△r q igure 2.5: A segment Ar connecting two point p and q on a parametric curver(t) 2.2 Arc length From Figure 2.5, we will derive an expression for the differential arc length ds of a parametric curve. First, let us express the vector Ar connecting two points p and q on an arc at parametric locations t and t+ At, respectively, as △r=p-q=r(t+△t)-r(t) p and q become infinitesimally close, the length of the segment connecting the two points approaches the arc length between the two points along the curve, r(t) andr(t+At). Or using Taylor's expansion on the norm(length)of the segment Ar and letting At-0, we can express the differential arc length as s~|△r|=r(t+△t)-r(t)=|△t+O(△2) Thus as△t→0 dt r DEfiniti d dt d Hence the rate of change at of the arc length s with respect to the parameter t is 2(t)+2(t)+22(t) (21)y z r(t) p q r(t + ∆t) ∆r x Figure 2.5: A segment ∆r connecting two point p and q on a parametric curve r(t). 2.2 Arc length From Figure 2.5, we will derive an expression for the differential arc length ds of a parametric curve. First, let us express the vector ∆r connecting two points p and q on an arc at parametric locations t and t + ∆t, respectively, as ∆r = p − q = r(t + ∆t) − r(t). As p and q become infinitesimally close, the length of the segment connecting the two points approaches the arc length between the two points along the curve, r(t) and r(t+ ∆t). Or using Taylor’s expansion on the norm (length) of the segment ∆r and letting ∆t → 0, we can express the differential arc length as ∆s ' |∆r| = |r(t + ∆t) − r(t)| = | dr dt ∆t + O(∆t 2 )| ' | dr dt|∆t. Thus as ∆t → 0 ds = | dr dt|dt = |r˙|dt. Definitions d dt ≡ ˙ d ds ≡ 0 Hence the rate of change ds dt of the arc length s with respect to the parameter t is ds dt = q x˙ 2 (t) + y˙ 2 (t) + z˙ 2 (t) (2.1) 7