DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§3. Some Global Results 31with equality holding if and only if a = b. Wetherefore have√A√πR 2 ≤ A + πR22≤ RL2 ,so 4πA ≤ L 2 .Now suppose equality holds here. Then we must have A = πR 2 and L =2πR. Itfollows thatthe curve C has the same breadth in all directions (since L now determines R). But equality mustalso hold in (∗), so the vectors α(s) = ( x(s), y(s) ) and ( y ′ (s), −x ′ (s) ) must be everywhere parallel.Since the first vector has length R and the second has length 1, we infer that(x(s), y(s))= R(y ′ (s), −x ′ (s) ) ,and so x(s) =Ry ′ (s). By our remark at the beginning of this paragraph, the same result will holdif we rotate the axes π/2; let y = y 0 be the line halfway between the enclosing horizontal lines l i .Now, substituting y − y 0 for x and −x for y, sowehavey(s) − y 0 = −Rx ′ (s), as well. Therefore,x(s) 2 + ( ) 2y(s) − y 0 = R 2 (x ′ (s) 2 + y ′ (s) 2 )=R 2 , and C is indeed a circle of radius R. □EXERCISES 1.31. a. Prove that the shortest path between two points on the unit sphere is the arc of a greatcircle connecting them. (Hint: Without loss of generality, take one point to be (0, 0, 1)and the other to be (sin u 0 , 0, cos u 0 ). Let α(t) =(sin u(t) cos v(t), sin u(t) sin v(t), cos u(t)),a ≤ t ≤ b, beanarbitrary path with u(a) =0,v(a) =0,u(b) =u 0 , v(b) =0, calculate thearclength of α, and show that it is smallest when v(t) =0for all t.)b. Prove that if P and Q are points on the unit sphere, then the shortest path between themhas length arccos(P · Q).2. Give a closed plane curve C with κ>0 that is not convex.3. Draw closed plane curves with rotation indices 0, 2, −2, and 3, respectively.*4. Suppose C is a simple closed plane curve with 0