. Differential and integral calculus, an introductory course for colleges and engineering schools. ep)2+p2 VdP* + p2dd*There are formulae in polar coordinates for determining convexand concave arcs and flexes, but these formulas are rather difficultto derive and to apply, and we do not give them. 98. Curves in Polar Coordinates. When the equation of thecurve can be brought to the form p = /(0), and when f(6) is asimple function, the general shape of the curve can be readilydetermined in many cases by inspection, and by calculating p fora few suitably chosen values of 6, and plotting these poin
. Differential and integral calculus, an introductory course for colleges and engineering schools. ep)2+p2 VdP* + p2dd*There are formulae in polar coordinates for determining convexand concave arcs and flexes, but these formulas are rather difficultto derive and to apply, and we do not give them. 98. Curves in Polar Coordinates. When the equation of thecurve can be brought to the form p = /(0), and when f(6) is asimple function, the general shape of the curve can be readilydetermined in many cases by inspection, and by calculating p fora few suitably chosen values of 6, and plotting these points. Allthis is explained in analytic geometry. Our formula (a) givesadditional information. Example 1. The Lemniscate of Bernoulli:P2= a2 cos this we get p = ± a Vcos20. We need consider only one branch, p = +aVcos20. p is real only when — j <0 = + ^ anc* therefore this branch lies wholly within the angle BOC. When0 = 0, p has its greatest value a, and when d]= ±2, p 0. Hence this branch forms a loop having at thepole two tangents BB and , since cos 2 0 = cos (—20),. 136 DIFFERENTIAL CALCULUS this branch is symmetrical about the initial line OA. By plotting a fewpoints in the angle BOA, the curve can be drawn with considerableaccuracy. The other branch p= — Vcos 2 0 is of the same size and shape and lies in the angle BOC. The curvehas a double point at the pole and eachbranch has a flex there. The curve has now been drawn with-out any aid from Calculus. Some ofthe foregoing conclusions, however, arenot quite warranted. For example, weare not yet quite sure that the lines BBand CC are actually tangents, and we arenot at all sure that the curve is roundedat the points A and A, as shown in the figure on page 135. It may havecusps at these points, with the line AA for cuspidal tangents, as in theaccompanying figure. To settle these doubts, we make use of formula
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