Calculus: Analytic Geometry and Calculus, with Vectors

7.3 Center and radius of curvature 431

x and ybe functions oft having two derivatives each over some interval
and let C be the curve traversed by the point P(t), having coordinates
x(t), y(t), as t increases over the interval. We suppose that there is no

t for whichx'(t) and y'(t) are both 0. Our results will be obtained with
the aid of the three familiar vectors

(7.381) r(t) = x(t)i + y(t)j

(7.382) v(t) = x'(t)i + Y'(t)j

(7.383) a(t) = x"(t)i -I- y"(t)j

The vector v(t) having its tail at P(t) is the forward tangent to C at P(t).

With each t we wish to associate an angle 0(t) that determines the direc-

tion of v(t), and this is a matter that must not be treated carelessly.

To restrict q5(t) to an interval like -ir < q5 <_ ir, so that 0 would be dis-

continuous when the vector rotates from northwest to southwest, would

defeat our purpose. To formulate general principles by which -0(t) can

be calculated may be beyond our capabilities. Let us then avoid possi-

ble unforeseen topological difficulties by restricting attention to curves C

for which it is clearly possible to determine q5(t) by the following proce-

dure. Let to be a particular t in the interval considered and let 0(to) be

the angle for which -ir < 0(to) 5 it and q5(to) is the ordinary trigono-

metric angle having its initial side on the positive x axis and its terminal

side on the vector through the origin parallel to v(to). Then, as t increases

(or decreases) from to, let ¢(t) vary with the vector v(t) in such a way that

0 is continuous. Some curves are less complicated than that shown in

Figure 7.384, and some are more complicated. Supposing that ¢(t) is

ON)

Figure 7.384

satisfactorily determined, we can put the formula (7.382) for v(t) in the

form

(7.385) Iv(t)I[cos 0(t)i + sin 0(t)j] = v(t),

where

(7.3851) x'(t) = Iv(t)Icos ¢(t), y'(t) = Iv(t)Isin 4(t)

and

(7.3852) Iv(t)I = {[x'(t)]2 + [y'(t)]2)/ 0.