Line Scaled Segment or Arc length parametrization for Line

Method 1, old code tweak

LineScaledCoordinate is not ReadProtected so I took and changed two lines:

LineScaledSegment[coord_?MatrixQ, rr0_] := Module[
      {dist, dist2, sta, sto, newpos, total, rr = rr0}
    , If[SameQ[coord, {}], Return[{}]]
    ; If[Equal[Length @ coord, 1], Return[coord[[1]]]]
    ; If[Greater[rr, 1], rr = 1]
    ; If[Less[rr, 0], rr = 0]
    ; dist = Map[Norm, Drop[coord + -RotateLeft[coord], -1]]
    ; total = Total @ dist
    ; If[LessEqual[total, $MachineEpsilon], Return[coord[[1]]]]
    ; DivideBy[dist, total]
    ; dist2 = FoldList[Plus, 0, dist]
    ; Part[dist2, Length[dist2]] = 1.
    ; sto = Part[
        Flatten[
          Position[dist2, PatternTest[_, GreaterEqual[#, rr] &]]
         ]
       , 1
       ]
    ; If[Equal[sto, 1], Return[ coord[[{1, 1}]]]] (*1 -> {1,1}*)
    ; sta = sto - 1
    ; newpos = Plus[coord[[sta]]
       , Divide[
          (rr + -Part[dist2, sta]) * (Part[coord, 
         sto] + -Part[coord, sta])
         , Part[dist2, sto] + -Part[dist2, sta]
         ]
       ]
    ; Append[coord[[;; sto - 1]], newpos] (*previously just newpos*)
    ]

path2 = RandomReal[10, {10^3, 2}];

t = 0;
LabeledSlider@Dynamic@t
Graphics[{
  Line@path2, Thick, Red,
  Line@Dynamic@LineScaledSegment[path2, t]
  }, ImageSize -> 500]

Modifying Kuba's code using Interpolation it seems maybe (??) to work faster...

coord=RandomReal[10,{10^3,2}];
dist=Map[Norm,Drop[coord+-RotateLeft[coord],-1]];
total=Total@dist;
DivideBy[dist,total];
dist2=FoldList[Plus,0,dist];
ff=Interpolation[Thread@{dist2,Range@Length@dist2},InterpolationOrder->0];
fxy=Interpolation[Thread@{dist2,coord},InterpolationOrder->1];

then

LabeledSlider[Dynamic@t]

Graphics[{Line@coord, Thick, Red, 
  Line@Dynamic@Append[coord[[;; IntegerPart[ff[t]] - 1]], fxy[t]]}, 
 PlotRange -> {{0, 10}, {0, 10}}]