DElete F[1] in the main code. 
Call it F in the init routine, but that can be moduled.
Define F[t[globally, as well as FDer FUnitDer FFront to be polar square root.

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Get["/Users/wagon/Desktop/exportFInt240.nb"];  (* move files *) 
FInt[240]=FInt240;

Initialization

Φ[fExpr_, max_, resPerLoop_]:=(fExpr1=fExpr/. t-> tSym;
ttq=Table[{tSym1, (fExpr1+  unit@D[fExpr1,tSym])/. tSym -> tSym1}, {tSym1, 2.π, max, π/resPerLoop}];
ρTranslate=Norm[ttq[[1,2]]];
int=Interpolation[ttq][tSym];
rot=N@RotationMatrix[-ArcTan@@(fExpr1+  unit@D[fExpr1,tSym]/. tSym-> 2π)];
InterpolatingFunction[int[[0,1]], int[[0,2]], int[[0,3]], int[[0,4]]/. {x_?NumericQ, y_} :> (rot. {x,y})-{ρTranslate-1,0}][t])

errorTable//Clear;
errorTable[1, nPoints_, wp_, pg_]:= errorTable[1, nPoints]= Table[{τ,bikeErrorPolarSquareRoot[τ,WorkingPrecision->wp,PrecisionGoal->pg]},{τ,2 π,100 π,((98π)/(nPoints-1))}]
errorTable[n_, nPoints_, wp_, pg_]:= errorTable[n, nPoints]= Table[{τ,bikeErrorBetter[FInt[n],FIntDer[n],{t,τ},WorkingPrecision->wp,PrecisionGoal->pg]},{τ,2 π,100 π,((98π)/(nPoints-1))}]

bikeErrorPolarSquareRootLowerBoundBestApproximation=(4 (π^(3/2)) (Sqrt[t])+8 (Sqrt[2]) (π^2) t+(t^2) (Sqrt[2+8 (t^2)])+(Sqrt[t]) (Sqrt[π+4 π (t^2)])+(Sqrt[2]) π (1+2 t (2 t+(Sqrt[1+4 (t^2)])))-(((2 π+t))^(3/2)) (Sqrt[2+8 (t^2)]) (Sqrt[2 π+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]])+ArcCot[((Sqrt[t])/(Sqrt[2 π]))] (2 (Sqrt[π]) (Sqrt[t])+4 (Sqrt[2]) π t+(Sqrt[2+8 (t^2)]) (t-(Sqrt[2 π+t]) (Sqrt[2 π+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]))))/((Sqrt[2 π+t]) (Sqrt[π+4 π (t^2)]) (Sqrt[1+4 (((2 π+t))^2)+4 ArcCot[((Sqrt[t])/(Sqrt[2 π]))] (4 π+2 t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))])]));




bikeErrorPolarSquareRootUpperBoundBestApproximation=δSquare^(1/2);
ω=.996;
τ=113.π;
bikeErrorPolarSquareRoot[τ]
ω bikeErrorPolarSquareRootLowerBoundBestApproximation+ (1-ω)bikeErrorPolarSquareRootUpperBoundBestApproximation  /. t-> τ
bikeErrorPolarSquareRootUpperBoundBestApproximation /. t-> τ;
bikeErrorPolarSquareRootLowerBoundBestApproximation /. t-> τ

bikeErrorPolarSquareRootLowerBoundBestApproximation=(4 (π^(3/2)) (Sqrt[t])+8 (Sqrt[2]) (π^2) t+(t^2) (Sqrt[2+8 (t^2)])+(Sqrt[t]) (Sqrt[π+4 π (t^2)])+(Sqrt[2]) π (1+2 t (2 t+(Sqrt[1+4 (t^2)])))-(((2 π+t))^(3/2)) (Sqrt[2+8 (t^2)]) (Sqrt[2 π+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]])+ArcCot[((Sqrt[t])/(Sqrt[2 π]))] (2 (Sqrt[π]) (Sqrt[t])+4 (Sqrt[2]) π t+(Sqrt[2+8 (t^2)]) (t-(Sqrt[2 π+t]) (Sqrt[2 π+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]))))/((Sqrt[2 π+t]) (Sqrt[π+4 π (t^2)]) (Sqrt[1+4 (((2 π+t))^2)+4 ArcCot[((Sqrt[t])/(Sqrt[2 π]))] (4 π+2 t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))])]));

