Plane Geometry - Bruce E. Shapiro

306 SECTION 52. ARC LENGTHAs we see in figure 52.1,AB ≤ AP 1 + P 1 P 2 + P 2 P 3 + · · · + P n−1 Bfor any points A = P 0 , P 1 , P 2 , . . . , P n−1 , P n = B. Now suppose we have any“smooth” curve γ connecting P 0 and P 1 , as shown in figure 52.2.Figure 52.2: The segment P 0 P n is the shortest path between P 0 and P n .Is it possible for the “length” of this curve to be shorter than the lengthof the line segment? We define the length of the curve by approximatingit with shorter and shorter line segments, and summing their lengths. Wedefine the length of the curve as the limit as the number of segments goesto ∞ and the length of each individual segment goes to zero.n−1∑Length(γ) = lim P i P i+1n→∞At this point we need to resort to some results from calculus. By a “smooth”curve, we mean any parameterized, differentiable function:Definition 52.1 Let I = [a, b] be any closed interval in R. Then a smoothcurve in the plane is any functionγ : I ↦→ R 2 ,i=0γ(t) = (x(t), y(t))where x(t) : I ↦→ R and y(t) : I ↦→ R are differentiable functions, and asmooth curve in space is any functionγ : I ↦→ R 3 ,γ(t) = (x(t), y(t), z(t))where x(t) : I ↦→ R, y(t) : I ↦→ R, and z(t) : I ↦→ R are differentiablefunctions. We will call the curves regular if γ ′ (t) ≠ 0 for all t in [a, b].« CC BY-NC-ND 3.0. Revised: 18 Nov 2012