. Differential and integral calculus, an introductory course for colleges and engineering schools. x = OD + PM = (a + 6)cos 6 + b sin PCM,ZPCM = 0 - ZNCD = 0 + 6 - |, sin PCM = - cos ( + 6) = -cos ^— 6. 130 Therefore DIFFERENTIAL CALCULUS Similarly The Hypocycloid. x = (a + 6) cos 0 — 6 cosy = (a + 6) sin 0 — 6 sin 6a + 6. z = OD+PM = (a -6) cos 0 + 6sinPCAT,ZPCM = - + ZiVCZ)=- + 0+|, sin PCM = cos (0 — 0) = cos a — 0. Therefore Similarly / 7 \ n ■ i cl — b nx = (a — b) cos 0+6 cos —=-— 0. 2/ = (a — 6) sin 0 — 6 sin a — b §94 (a) (b) Observe that the equations of the hypocycloid may be obtaine
. Differential and integral calculus, an introductory course for colleges and engineering schools. x = OD + PM = (a + 6)cos 6 + b sin PCM,ZPCM = 0 - ZNCD = 0 + 6 - |, sin PCM = - cos ( + 6) = -cos ^— 6. 130 Therefore DIFFERENTIAL CALCULUS Similarly The Hypocycloid. x = (a + 6) cos 0 — 6 cosy = (a + 6) sin 0 — 6 sin 6a + 6. z = OD+PM = (a -6) cos 0 + 6sinPCAT,ZPCM = - + ZiVCZ)=- + 0+|, sin PCM = cos (0 — 0) = cos a — 0. Therefore Similarly / 7 \ n ■ i cl — b nx = (a — b) cos 0+6 cos —=-— 0. 2/ = (a — 6) sin 0 — 6 sin a — b §94 (a) (b) Observe that the equations of the hypocycloid may be obtainedfrom those of the epicycloid by changing 6 into — 6, and vice versa. If we write m = —r— in the one case and m = —=■— in theb o other, we get the equations of the two curves in more compactform, viz., t = mcosd— cosmd, V j- = rasin0 — smra0, r = mcosd + cosm0, V r = msmd — sinra0. b 95 CYCLOIDAL CURVES 131 It is geometrically evident that each curve consists of a series ofarches, and has a cusp at each point of contact of the generatingpoint, P, with the fixed circle. When 6 is a divisor of a, the num-ber of arches and of cusps = r> and there are no double points. When b is not a divisor of a, the curve has double points. When aand b are commensurable, th
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