Archimedean Spirals

A spiral of Archimedes is a curve described by a point which travels along a straight line with constant speed whilst the line rotates at a constant angular rate about a fixed point on it. In symbols: r = u.t, q = n.t, so that we have the equation r = k.q, where the constant k = u/n (with units of length/angle). The distance between points on successive turns of the spiral is constant, namely 2.p.k.


The figure shows the curve for positive q. For negative q reflect the curve in the y axis. [Lamb p.308]

Equiangular Spiral

An equiangular spiral has the property that the curve makes a constant angle with the radius. Denoting this angle by a we have r~q = r.cota, the solution of which is r = k.exp(q.cota). When q = 0 we have r = k. As q increases we get a positive (anticlockwise) spiral taking the point further from O, while as the angle takes negative values we get a negative spiral taking the point closer and closer to O. If k = 1 then the values of r for negative q are the reciprocals of the corresponding values for positive q.

Since r~s = cosa the length of the curve between the radii r and r' is (r'−r).seca [Lamb pp.307-8]

In an equiangular spiral the angle of the normal is y = q + a. So y~s = q~s = (sina)/r, or the radius of curvature is c = r/sina. Hence the radius of curvature subtends a right angle at the origin. [Lamb p.337] The tangents and normals at intervals of 90° in q define a spiralling pattern of rectangles, each containing a section of the spiral that is geometrically similar to all the others.

If a = p/4 (45°) then cota = 1 and so r = k.e^q. In this case seca = √2.

If a = p/3 (60°) then cota = 0.57735 = 1/√3.

There is an equangular spiral that is its own evolute, for which a is the root of exp(1.5p.cota) = tana, approximately 74°39'. In this the tangent strikes the next coil of the spiral normally. [Cundy and Rollett p.71]