SPIRAL  , in

mathematics, the locus of the extremity of a line ((it 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 conveniently 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 . I) is such that as the vector angle increases arithmetically, the radius vector increases .g fiG /-/G 3 . geometrically; this definition leads to an equation of the form r=Aead, 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 a . 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 =aO (fig . 2) ; the hyperbolic or reciprocal spiral, r =aC-I (fig . 3) ; and the lituus, r = a0- (fig . 4) .

The first-named was discovered by

Conon, whose studies were completed by Archimedes . Its " p, r" equation is p=r2h' (a2+r2), 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-2+a-2, 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 isp-2=Ar 2+B,inwhichAand B can have any values . If B =o, we have p = r-%i A, and the locus is the equiangular spiral . If A=1 we have p-2=r 2+B, which leads to the polar equation rO =IN B, i.e. the reciprocal spiral . The more general investigation is as follows: Writing u=r= we have p—2=Aug+B, and since p-2 = u2 + (du/dO )2 (see INFINITESIMAL CALCULUS), then Aug+B=u2+ (du/dO)2, i.e . (du/dO)2=(A—I )u2+B . The right-hand side may be written as C2 (u'2+D2), C2 (u2—D2), C2 (D2—u2) according as A—1 and B are both positive, A-I positive and B negative, and as A — r negative and B positive .

On integration these three forms yield the polar equations u=C sin hDO, u=C

cos hDO, and u=C sin DO .