SPIRAL

This article appears in Volume V25, Page 692 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: SOU-STE

SPIRAL , in

mathematics

MATHEMATICS (Gr. /saBnľar1Kil, Sc. vOcvn or E7rlQTt'ľn; from p iN.La, "learning" or "science ")

, the locus of the extremity of a line ((it

radius

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

RADIUS

vector and the vector

angle

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form,
angus
- ANGUS, EARLS OF
, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

; 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

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form,
angus
- ANGUS, EARLS OF
, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

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

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

ANALOGY (Gr. avaXo-yLa, proportion)

of its equation (polar) to that (Cartesian) of a hyperbola between the asymptotes, is the inverse of the Archimedean. Its p, r equation is p2 =-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 spiralstermed

Cotes

COTES, ROGER (1682-1716)

's spirals appear as the path of a particle moving under the influence of a central force varying as the inverse

cube

CUBE (Gr. K46os, a cube)

of the distance (see

MECHANICS

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 p2=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=(AI )u2+B. The right-hand side may be written as C2 (u'2+D2), C2 (u2D2), C2 (D2u2) according as A1 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. Of

interest

INTEREST

is the spiral r=a02/(02I), which has the circle r=a as an asymptote in addition to a linear asymptote

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