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


Fermat Spiral

\displaystyle r^2 = a^2 \theta


Logarithmic Spiral

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


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.


Hyperbolic Spiral

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


Lituus Spiral

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


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.


Circle Involute


Cotes Spiral

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


Nielsen Spiral


Polygonal Spiral


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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