# Definition:Spiral

## 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:

$r = a \theta$

### Reciprocal Spiral

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

$r = \dfrac a \theta$

### Fermat's Spiral

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

$r^2 = \dfrac a \theta$

### Logarithmic Spiral

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

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

### 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.

### Lituus

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

$r^2 \theta = a^2$