** Next:** 8.4 The Peano Curve and Fractal Curves
**Up:** 8 Special Plane Curves
** Previous:** 8.2 Roulettes (Spirograph Curves)

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).

*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.