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Archimedean Spiral

above: left: Archimedes's Spiral. Right: Archemedean spirals.

Mathematica icon Mathematica Notebook for This Page.

History

Studied by Archimedes↗ (~287BC-~212BC).

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. (*XahNote: What's 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.)

above: Archimedean spirals. movie icon r == θ^n, n from -1.6 to 0; n from 0 to 2; gcf icon archimedean_spiral.gcf.

Archimedean spiral

Formulas

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's no scaling.

The inverse curve of Archimedes' spiral with inversion circle of radius 1 at center is the reciprocal spiral. In the following images, the red curve is the reciprocal spiral, the purple is the Archimedes' spiral. The yellow curve is the inversion circle.

gcf icon archimedes_spiral_inver.gcf

Inversion of Archimedes's spiral

The inverse curve of Fermat's spiral with inversion circle of radius 1 at center is the lituus. The following shows a Lituus and Fermat's spiral. The red curve is the Fermat's spiral. The blue curve is its inversion, which is a lituus scaled by 5^2. The yellow circle is the inversion circle with radius 5. Note that points inside the circle gets mapped to outside of the circle. The closer the point is to the origin, the farther is its corresponding point outside the circle.