. Differential and integral calculus. F CUR-EVOLUTE AND INVOLUTE. History. — Huygens, in the third chapter of his Horologium Oscillato-rium (1673), defines evolutes and involutes, proves some of their more ele-mentary properties, and illustrates his method by finding the evolutes ofthe cycloid and the parabola. 132. The Measure of the Curvature, or more simply, The Curva-ture of a curve, is the ratio of the rate of change of its directionto the rate of change of its length. Let a be the angle whichthe tangent to the curveMN at P, Fig. 25, makeswith the Xaxis and let NP= s; then, since the dire
. Differential and integral calculus. F CUR-EVOLUTE AND INVOLUTE. History. — Huygens, in the third chapter of his Horologium Oscillato-rium (1673), defines evolutes and involutes, proves some of their more ele-mentary properties, and illustrates his method by finding the evolutes ofthe cycloid and the parabola. 132. The Measure of the Curvature, or more simply, The Curva-ture of a curve, is the ratio of the rate of change of its directionto the rate of change of its length. Let a be the angle whichthe tangent to the curveMN at P, Fig. 25, makeswith the Xaxis and let NP= s; then, since the direc-tion of a curve at a pointis the same as that of thetangent, we have da = rateFig. 25. of change of direction of the curve, and ds = rate ofchange of the length of the curve. Hence, by definition, da in which k represents the curvature of MNat the point P. To show how this ratio measures the curvature let us sup-pose the curve MN to be generated by the point P movingwith any velocity v and carrying its tangent along with it as it. Curvature Evolute and Involute 181 moves. Let us further suppose the curve SL (tangent toMN and PT at P) to be generated by a coincident point Pmoving with the same velocity v. Then, by definition, thecurvature, of SL at P is , da in which do! is the rate of change of the direction of PT, thetangent to SL, at P, and ds is the rate of change of the lengthSP = s. But since the generating points move with the samevelocity, we have v = ds = ds §17; , k da. hence, -/ = T~r k da , the curvature of two curves at any two points are to eachother as the rates of change of their direction. For example, let da = 300 a second and da! = 6o° a second;then - = 3— = -Kf 6o° 2 or k = 2 k, , the curvature of one curve is twice that of the other. 133. Circle of Curvature. Radius of Curvature. The circle tangent to a curve at a point, and having the samecurvature as the curve at the point, is called the Circle of Curvature,for that point. The Radius of Cu
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