1911 Encyclopædia Britannica/Spiral

SPIRAL, in mathematics, the locus of the extremity of a line (01 radius vector) which varies in length as it revolves about a fixed point (or origin). Here we consider some of the more important plane spirals. Obviously such curves are con- veniently expressed by polar equations, i.e. equations which directly state a relation existing between the radius vector and the vector angle; another form is the " p, r " equation, wherein r is the radius vector of a point, and p the length of the perpendicular from the origin to the tangent at that point.

The equiangular or logarithmic spiral (fig. 1) is such that as the vectoF angle increases arithmetically, the radius vector increases

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geometrically; this definition leads to an equation of the form r — Ae^, where e is the base of natural logarithms and A, B are constants. Another definition is that the tangent makes a constant angle (a, say) with the radius vector; this leads to p — r sin o. This curve has the property that its positive pedals, inverse, polar reciprocal and evolutes are all equal equiangular spirals. A group of spirals are included in the "parabolic spirals " given by the equation r = aO"; the more important are the Archimedean spiral, r — a6 (fig. 2) ; the hyperbolic or reciprocal spiral, r = a(?~i (fig. 3) ; and the lituus, r = ad~i (fig. 4). The first-named was discovered by Conon, whose studies were completed by Archimedes. Its " p, r " equation is p = r 2 /V (a 2 +r 2 ), and the angle between the radius vector and the tangent equals the vector angle. The second, called hyperbolic on account of the analogy of its equation (polar) to that (Cartesian) of a hyperbola between the asymptotes, is the inverse of the Archimedean. Its p, r equation is p~ 2 =r~i-\-a~*, and it has an asymptote at the distance a above the initial line. The lituus has the initial line as asymptote. Another group of spirals — termed Cotes's spirals — appear as the path of a particle moving under the influence of a central force varying as the inverse cube of the distance (see Mechanics). Their general equation is p~? = Ar~' + B,in which A and B can have any values. If B =0, we have p = r^ A, and the locus is the equiangular spiral. If A = i we have p~2=r~z-\-B, which leads to the polar equation r0 = i/VB, i.e. the reciprocal spiral. The more general investigation is as follows: Writing u = r~i we have p~ 2 =Au 2 + B, and since p- 2 = u 2 + (du/d8) 2 (see Infinitesimal Calculus), then Au 2 + B=u 2 + (du/d0) 2 , i.e. (du/de) 2 = (A-i)u 2 + B. The right-hand side may be written as C 2 (u 2 + D r ), C 2 (zt 2 — D 2 ), C 2 (D 2 — m 2 ) according as A — I and B are both positive, A — I positive and B negative, and as A — 1 negative and B positive. On integration these three forms yield the polar equations u = C sin hDO, w = C cos hDe, and k = C sin D6. Of interest is the spiral r = a0 2 /(8 2 — i), which has the circle r = a as an asymptote in addition to a linear asvmptote