inverse

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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:

Moutard introduced the notion, in 1860.

An anallagmatic curve can be identified with a cyclic.

Other interesting inverse relations are the following.

curve 1 center of
inversion
(curve 1)
center of
inversion
(curve 2)
curve 2
Archimedean spiral (parameter a) pole pole Archimedean spiral (parameter -a)
Cayley's sextic focus   Tschirnhausen's cubic
circle center center circle
cissoid (MacTutor: cardioid) cusp vertex parabola
cochleoid pole - quadratrix
conic focus pole limašon
ellipse focus pole or node ordinary limašon
epi spiral pole pole rhodonea
Fermat's spiral pole pole lituus
folium simple   top of the knot Tschirnhausen's cubic
hyperbola focus pole or node limašon (with a noose)
 rectangular - center center lemniscate
 rectangular - vertex node (right) strophoid
asymptote angle: φ/3 vertex node trisectrix of Maclaurin
line not on line on circle circle
parabola focus cusp cardioid
sinusoidal spiral (parameter a) pole pole sinusoidal spiral (parameter -a)
trisectrix of Maclaurin focus - Tschirnhausen's cubic

 


notes

1) Without change, from allagma (Gr.) = change.