Mathematical Spirals

The equations of the basic spirals are given in polar coordinates. The $\displaystyle r$ is the radius and $\displaystyle \theta$ is the angle measured from the positive horizontal axis.The $\displaystyle a$ and $\displaystyle b$ are constants.

Archimedes Spiral

$\displaystyle r=a\theta$

http://mathworld.wolfram.com/ArchimedesSpiral.html

Fermat Spiral

$\displaystyle r^2 = a^2 \theta$

http://mathworld.wolfram.com/FermatsSpiral.html

Logarithmic Spiral

$\displaystyle r = a e^{b \theta}$

http://mathworld.wolfram.com/LogarithmicSpiral.html

Golden Spiral

The equation in polar coordinates is the same as the logarithmic spiral but the the constant $\displaystyle b$ has a special value

$\displaystyle r = a e^{b \theta}$

$\displaystyle |b| = \frac{ln\phi}{\pi / 2}$

The constant $\displaystyle b$ is in units of radians. $\displaystyle \phi$ is the golden number.

http://mathworld.wolfram.com/GoldenSpiral.html

Hyperbolic Spiral

$\displaystyle r = \frac{a}{\theta}$

http://mathworld.wolfram.com/HyperbolicSpiral.html

Lituus Spiral

$\displaystyle r^2 = \frac{a^2}{\theta}$

http://mathworld.wolfram.com/Lituus.html

Cornu Spiral

The Cornu spiral is also known as the clothoid or Euler’s spiral. The mathematical equation is not as simple as the others shown above. Please refer to the link below.

http://mathworld.wolfram.com/CornuSpiral.html

Circle Involute

http://mathworld.wolfram.com/CircleInvolute.html

http://mathworld.wolfram.com/Involute.html

Cotes Spiral

A spiral that gives the solution to the central orbit problem under a radial force law. Please refer to the link below.

http://mathworld.wolfram.com/CotesSpiral.html

Nielsen Spiral

http://mathworld.wolfram.com/NielsensSpiral.html

Polygonal Spiral

http://mathworld.wolfram.com/PolygonalSpiral.html