Calculus: Analytic Geometry and Calculus, with Vectors

430 Curves, lengths, and curvatures

A similar procedure, in which the multipliers -x'(t + At) and x'(t) are
used, gives the second of the formulas (7.35). Formulas for andn
are now obtained from (7.35) by dividingand transposing. As we shall
see, formulas for X and Y are obtained bydividing (7.35) by At and taking
limits as At, 0. Since (7.343) and (7.351) show that

[x'(t)]2 + [Y'(t)]2

lim - = x'(t)y"(t) - x"(t)y'(t),

At

lim 0

nt--0Ot

D

we find from (7.35) that

(7.361) X - x(t) =

l i o [ - x(t)]

[x'(t)]2 + [Y'(t)]2 Y, (t)

x'(t)y"(t) - x"(t)y'(t)

(7.362) Y - y(t) = lim [7] - Y(t)]A [XI(t)12 + [y'(t)]2 x (t)

x'(t)Y"(t) - x"(t)y'(t)

.

Transposing the terms x(t) and y(t) gives formulas

(7.363) [x'(t)]2 + [y'(t)]2 ,

X = x(t) -x'(t)y"(t) - x"(t)Y'(t)Y (t)

(7.364) [XI(t)12 + [y'(t)]2

Y = Y(t) +x'(t)y"(t)- x"(t)Y'(t)x (t)

for the coordinates (X,Y) of the center of curvature of C at the point

(x(t), y(t)), but these new formulas are sometimes less useful than their

parents. The definition of radius of curvature p implies that

and hence

(7.37)

p = [X - x(t)]2 + [Y - y(t)12

1[x(t)]2+ [y'(t)]2]H

P = Ix'(t)y"(t) - x"(t)y'(01

In case x(t) = t so x'(t) = 1 and x"(t) = 0, it is customary to replace t

by x and write

(7.371)

r

= + [Y'x)]2'` =l1 +(

IY (x)I Idyl

dx2

To steer our thoughts toward another problem involving curvature,

we consider the rate of change of direction of an automobile which trav-

erses a level road that winds around swamps and between mountains.

At each time t the rate depends upon the speed of the automobile and

upon another number which is called the curvature of the road at the

position of the automobile. To be more precise about this matter, let