Definition:Spiral

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Definition

A spiral is a plane curve, or part of a plane curve, which can be expressed in polar coordinates in the form:

$r = \map f \theta$

where $f$ is either (strictly) increasing or (strictly) decreasing.


Hence a spiral is a plane curve which emanates from a central point, getting progressively farther away (or closer in) as it revolves around that point.


Archimedean Spiral

The Archimedean spiral is the locus of the equation expressed in Polar coordinates as:

$r = a \theta$


ArchimedeanSpiral.png


Reciprocal Spiral

The reciprocal spiral is the locus of the equation expressed in Polar coordinates as:

$r = \dfrac a \theta$


ReciprocalSpiral-positive.png


Fermat's Spiral

Fermat's spiral is the locus of the equation expressed in Polar coordinates as:

$r^2 = \dfrac a \theta$


FermatSpiral-positive.png


Logarithmic Spiral

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

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


LogarithmicSpiral.png


Cornu Spiral

The Cornu spiral is the locus $C$ of the equation expressed in intrinsic coordinates as:

$s = a^2 \kappa$

where:

$s$ denotes the length of arc at a point of $C$ from the origin
$\kappa$ denotes the curvature of $C$ at that point.


CornuSpiral.svg


Lituus

The lituus is the locus of the equation expressed in Polar coordinates as:

$r^2 \theta = a^2$


Lituus.png


Sources