. Differential and integral calculus. ., 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 Curvature is the radius of the circleof curvature, and the Center of Curvature is the center of thiscircle. Thus, Fig. 26, if the circle CPPr has the same curvature asthe curve NM at the point P, CPP is the circle of curvaturefor that point; OP, the radius of CPP, is the radius of curva-ture, a
. Differential and integral calculus. ., 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 Curvature is the radius of the circleof curvature, and the Center of Curvature is the center of thiscircle. Thus, Fig. 26, if the circle CPPr has the same curvature asthe curve NM at the point P, CPP is the circle of curvaturefor that point; OP, the radius of CPP, is the radius of curva-ture, and the center O, the center of curvature. It is obvious that the circle of curvature and radius of curva- 182 Differential Calculus ture vary from point to point as the curvature of the curvechanges. Since, by definition, the circle of curvature CPP and thecurve NM have the same curvature at P, we have, § 132, dads for the curvature of the circle at P. But if we suppose thelength of the arc PP(= ds) to be the velocity with which the. generating point passes through P, then POP = a — a repre-sents the angular velocity of the tangent, , the rate of changeof its direction. Hence, POP1 = da. Let OP = p, then from the circle, arc PP POP = , Hence, da = P = ds P ds 1 da , the radius of curvature is the reciprocal of the curvature. Curvature Evolute and Involute 183 Cor. i. If p = -j-,- be the radius of curvature at any other aa. J point of the same curve, or at some point of another curve, ds u p da we have / j p dsda7 aVp dS , S = -r p (la ~ds Hence, the curvatures of a curve at any two points are inverselv as the radii of curvature at the points. 134.* Expressions for the Radius of Curvature,i. In Terms of Rectangular Coordinates. Remembering that ds = {doc2, -f- dy)-, § 18, and that dya = tan *—- , § io, we havedx dsP = da _ (dx2 + dff ~ ^tan-^dx (dx* + d/)\ <Py dxdf * Equation (i) § 134, was first given by John Bernoulli in 1701. 184 Differential Calcu
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
