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SPIRAL
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SPIRAL , in See also:mathematics, the See also:locus of the extremity of a See also:line ((it See also:radius vector) which varies in length as it revolves about a fixed point (or origin). Here we consider some of the more important See also:plane spirals. Obviously such curves are conveniently expressed by polar equations, i.e. equations which directly See also:state a relation existing between the radius vector and the vector See also:angle; another See also:form is the " p, r " See also: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 See also:definition leads to an equation of the form r=Aead, where e is the See also:base of natural logarithms and A, B are constants. Another definition is that the tangent makes a See also:constant angle (a, say) with the radius vector; this leads to p=r See also:sin a. This See also:curve has the See also:property that its See also:positive pedals, inverse, polar reciprocal and evolutes are all equal equiangular spirals. A See also: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 See also:lituus, r = a0- (fig. 4). The first-named was discovered by See also:Conon, whose studies were completed by See also:Archimedes. Its " p, r" equation is p=r2h' (See also:a2+See also:r2), and the angle between the radius vector and the tangent equals the vector angle. The second, called hyperbolic on See also:account of the See also:analogy of its equation (polar) to that (Cartesian) of a See also: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 See also:Cotes's spirals —appear as the path of a particle moving under the See also:influence of a central force varying as the inverse See also:cube of the distance (see See also:MECHANICS). Their See also: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: See also: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-See also:hand See also: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 See also:cos hDO, and u=C sin DO. Additional information and CommentsThere are no comments yet for this article.
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