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

ArchimedesSpiral_1000

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

Fermat Spiral

\displaystyle r^2 = a^2 \theta

FermatsSpiral_1000

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

Logarithmic Spiral

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

LogarithmicSpiral_800

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.

GoldenSpiral_1000

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

Hyperbolic Spiral

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

HyperbolicSpiral_751

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

Lituus Spiral

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

Lituus_901

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.

CornuSpiral_700

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

Circle Involute

CircleInvolute_800

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.

CotesSpiral_1100

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

Nielsen Spiral

NielsensSpiral_1000

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

Polygonal Spiral

PolygonalSpiral_800

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

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About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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