Calculus: Analytic Geometry and Calculus, with Vectors

434 Curves, lengths, and curvatures

and Y depend upon t. Supposing P moves along the graph of y = kx2 in such a

way that x = t and y = kt2, show that the equationsof the evolute are

X = -4k2t3, Y = Zk+ 3kt2.

7 This problem involves a little story. A string is wound in a clockwise

direction around a circular spool of radius a with an end at the point (a,0).

When the string is unwound, being kept stretched during the process, the end

of the string traces the spiral curve C shown in Figure 7.391. When aO units

Figure 7.391

of string have been unwound, this part of the string is tangent to the spool at

the point Q(X,Y) for which

X = a cos 8, Y= a sin 0

and the end of the string is at the point P(x,y) for which

x=x(8) =acos8+aOsin9

y = y(O) = a sin 0-aOcos6.

It can be observed that Q moves around the circle just as rapidly as the distance

from Q to P increases. It is not unreasonable to guess (or at least to consider the

possibility) that the center and radius of curvature of C at P are Q and aO. On

the other hand, a skeptic can be uncertain whether QP is perpendicular to the

tangent to C at P. The situation demands clarification. Start with the equa-

tions of C and find the center and radius of curvature of C at P. Ans.: (X,Y)

and a6. Remark: The curve C is called the involute of the spool. The spool

is the evolute of its involute.

8 Determine the radius of curvature of the "standard ellipse"

x y2

a2-} b2=1 or x=acos0, y=bsin6

by use of the first "standard equation" and then by use of the latter parametric

equations.

(^9) Using the parametric equations of Problem 8, find the evolute of the
standard ellipse.