inverse

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 inversion, where 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 ¹⁾. Examples include:

• cardioid
• Cartesian oval
• Cassini oval
• circle (center of inversion not on circle) 
• logarithmic spiral (pole)
• limaçon
• sinusoidal spiral (pole)
• strophoid (pole)
• trisectrix of MacLaurin

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.