. Differential and integral calculus. ters ofcurvature form the Evolute. Thus, if the curve N OS is the locus of the center of curva-ture of the curve NPM, NOS is the evolute of NPM, andNPM is the Involute. See Fig. 27. 138. Equation of the Evolute. Let y=f(x) be the equation of involute NPM (Fig. 27), andlet O be the center of curvature corresponding to any point P on the involute. Let (x, y), (x, y)tbe the coordinates of P and O, re-spectively. It is required to deter-mine (1st) the values of x andy interms of x and y, and (2d) to de-termine from these values in con-junction with the equatio
. Differential and integral calculus. ters ofcurvature form the Evolute. Thus, if the curve N OS is the locus of the center of curva-ture of the curve NPM, NOS is the evolute of NPM, andNPM is the Involute. See Fig. 27. 138. Equation of the Evolute. Let y=f(x) be the equation of involute NPM (Fig. 27), andlet O be the center of curvature corresponding to any point P on the involute. Let (x, y), (x, y)tbe the coordinates of P and O, re-spectively. It is required to deter-mine (1st) the values of x andy interms of x and y, and (2d) to de-termine from these values in con-junction with the equation of theinvolute, y =f(x), a general relationbetween x and y, , to determinethe equation of the evolute. 1. To determine the coordinates of the center of curvature {x\ y).Let 0P=p, and draw PD _L OB. Since OP is _L to thetangent PT, we have DOP = a. We have, therefore, OB= OC-PD and OB = PC + OD. But OB = x>, OC=x,P£> = Psma; and 0B=/,PC = y, OD = p cos a. Whence, substituting, we have x = x — p sin a \ y = y + p cos a /. Fig. 27. (a) or since sin a _* dy ds vV*2 + dy* dxand cos a = — = dx ds -yjdx*1 + dy^ these values together with the value of p substituted in (a) give Curvature Evolute and Involute 189 i + fdy\\dy \dx dx . x=x~-—wf-1— w dx* *-»+-^ •••••• w <fr2 2. To deteiuiine the equation of the we now combine with (i) and (2) the equation of theinvolute, y=A*) (3) eliminating the variable coordinates x and y of the involute,we shall have a resulting equation in xr and y. The equationthus obtained is obviously that of the evolute. To illustrate, let us find the equation of the evolute of theparabola, f- = 4 ax. Differentiating we obtain dy= fa\ <Py_ _±xfa_ dx V x dx* 2 V xs Hence, x = x + {- fLLLf , *\£ , Jl/ — 2 tf , ^ = 3^ + 2 a, x — • 3 Also, y = y — *V .V3 ; /=_^,.r = -(4«y)S. 190 Differential Calculus
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
