Mathematica: Arc Length Speed

WolframLang. Arc Length Speed

WolframLang. Arc length using NIntegrate is faster than builtin ArcLength with machine number input.

(* 
2024-03-30
my arc length function, originally written around 1995.
 *)

Clear[ xnArcLength, xnArcLength2];

xnArcLength::usage = "xnArcLength[{fx, fy}, {t, a, b} ] return arc length of the curve {fx, fy} from a to b using NIntegrate.";

xnArcLength[ parametrizedCurve_?VectorQ, {t_Symbol, tmin_, tmax_}]:= NIntegrate[ Evaluate@ Sqrt @ Total @ ( D[ parametrizedCurve, t]^2 ) , {t, N@ tmin, N@ tmax} ]

xnArcLength2::usage = "xnArcLength2[{fx, fy}, {t, a, b} ] return arc length of the curve {fx, fy} from a to b using NDSolve.";

xnArcLength2[ parametrizedCurve_?VectorQ, {t_, tmin_, tmax_}]:=
Module[ {a,b,s,f},
 {a,b} = N@{tmin,tmax};
 s = NDSolve[
  { f'[t] == Sqrt[Plus@@(D[ parametrizedCurve, t]^2)],
   f@a == 0
  }, f, {t,a,b}
 ][[1,1,2]];
 s[b]
]