Smithsonian miscellaneous collections . ). Polar Equation of Ellipse, r = FP, 6 = angle . XFP, a(j -e^) I — e cos r = 0P, d = angle XOP, VI - e^ cos^ 6 Equation of Ellipse in terms of p, the perpendicular from F upon thetangent at P, and r, the radius vector FP: 1 = f _ Lp^ r a I == semi latus rectum. Hyperbola (Fig. s). O, Center; F, F, of hyperbola, origin at O, ^ y- _a2 ~ 62 ~ ^ X = OM, y = MP, a = OA= OA. Parametric Equations of h\perbola, X = a sec <f), y = b tan 0. </) = angle XOP, where P is the point where the ordinate at T meets
Smithsonian miscellaneous collections . ). Polar Equation of Ellipse, r = FP, 6 = angle . XFP, a(j -e^) I — e cos r = 0P, d = angle XOP, VI - e^ cos^ 6 Equation of Ellipse in terms of p, the perpendicular from F upon thetangent at P, and r, the radius vector FP: 1 = f _ Lp^ r a I == semi latus rectum. Hyperbola (Fig. s). O, Center; F, F, of hyperbola, origin at O, ^ y- _a2 ~ 62 ~ ^ X = OM, y = MP, a = OA= OA. Parametric Equations of h\perbola, X = a sec <f), y = b tan 0. </) = angle XOP, where P is the point where the ordinate at T meets thecircle of radius a, center O. GEOMETRY OF = OF = ea. e = eccentricity Va- + b 49. Fig. 5 FL = - = aie^ - i) = semi latus FP = ex + a, FP = ex- a, FP - FP = 2a. T = angle XTP. tan r = bx a\/x~ — a^ NM = ^, ON = e% OT = ~, OT = -,a/- X y X X or ON />n = - -ah PS- <? ,os-^--. y/e^x^ y/e^x — a^ OJJ = Asymptote. tan XOU = - • a b =- distance of vertex A from asymptote. 50 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS Radius of curvature of hyperbola, P = abangle FPT = angle FFT. angle FPN = co = - - FFN = co= - + FFT. aeytan CO = -^. -\/g^.V — a^ p cos CO FP FPCoordinates of center of curvature, Equation of Evolute of hyperbola, (f-(f In a rectangular hyperbola b = a; the asymptotes are perpendicular toeach other. Equation of rectangular hyperbola with asymptotes as axes andorigin at 0: xy = -. Length of arc of hyperbola,6 / • ^^^ v^ ^ A _ I ^„„ ^ _ an L I ^ ^=^, k = 1, tan 0 leyo VI — ^^ sin^ 0 e Polar Equation of hyperbola: r = FP, e = XFP, r ^a ^ ^ r = OP, (9 = ZOP, r2 e cos (7-1e- COS 6-1
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