. Differential and integral calculus, an introductory course for colleges and engineering schools. x = a cos 6 + b (cos 6 — cos a + b •) = a cos e+26sin^+1)9sing, y = a sin 6 + Msin 6 — sin —7— g) x = a cos 6 + ad sin (^-=- + 1J = asin0— 2 b cosThese may be written sin ad26 aS_26 aQ_26 §96 • CYCLOIDAL CURVES 133 . ad cm y = a sin 6 — a^cosf^r + 1J0 Sm2 6 ad26 Now, when b = oo, . ad sin l^-r + 1J0 = sin 6, cos (~-r + 1 )6 = cos 0, and — = 1 26Hence the parametric equations of the involute of the circle are x = a(cos 6 + 6 sin 6),) , n y = a(sin0 — 0cos 6). CHAPTER XIII CURVES GIVEN BY POLAR EQU
. Differential and integral calculus, an introductory course for colleges and engineering schools. x = a cos 6 + b (cos 6 — cos a + b •) = a cos e+26sin^+1)9sing, y = a sin 6 + Msin 6 — sin —7— g) x = a cos 6 + ad sin (^-=- + 1J = asin0— 2 b cosThese may be written sin ad26 aS_26 aQ_26 §96 • CYCLOIDAL CURVES 133 . ad cm y = a sin 6 — a^cosf^r + 1J0 Sm2 6 ad26 Now, when b = oo, . ad sin l^-r + 1J0 = sin 6, cos (~-r + 1 )6 = cos 0, and — = 1 26Hence the parametric equations of the involute of the circle are x = a(cos 6 + 6 sin 6),) , n y = a(sin0 — 0cos 6). CHAPTER XIII CURVES GIVEN BY POLAR EQUATIONS 97. The Tangent in Polar Coordinates. Let p, 0 be the polar coordinates of the curve point P,and let be the angle between theradius vector and the tangent at seek to express tan in termsof p and 0. Let the coordinates ofQ be p + Ap and 0 + A0. Let PBbe drawn perpendicular to OQ. Then tan 0 = 757:, P£ = p sin A0,andBQ = OQ - OB = p + Ap - p cos Therefore, tan/3 p sin A0 sin A0 A0 Ap + p(l - cosA0) Ap 1 — cos A0A0 + p A0 Now let Q approach P as a limit, and make use of I and II ofArt. 11, and there results pdd (a) tan Dep = PDpd = dp cot0 = -^ = A, log From these equations it is seen that Dep (or DP0) serves the samepurpose in polar coordinates that Dxy does in Cartesian coordi-nates, viz., determines the direction of the tangent line. 134 » CURVES GIVEN BY POLAR EQUATIONS 135 Problem 1. From the figure show that when the tangent is 7T — 0 Problem 2. From the figure show that ~ , p+ tan d Dep pdd + tan0dp Dxy = tan a = -^ —^ = ^ 4 f • iV — p tan 6 dp— p tan 0d0 Problem 3. Obtain the same expressions for Dxy by differentiating x = p cos 0, ?/ = p sin 0. Problem 4. Let p be the length of the perpendicular from the pole upon the tangent. Show that V(Dep)2+p2 VdP* + p2dd*There are formulae in polar coordinates for determining convexand concave arcs and flexes, but these formulas are rather difficultto de
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