When a curve C1 is given by a Cartesian function y = f(x), the inverse function is
defined as the function g for which g(f(x)) = x. The corresponding inverse
curve C2 has the same form as C1, only the x- and y-axis are interchanged.
A more interesting case is polar
each point is inverted along the line through the center of inversion.
Given a curve C1, draw a line l through O. Line l intersects C1 in a point P. Now
construct points Q of C2 so that OP * OQ = 1.
When curve C1 has been defined as r = f(φ), then the polar inverse
- with O as the center of inversion - is a curve C2, which has as polar equation: r = 1 / f(φ).
When C2 is the inverse of C1, then C1 is the inverse of C2.
Some curves have several curves inversely related to them. Each inverse then has a different center of inversion.
The first mathematician who discussed the curves was Steiner (1824).
A curve which is invariant under inversion - given a certain point of inversion - is
called an anallagmatic curve
Moutard introduced the notion, in 1860.
An anallagmatic curve can be identified with a cyclic.
Other interesting inverse relations are the following.
1) Without change, from
allagma (Gr.) = change.