. Differential and integral calculus, an introductory course for colleges and engineering schools. that = \ (t — 6) and that Dx2y = - 128 V1 94 CYCLOIDAL CURVES 129 Problem 2. If the origin be taken at H, the highest point of the cycloid,and if we write d = w+ d, show that the equations of the cycloid arex= a(d +sin0)j y= a( — l+cosd).Problem 3. Obtain the x- and ^/-equation of the cycloid. 94. The Epi- and Hypo-cycloids. When a circle rolls on afixed circle, a fixed point in the circumference of the rolling circledescribes a curve termed an epicycloid or a hypocycloid, accordingas the moving
. Differential and integral calculus, an introductory course for colleges and engineering schools. that = \ (t — 6) and that Dx2y = - 128 V1 94 CYCLOIDAL CURVES 129 Problem 2. If the origin be taken at H, the highest point of the cycloid,and if we write d = w+ d, show that the equations of the cycloid arex= a(d +sin0)j y= a( — l+cosd).Problem 3. Obtain the x- and ^/-equation of the cycloid. 94. The Epi- and Hypo-cycloids. When a circle rolls on afixed circle, a fixed point in the circumference of the rolling circledescribes a curve termed an epicycloid or a hypocycloid, accordingas the moving circle is outside or inside the fixed circle. To obtain the parametric equations of these curves, we take theorigin at the center of the fixed circle, and choose for OX a linethrough A, one of the points of coincidence of the generatingpoint, P, with the fixed circle. Let a be the radius of the fixedcircle and b that of the generating circle. In both figures, 0B= x, PB= y, arc AN = ad, arc PN = b. Hence, since arc AN = arc PN, b = ad and 0 = r8. The 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
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