Calculus: Analytic Geometry and Calculus, with Vectors

7.3 Center and radius of curvature 435

10 As an alternative to (or in addition to) Problem 8, find the radius of curva-
ture of the "standard hyperbola."
11 A glance at the hypocycloid having the equations

x= acos3t, y= asin' t

indicates that the radius of curvature p should be a maximum at points (x,y)
for which IxI = Iyi and a minimum at points for which xy = 0. Can we believe
our eyes? f4ns.: No, because p has no minimum. When t 5x6 n-7r/2, calculations
give p = 2 sin 2t1. Thus p is a maximum when 2t = n +^17 or t =

\2 + D Tr and hence when Isin tI = Icos ti or IxI = Iyl. We see that p - *0 as

-a 0, but the curvature at the cusps is undefined.

12 When a flexible cord or chain (the Latin word for chain is catena) is sus-

pended from its ends in a parallel force field, it hangs in a curve (or point set)

called a catenary. Differential equations textbooks show that a rectangular

coordinate system can be chosen in such a way that the equation of the catenary

is

y =

2

a(exla + e -la).

Find the radius of curvature of this catenary at the point (x,y). 11ns.: y2/a.

13 Find the radius of curvature of the cycloid having the equation

x=a(O-cosB), y=a(l-cos8).

14 Find the evolute of the cycloid of Problem 13.

15 Persons who picnic beside lakes several miles long can wonder whether

poor visibility instead of curvature of the earth is responsible for invisibility

of distant boats and shores. This problem involving curvature can be solved

very simply. Figure 7.392 shows the center of a spherical earth at C on the

Figure 7.392

positive y axis, the x axis being tangent to the surface of the earth at 0. The

line from C to Q(x,0) intersects the surface of the earth at P. The number I PQI

is the height Is of an object which is visible from the point 0 on the earth x miles

away. The simple calculation

x2

(1) h= R2--x2-R= R2+x2+R