Logarithmic Spiral

The logarithmic spiral is a spiral whose polar equation is given by

(1)

where is the distance from the origin, is the angle from the x-axis, and and are arbitrary constants. The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira mirabilis. It can be expressed parametrically as

			(2)
			(3)

This spiral is related to Fibonacci numbers, the golden ratio, and the golden rectangle, and is sometimes called the golden spiral.

The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3).

The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw it true to form) together with the words "eadem mutata resurgo" ("I shall arise the same though changed"). Torricelli worked on it independently and found the length of the curve (MacTutor Archive).

The rate of change of radius is

(4)

and the angle between the tangent and radial line at the point is

(5)

So, as , and the spiral approaches a circle.

If is any point on the spiral, then the length of the spiral from to the origin is finite. In fact, from the point which is at distance from the origin measured along a radius vector, the distance from to the pole along the spiral is just the arc length. In addition, any radius from the origin meets the spiral at distances which are in geometric progression (MacTutor Archive).

The arc length (as measured from the origin, ), curvature, and tangential angle of the logarithmic spiral are given by

			(6)
			(7)
			(8)

The Cesàro equation is then given by

(9)

On the surface of a sphere, the analog is a loxodrome.