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 real numbers. The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira mirabilis. It can be expressed parametrically as

			(2)
			(3)

The golden spiral is a special case related to Fibonacci numbers, the golden ratio, and the golden rectangle.

A polygonal approximation to a logarithmic spiral can be constructed using equally spaced rays with angular separation . Starting at a point on one ray, draw each line segment at a fixed angle to the outward radial direction until it meets the next ray. For with , the resulting line segments approach a logarithmic spiral with as with held fixed. The perpendicular construction instead approaches a circle.

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).

For , 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.

For , the arc length from the limiting pole at the origin, curvature, and a continuous function giving the tangential angle are

			(6)
			(7)
			(8)

The pole is approached as , and the arc length to any point on the spiral is finite. Any radius from the origin meets the spiral at distances in geometric progression (MacTutor Archive).

The Cesàro equation, with the same arc length origin, is

(9)

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