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geometry - Why are these methods giving me different results? (Trying to test SplineFit)


I am creating a polymer(where each monomer is of equal length) using this method:



angles = RandomVariate[NormalDistribution[0, 0.2], 40];
a1 = 1;
anglePath1[angles_, a_] := FoldList[# + a {Cos@#2, Sin@#2} &, {0, 0}, Accumulate@angles]
p1 = anglePath1[angles, a1];

Then, I use SplineFit


Needs["Splines`"];
spt = SplineFit[p1, Cubic];
lt = spt[[2, -1]];


Then, I break the polymer into monomer of equal length. To do so, I calculate the arc length of the polymer:


dz = .0000001;
arc = NIntegrate[Norm[(spt[z + dz] - spt[z])/dz], {z, 0, lt}]

mesh = Solve [arc/a1 == div];
meshf = Round[div /. mesh];

After this I extract co-ordinates:


plot = ParametricPlot[spt[t], {t, 0, lt}, 
         MeshFunctions -> {"ArcLength"}, Mesh -> {meshf[[1]]}, 

         MeshStyle -> {PointSize[0.01], Red}]

coord = Cases[Normal@plot, Point[p_] :> p, Infinity];

So, now I calculate distance between each points:


(*distance calculated using initially created co-ordinates*)
Norm /@ Differences@p1

(*distance calculated using co-ordinates grabbed from spline fitted curve*)
Norm /@ Differences@coord


These two distances are different and I don't understand why.


The method calculating arc length and grabbing co-ordinates are from @MichaelE2 answer from this post:



Answer



Your points aren't in order. Change:


coord = Cases[Normal@plot, Point[p_] :> p, Infinity]

by


coord = Sort@Cases[Normal@plot, Point[p_] :> p, Infinity]


Then you'll get


(Norm /@ Differences@p1)[[1 ;; 5]]
(Norm /@ Differences@coord)[[1 ;; 5]]

(*
{1., 1., 1., 1., 1.}
{0.976258, 0.97659, 0.975463, 0.976125, 0.976947}
*)

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