When a curve C_{1} 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 C_{2} has the same form as C_{1}, only the x- and y-axis are interchanged.
A more interesting case is polar
inversion, where
each point is inverted along the line through the center of inversion.
Given a curve C_{1}, draw a line l through O. Line l intersects C1 in a point P. Now
construct points Q of C_{2} so that OP * OQ = 1.
When curve C_{1} has been defined as r = f(φ), then the **polar inverse**
- with O as the center of inversion - is a curve C_{2}, which has as polar equation: r = 1 / f(φ).
When C_{2} is the inverse of C_{1}, then C_{1} is the inverse of C_{2}.
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****
**^{1)}.
Examples include:
*Moutard* introduced the notion, in 1860.
An anallagmatic curve can be identified with a cyclic.
Other interesting inverse relations are the following.
**notes**
1) Without change, from
allagma (Gr.) = change. |