Definition:Logarithmic Spiral

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Definition

The logarithmic spiral is the locus of the equation expressed in polar coordinates as:

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


LogarithmicSpiral.png


Also presented as

The logarithmic spiral can also be presented in the forms:

\(\ds \map \ln {\dfrac r a}\) \(=\) \(\ds \theta \cot b\)
\(\ds r\) \(=\) \(\ds a e^{\theta \cot b}\)
\(\ds \ln r\) \(=\) \(\ds a \theta\)


Also known as

Other names for the logarithmic spiral include:

equiangular spiral
exponential spiral
logistic spiral.


Also see

  • Results about the logarithmic spiral can be found here.


Historical Note

The logarithmic spiral was investigated in some depth by Jacob Bernoulli.

He directed that on his death a logarithmic spiral would be engraved on his tombstone, with the motto Eadem mutata resurgo (Though changed I arise the same).


Sources

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