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8.3 Spirals

A number of interesting curves have polar equation r=f(), where f is a monotonic function (always increasing or decreasing). This property leads to a spiral shape. The logarithmic spiral or Bernoulli spiral (Figure 1, left) is self-similar: by rotation the curve can be made to match any scaled copy of itself. Its equation is r=k; the angle between the radius from the origin and the tangent to the curve is constant, and equal to =arccot a. A curve parametrized by arclength and such that the radius curvature is proportional to the parameter at each point is a Bernoulli spiral.

  
Figure 1: The Bernoulli or logarithmic spiral (left), the Archimedes or linear spiral (middle), and the Cornu spiral (right).

In the Archimedean spiral or linear spiral (Figure 1, middle), it is the spacing between intersections along a ray from the origin that is constant. The equation of this spiral is r=a; by scaling one can take a=1. It has an inner endpoint, in contrast with the logarithmic spiral, which spirals down to the origin without reaching it. The Cornu spiral or clothoid (Figure 1, right), important in optics and engineering, has the following parametric representation in Cartesian coordinates:

(C and S are the so-called Fresnel integrals; see the Standard Math Tables and Formulas.) A curve parametrized by arclength and such that the radius curvature is inversely proportional to the parameter at each point is a Cornu spiral (compare the Bernoulli spiral).

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Silvio Levy
Wed Oct 4 16:41:25 PDT 1995

This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.