Plane Geometry - Bruce E. Shapiro

SECTION 52. ARC LENGTH 307Now we can take the limit using calculus, and the result is the following.A similar result can be stated in the plane.Theorem 52.2 Let γ : [a, b] ↦→ R 3 be a smooth curve.length of γ, denoted by L(γ) iswhere|γ ′ (t)| =L(γ) =∫ baThen the arc|γ ′ (t)|dt (52.1)»x ′ (t) 2 + y ′ (t) 2 + z ′ (t) 2Example 52.1 Let γ be the line segment from (a, b) to (c, d) in the realplane. Then the line containing these two points (assume that c > a andb > d) isy = b + d − b (x − a)c − aHence a parameterization of this curve isÅγ(t) = t, = b + d − b ãc − a (t − a)on the interval [a, c]. Differentiating,Åγ ′ (t) = 1, d − b ãc − a|γ ′ (t)| 2 (d − b)2= 1 +(c − a) 2√(c −|γ ′ a)2 + (d − b)(t)| =2c − aTherefore the arc length from (a, b) to (c, d) is∫ c√(c − a)2 + (d − b)L(γ) =2dta c − a»= (c − a) 2 + (d − b) 2which is what we would expect from the Pythagorean theorem.Theorem 52.3 Let γ : (a, b) ↦→ R 2 be any regular curve in the real plane.Then the line segment connecting A = γ(a) to B = γ(b) is the curve ofshortest distance between A and B, i.e.,d(γ(a), γ(b)) ≤ L(γ)Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.