An elementary course of infinitesimal calculus . Fig. 102. The dotted part of the curve in Fig. 102 corresponds tonegative values of 0. 141. The Llma^on, and Cardiold. If a point 0 on the circumference of a fixed circle ofradius |a be taken as pole, and the diameter through 0 asinitial line, the radius vector of any point Q on the cir-cumference is given by r = acos^ (1). If on this radius we take two points P, P at equal constantdistances c firom Q, the locus of these points is called a lima9on. Its equation is evidently r = acos ^ + c (2). This includes the paths both of P and of P, if 6 ran
An elementary course of infinitesimal calculus . Fig. 102. The dotted part of the curve in Fig. 102 corresponds tonegative values of 0. 141. The Llma^on, and Cardiold. If a point 0 on the circumference of a fixed circle ofradius |a be taken as pole, and the diameter through 0 asinitial line, the radius vector of any point Q on the cir-cumference is given by r = acos^ (1). If on this radius we take two points P, P at equal constantdistances c firom Q, the locus of these points is called a lima9on. Its equation is evidently r = acos ^ + c (2). This includes the paths both of P and of P, if 6 range from0 to 2ir. 140-141] SPECIAL CURVES. 369 If c a, r cannot vanish; see the curvetraced by Pj, Pj in the Pig. 103. In the critical case of c = ft, the loop shrinks into a locus is now called a cardioid or heart-shaped equation is r- = a(l+cos^) (3). See the curve traced by Pj, Pi in the figure. Also Fig. 89,p. 355. L. 24 370 INFINITESIMAL CALCULUS. [CH. IX 142. The curves r = acosn0. A number of important curves are included in the type r = a^cosn0 : (1). Thus if w = + 1, we have the circle r = acos 0 (2) and the straight line rcos 6 = a (3). If n = + 2 we have the lemniscate of Bernoulli 7^ = 0=003 2^ (4), and the rectangular hyperbola /?I cos 20 = a (5). The equation (4) makes r real for values of 6 between?t Jir, imaginary for values between ^ and fir, and so on. Alsor is a maximum for ^ = 0, 6 = ir, etc. It follows that thelemniscate consists of two loops, with a node at the origin. SeeFig. 113, p. 389. If n = + ^, we have the cardioid ri = aicos^O, or r=Ja(l + cos0) (6), and the parabola 2a r^eosi0 = a^, or r = q ;: (7). 1 + cos p ^ ^ The curves corresponding
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