Archimedean Spiral

History

Studied by Archimedes (~287 BC to ~212 BC).

The reason parabolic spiral and hyperbolic spiral are so named is because their equation in polar system r*θ == 1 and r^2 == θ resembles the equation for hyperbola x*y == 1 and parabola x^2 == y in rectangular coordinates system. What is Fermat's involvement with parabolic spiral?

Hyperbolic spiral is also called reciprocal spiral, because it is the inverse curve of Archemedes' spiral, with inversion center at the center. The inversion curve of all Archemedean spirals with respect to a circle on center is another Archemedean spiral. (see below)

Description

Archimedean spiral is defined by the polar equation r == θ^n. Special names are given for some value of n.

n == 1, we have r == θ, Archimedes' spiral.
n== 1/2, we have r == Sqrt[θ], Fermat's spiral. (aka parabolic spiral)
n== 0, we have r = 1, circle.
n== -1/2, we have r == 1/Sqrt[θ], Lituus.
n== -1, we have r = 1/θ, reciprocal spiral. (aka hyperbolic spiral.)

archimedean spiral — Archimedean spirals

archimedeanSpiral1 — r == θ^n, n from -1.6 to 0

Formula

Polar equation: r == θ^n
Rectangular, Parametric: t^n*{Cos[t],Sin[t]}

Properties

Inversion

The inverse curve of a Archimedean spiral with respect to the center is another Archimedean spiral scaled. Archimedean spiral in parametric form is {t^n*Cos[t], t^n*Sin[t]}. The inversion at origin with radius b of a point {x,y} is {(b^2*x)/(x^2 + y^2), (b^2*y)/(x^2 + y^2)}. Apply this to the parametric form and simply we get b^2*{Cos[t]*t^-n, Sin[t]*t^-n}, which is in polar form r==b^2*θ^(-n). When b==1, there is no scaling.

The inverse curve of Archimedes' spiral with inversion circle of radius 1 at center is the reciprocal spiral.