Plane and solid analytic geometry; an elementary textbook . Fig. 106. 204 ANALYTIC GEOMETRY [Ch. XV, § 108 on the circumference of a circle which rolls on the out-side of a fixed circle is called an epicycloid. Let the student show that the equations of the epi-cycloid are a + bs x — (a-\-b~) cos (j> — b cosy = (« + b) sin — b sin b a + b * 108. The cardioid. — An important special case of theepicycloid is the cardioid, in which a = b; but instead of. Fig. 107. obtaining its equation as a special case of that curve,we shall find it easier to obtain its polar equation at oncefrom the definit
Plane and solid analytic geometry; an elementary textbook . Fig. 106. 204 ANALYTIC GEOMETRY [Ch. XV, § 108 on the circumference of a circle which rolls on the out-side of a fixed circle is called an epicycloid. Let the student show that the equations of the epi-cycloid are a + bs x — (a-\-b~) cos (j> — b cosy = (« + b) sin — b sin b a + b * 108. The cardioid. — An important special case of theepicycloid is the cardioid, in which a = b; but instead of. Fig. 107. obtaining its equation as a special case of that curve,we shall find it easier to obtain its polar equation at oncefrom the definition. Let 0, the original point of contact of the two circles, Ch. XV, § 110] HIGHER PLANE CURVES 205 be chosen as the polar origin, and the diameter CA con-tinued through this point as the polar axis. Let thesecond circle roll on the first until the line of centres CCmakes an angle 0G7O(= ) with its original position,and let P be the new position of the point of OP = p, and the angle AOP = 0. From 0 and Pdrop the perpendiculars OM and PN o\\ CC. Evidentlythe arc OK equals the arc PK, and the angles PCC andOCC are therefore equal. Then OM=PN, and OP isparallel to CC. Then 6 = <£. Now CC = CM + MN + NC = 2 a, or 2 CM+ OP = 2 a, or 2 a cos 6 -f p = 2 a, or jo = 2 a (1 - cos 0). 109. The catenary. — The curve assumed by a perfectlyflexible chain of uniform weight per linear unit, whensuspended at its ends, is called a cat
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