# Archimedean Spiral

## History

Studied by Archimedes (~287 BC – ~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'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.)

## 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's no scaling.

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

The inverse curve of Fermat's spiral with inversion circle of radius 1 at center is the lituus.

### Nested Tangent Circles

The tangent circles of Archimedes's spiral are all nested. need to proof that archimedes spiral's osculating circles are nested inside each other.

## Related Web Sites

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

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