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 either:

emanates from a central point, getting progressively farther away

or:

comes in from the point at infinity, getting progressively closer in

as it revolves around the central 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:

$r^2 = a^2 \theta$

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 Cesàro form 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:

$r^2 = \dfrac {a^2} \theta$

Also see

• Results about spirals can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): spiral: 1.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): spiral
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): spiral
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): spiral