7.3 Center and radius of curvature 433Since do/ds = b'(t)/s'(t) = ¢'(t)/jv(t)I, the formulas (7.387) and (7.3852)
yield the formula
- K =
x'(t)y"(t) - x"(t)y'(t)
(7.
[[x (t)]2 + [y'(t)]21from which the curvature of C at P(t) can be calculated without reference
to other formulas. Perhaps we should take notice of the fact that a curve
C is an ordered set of points, and that we can misunderstand (7.389)
when we forget this fact. In particular, the sign in (7.389) will be wrong
if we put the coordinate system on C backwards so that s decreases as t
increases.
We conclude with a fundamental observation. As formulas (7.37)
and (7.389) show, the radius of curvature p and the absolute value SKI of
the curvature K are reciprocals wherever we have defined both of them.
Problems 7.39
1 By use of the equationsx = acost, y = a sin t,
show that the curvature of the curve C consisting of a circle of radius a traced in
the positive direction is identically 1/a. Then by use of the equationsx = a cos t, y = -a sin t,
show that the curvature of the curve r (capital gamma) consisting of a circle
of radius a traced in the negative direction is identically -1/a.
2 Hindsight can be very good. Look at (7.388). Then run with constant
speed and in the positive direction around a circle of radius a and observe that
0 increases at a constant rate. Then reverse the direction of the run and observe
that 0 decreases at the same constant rate.
3 Determine the radius of curvature of (that is, at points of) the parabola
having the equation y = kx2. Find the minimum radius of curvature. Sketch
a graph for the case in which k = 1 and determine whether the answer seems to
be correct.
4 Show that the normals to the graph of y = x2 at the points (0,0) and (0.01,
0.0001) intersect at the point (0, 0.5001). Show that this intersection is at dis-
tance 0.0001 from the center of curvature of the graph at the point (0,0).
5 A glance at the graph of y = log x suggests that the absolute value of the
curvature is greatest and that the radius of curvature is least when x is somewhere
between and 2. Find the x for which the radius of curvature attains its mini-
mum value. Ans.: V-212 = 0.707.
6 When a point P(x,y) moves along an arc or curve C having equations
x = x(t), y = y(t) that satisfy appropriate conditions, the center (X,Y) of curva-
ture moves along an arc that is called the evolute of C. The formulas (7.363) and
(7.364), which we should be able to use but need not remember, show how X