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For other uses, see Spiral (disambiguation).

In mathematics, a spiral is a curve which turns around some central point or axis, getting progressively closer to or farther from it, depending on which way one follows the curve.

Two-dimensional spirals

A two-dimensional spiral may be described using polar coordinates by saying that the radius r is a continuous monotonic function of θ. The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).

Some of the more important sorts of two-dimensional spirals include:

• The Archimedean spiral: r = a + bθ
• The Cornu spiral or clothoid
• Fermat's spiral: r = θ^1/2
• The hyperbolic spiral: r = a/θ
• The lituus: r = 1/θ^1/2
• The logarithmic spiral: r = ab^θ; approximations of this are found in nature
• The Fibonacci spiral and golden spiral: special cases of the logarithmic spiral.

Three-dimensional spirals

As in the two-dimensional case, r is a continuous monotonic function of θ.

For simple 3-d spirals, the third variable, h (height), is also a continuous, monotonic function of θ.

For example, a conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of θ.

For compound 3-d spirals, such as the spherical spiral described below, h increases with θ on one side of a point, and decreases with θ on the other side.

The helix and vortex can be viewed as a kind of three-dimensional spiral.

For a helix with thickness, see spring (math).

Spherical spiral

A spherical spiral (rhumb line) is the curve on a sphere traced by a ship traveling from one pole to the other while keeping a fixed angle (but not a right angle) with respect to the meridians of longitude, i.e. keeping the same bearing. The curve has an infinite number of revolutions, with the distance between them decreasing as the curve approaches either of the poles.

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