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I know a useful function LineScaledCoordinate through animate-point-going-around-a-triangle, so does it have an inverse function?

Needs["GraphUtilities`"]; pts = {{0, 0}, {1, 0}, {0.6, 0.4}, {0.1, 0.5}}; P = LineScaledCoordinate[pts, 0.3]

Suppose this function is called pathParameter, pathParameter[pts, P] should return 0.3.

Related link: Line Scaled Segment or Arc length parametrization for Line

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    $\begingroup$ This function pathParameter[pts, P] would require Element[P,Line[pts] ? $\endgroup$ Commented Nov 15, 2021 at 9:01
  • $\begingroup$ @UlrichNeumann P should on Line[pts], if P is far away from the line, the returned value should greater than 1 or less than 0. $\endgroup$ Commented Nov 15, 2021 at 10:40

3 Answers 3

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onSegment[{p1_, p2_}, p_, tol_:10^(-10)]:= (Det[{p2 - p1, p - p1}] < tol) && ((p - p1) . (p2 - p1) > 0 && (p2 - p) . (p2 - p1) >= 0 || And @@ MapThread[Equal, {p, p1}]) pathParameter[coord_?MatrixQ, p:{x_, y_}, tol_:10^(-10)] := Block[{segs, pos, dist, pdist}, segs = Partition[coord, 2, 1]; pos = Position[segs, _?(onSegment[#, p, tol] &), {1} , 1, Heads->False]; If[Length[pos] == 0, Return[-1]]; dist = Map[Norm, Drop[coord - RotateLeft[coord], -1]]; pdist = Total[dist[[;;pos[[1,1]] - 1]]] + Norm[p - segs[[pos[[1, 1]], 1]]]; pdist / Total[dist] ]
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ClearAll[invLSC] invLSC[pl_, p_] /; Element[p, Line @ pl] := Module[{arclengths, index, mr = MeshRegion[pl, Line[Range[Length @ pl]]]}, index = Last @ Region`Mesh`MeshNearestCellIndex[mr, p]; arclengths = PropertyValue[{mr, 1}, MeshCellMeasure]; (Total[arclengths[[;; index - 1]]] + Norm[pl[[index]] - p]) / Total[arclengths]] invLSC[___] := -1

Examples:

2D

Needs["GraphUtilities`"]; pts = {{0, 0}, {1, 0}, {0.6, 0.4}, {0.1, 0.5}}; {P = LineScaledCoordinate[pts, .3], invLSC[pts, P]} 
{{2.37301, 6.88155}, 0.3} 
{P2 = LineScaledCoordinate[pts, .8], invLSC[pts, P2]} 
{{4.41186, 4.25237}, 0.8} 
{SeedRandom[1]; P3 = RandomReal[1, 2], invLSC[pts, P3]} 
{{0.817389, 0.11142}, -1} 
SeedRandom[1]; pts = RandomReal[10, {15, 2}]; q = RandomReal[1, 10]; Grid[Prepend[{#, pt = LineScaledCoordinate[pts, #], invLSC[pts, pt]} & /@ q, Style[#, 14] & /@ {"q", "pt = LineScaledCoordinate[pts, q]", "invLSC[pts, pt]"}], Dividers -> All]

enter image description here

pl = RandomPoint[Line @ pts, 10]; Grid[Prepend[{#, s = invLSC[pts, #], LineScaledCoordinate[pts, s]} & /@ pl, Style[#, 14] & /@ {"pt", "s = invLSC[pts, pt]", "LineScaledCoordinate[pts, s]"}], Dividers -> All]

enter image description here

3D

SeedRandom[1]; pts = RandomReal[10, {15, 3}]; q = RandomReal[1, 10]; Grid[Prepend[{#, pt = LineScaledCoordinate[pts, #], invLSC[pts, pt]} & /@ q, Style[#, 14] & /@ {"q", "pt = LineScaledCoordinate[pts, q]", "invLSC[pts, pt]"}], Dividers -> All]

enter image description here

pl = RandomPoint[Line @ pts, 10]; Grid[Prepend[{#, s = invLSC[pts, #], LineScaledCoordinate[pts, s]} & /@ pl, Style[#, 14] & /@ {"pt", "s = invLSC[pts, pt]", "LineScaledCoordinate[pts, s]"}], Dividers -> All]

enter image description here

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Needs["GraphUtilities`"]; pts = {{0, 0}, {1, 0}, {0.6, 0.4}, {0.1, 0.5}}; P = LineScaledCoordinate[pts, 0.3]; x[t_?NumericQ] := LineScaledCoordinate[pts, t][[1]] y[t_?NumericQ] := LineScaledCoordinate[pts, t][[2]]; sol = NMinimize[{({x[t], y[t]} - P) . ({x[t], y[t]} - P), 0 <= t <= 1}, t]; t /. sol[[2]]

0.3

SeedRandom[111]; Needs["GraphUtilities`"]; pts = {{0, 0}, {1, 0}, {0.6, 0.4}, {0.1, 0.5}}; randoms = RandomReal[{0, 1}, 20] points = LineScaledCoordinate[pts, #] & /@ randoms; inverse[pts_, P_] := Module[{x, y, sol, t}, x[t_?NumericQ] := LineScaledCoordinate[pts, t][[1]]; y[t_?NumericQ] := LineScaledCoordinate[pts, t][[2]]; sol = NMinimize[{({x[t], y[t]} - P) . ({x[t], y[t]} - P), 0 <= t <= 1}, t]; t /. sol[[2]]]; inverse[pts, #] & /@ points

{0.183729, 0.779252, 0.755279, 0.203977, 0.298739, 0.538596, 0.00195475, 0.715802, 0.169596, 0.858267, 0.212156, 0.304449, 0.339743, 0.42689, 0.318241, 0.016407, 0.398779, 0.998553, 0.240465, 0.718291}

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  • $\begingroup$ Thanks for your answer, but it's too slow to be practical, I need to do it in real time. $\endgroup$ Commented Nov 15, 2021 at 10:47

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