Page 51 - Standard Handbook Of Petroleum & Natural Gas Engineering

40 Mathematics

In polar coordinates (Figure 1-34), r = f(8), where r is the radius vector and
8 is the polar angle, and

so that by x = p cos 8, y = p sin 8 and K = 1/R = de/ds, then

ds
R=-= [r2 +(r’)2]9YS
de r2 - rr” + 2(r’)‘
If the equation of the circle is
R2 = (x - a) + (y - p)‘

by differentiation and simplification

Y’C1+ (Y’)‘]
a=x-
Y”

and

The evolute is the locus of the centers of curvature, with variables a and p,
and the parameter x (y, y’, and y” all being functions of x). If f(x) is the evolute
of g(x), g(x) is the involute of f(x).
Indefinite Integrals

Integration by Parts makes use of the differential of a product
d(uv) = u dv + v du

V

Figure 1-34. Radius of curvature in polar coordinates.