,” from the x-axis to the y-axis, both of which are one-dimensional sets of points. In higher mathematics we also deal with transformations from a two-dimensional set of points to another two-dimensional set, that is, from one plane to another. One of the most interesting transformations of this kind is the transformation of inversion, or more precisely, inversion in the unit circle. Given a circle with center O and radius 1, a point P whose distance from O is OP = r is “mapped” to a point Q, lying on the same ray from O as P, whose distance from O is OQ = 1/r (Fig. 12.1). In this way, a one-to-one correspondence is established between the points of the original plane and those of the new plane: every point of the one plane is mapped onto a point of the other.¹ There is only one exception to this rule: the point O itself.

I could be bounded in a nutshell, and count myself a king of infinite space.

— William Shakespeare (1564–1616), Hamlet, Act II, Scene 2