Calculus: Analytic Geometry and Calculus, with Vectors

¢28 Curves, lengths, and curvatures

be a stupid waste of time if it were not for the fact the symbols are useful. They
appear even in quite elementary physicsand engineering, and mathematicians
have a responsibility to tell what they mean.

7.3 Center and radius of curvature

"differential geometry" provide information about matters relating to

curvature of curves that lie in E3. In this section, we give most of our

attention to curves that lie in an xy plane. Our approach to the subject

sacrifices brevity to place emphasis upon elementary geometric ideas

that can be of interest to everyone and are needed by engineers and others

who study the bending of beams.

Let x and y, or x(t) and y(t), have continuous second derivatives over

some open interval a < t < b in which t and t + At are always supposed

to lie. Let P(t) denote the point with coordinates (x(t), y(t)) and let

r(t) be the vector running from the origin to P(t) so that

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

(7.312) v(t) = x'(t)i + y'(t)j

(7.313) a(t) = x"(t)i + y"(t)j.

Let C be the curve traversed by P(t) and the tip of r(t) as t increases, so

that v(t) is tangent to C at P(t) when v(t) 5zl- 0. Henceforth we consider

only values of t for which

i j k

(7.32) v(t) x a(t) = x'(t) y(t) 0 = [x'(t)y"(t) - x"(t)y'(t)lk 0 0.

x"(t) y"(t) 0

Since Iv x al = lvi lal Isin Bl, this means that v and a are nonzero vectors

which do not lie on parallel lines. This and the first of the formulas

v(t + At) - v(t)

= a(t),

v(t + At) - v(t)

Fg- 0

W-0

(7.321) lm'

At

v(t)x

At

implies existence of a S > 0 such that the second holds when 0 < lAti < S.

Since V(t) X v(t) = 0, we conclude that v(t) x v(t + At) 5,4- 0 and hence

Figure 7.33

C Supposing that 0 < l,tl < S, we

that the tangents v(t) and v(t + At) do

not lie on parallel lines when 0 < lotl < S.

construct Figure 7.33 and look at it.

Since the tangents v(t) and v(t + At)

(XY) P(t+ot)= to r at P(t) and P(,, + At) do not lie

( y(t+ot)) on parallel lines, the normals to Cat

these points must intersect at a point

Q ,,I) having coordinates t (xi) and

P(t)=P(At)1 y(t)) 71 (eta). If C is a circle, then the