Calculus: Analytic Geometry and Calculus, with Vectors

412 Curves, lengths, and curvatures

3 Show that the projection on the xy plane of a curve in Es is a curve in E2.

4 Discover continuous functions x and y such that as t increases over the

interval 0 5 t S 4, the point (x(t), y(t)) goes once in the positive direction

around the square having vertices at the points (0,0), (0,1), (1,1), and (0,1).

Hint: Write the equations which tell where a pencil point will be at time t if it

traverses the square with unit speed. One ans.: x(t) = t and y(t) = 0 when

0<t<1;x(t)=l and y(t)=t-1when 15t<=2;x(t)=3-tand y(t)=1

when 2:5t53;x(t)=0and y(t)=4-twhen 35t<-4.

5 Let L, be the length of the central path from f1 to B in Figure 7.191 and

let L2 be the length of the upper path. Find L2 - Ll. Ans.: (7r - 2)h.

Figure 7.191 Figure 7.192

6 Supposing that the inner and outer circles of Figure 7.192 have radii e

and R, find the length of the curve which begins and ends at -4.

7 Draw a square and from each corner draw a line segment toward the

center but reaching only halfway to the center. Then insert arrows to specify

a curve which forms the boundary of the inner region. Find the length of the

curve.

(^8) This problem requires us to think about and calculate the lengths L, and
L2 of two of the paths by which an insect might crawl from the bottom B to the
Figure 7.193
B
top T of the three-dimensional ring or anchor ring or
torus of Figure 7.193. The first path, of length L1,
is a semicircle of radius b + a which lies on the outer
circumference of the torus. The second path, of
length L2, consists of a semicircle of radius a, and
then a semicircle of radius b - a which lies on the
inner circumference of the torus, and finally another
semicircle of a radius a. Try to guess which of the
paths is shorter and then calculate L, and L2. Ans.:
L, _ ir(a -I- b), L2 = ira + ir(b - a) -I- ira,
so the two paths have equal lengths. Remark: If a
curve G lies in a set S and joins two points P, and
P2 of S, and if the length of G is less than or equal to the length of each other
curve C in S which joins P, and P2 then G is called a geodesic in S. As is easily
imagined, the study of geodesics on a torus is an honorable part of an interesting
subject.
(^9) Creators of interesting tales for children say that, before the time when
Columbus sailed across the ocean wet, everybody thought that the earth was
flat. It has in fact been widely known for more than two thousand years that