(* next uses the improvement on α to get the lower bound *) bikeErrorPolarSquareRootLowerBoundBestApproximation=(-((-((2 (Sqrt[π])+(Sqrt[2]) t (Sqrt[(1/t)+4 t])) Cos[t])+(Sqrt[2]) (Sqrt[t]) (Sqrt[(1/t)+4 t]) (Sqrt[2 π-(π/(t^2))+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]) Cos[(π/(t^2))-t-ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]+4 (Sqrt[π]) t Sin[t]) (2 (π (2-(1/(t^2)))+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]) Cos[(π/(t^2))-t-ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]-Sin[(π/(t^2))-t-ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]))-(Cos[(π/(t^2))-t-ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]+2 (π (2-(1/(t^2)))+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]) Sin[(π/(t^2))-t-ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]) (4 (Sqrt[π]) t Cos[t]+(2 (Sqrt[π])+(Sqrt[2]) t (Sqrt[(1/t)+4 t])) Sin[t]+(Sqrt[2]) (Sqrt[t]) (Sqrt[(1/t)+4 t]) (Sqrt[2 π-(π/(t^2))+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]) Sin[(π/(t^2))-t-ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]))/(2 (Sqrt[π]) (Sqrt[t]) (Sqrt[(1/t)+4 t]) (Sqrt[π (2-(1/(t^2)))+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]) √(((t^4)+4 (t^6)+4 (π^2) (((1-2 (t^2)))^2)+8 π (t^3) (-1+2 (t^2))+8 (t^2) ((t^3)+π (-1+2 (t^2))) ArcCot[((Sqrt[t])/(Sqrt[2 π]))]+4 (t^4) ((ArcCot[((Sqrt[t])/(Sqrt[2 π]))])^2))/((t^5)+π (t^2) (-1+2 (t^2))+(t^4) ArcCot[((Sqrt[t])/(Sqrt[2 π]))])));

conjugate[x_]:=((Expand[Numerator[x] (Numerator[x]/. (Sqrt[z_]):>-(Sqrt[z]))])/(Expand[Denominator[x] (Numerator[x]/. (Sqrt[z_]):>-(Sqrt[z]))]));
eProduct[1]=(π/(3 (t^2)));
eProduct[n_]:=eProduct[n]=(eProduct[n-1]/. t->t+2 π) (1+(π/t)/. t->t+2 π);
eProductSym[1]:=(π/(3(t^2)));
eProductSym[n_]:=((π Pochhammer[((3 π+t)/(2 π)),n-1])/(3 (((2 (n-1) π+t))^2) Pochhammer[1+(t/(2 π)),n-1]))

distancePointLine[{{x1_, y1_},{x2_, y2_}}, {x0_,y0_}]:= (((x2-x1)(y1-y0)-(x1-x0)(y2-y1))/(Sqrt[(((x2-x1))^2)+(((y2-y1))^2)]));
distancePointLine1[{L1_,L2_}, P_]:= ((Det[{L2-L1, L1-P}])/(Sqrt[(L1-L2).(L1-L2)]));


$MaxExtraPrecision = 1000;
ED=EuclideanDistance;
at=AbsoluteThickness;
ap=AbsolutePointSize;
N20[z_, τ_:10^10]:= Module[{g=getSymbol[z]}, If[g==={}, N[z,20],  N[z/. g->τ, 20]]];
getSymbol[z_] := Module[{dc=DeleteCases[Union[Cases[z, _Symbol,∞]],π]}, If[dc==={}, dc, dc[[1]]]];


EDSquaredDot[P1_, P2_]:= (P1-P2).(P1-P2);
intDomMax[z_] := z[[0,1,1,2]];  intDomMin[z_] := z[[0,1,1,1]];
intDom[z_] := {intDomMin[z],intDomMax[z]};
intCount[z_] := Length@z[[0,3,1]];
SetOptions[Interpolation,InterpolationOrder->3];
cols=Flatten[Table[{Red,Green,Blue,Cyan,Magenta,Yellow,Black},10]];
unit[z_] := z/(Sqrt[z.z]);



limit∞[z_]:= Limit[z, getSymbol[z]->∞];
limit0[z_]:= Limit[z, getSymbol[z]->0, Direction-> -1];
series[q_,n_]:=Simplify[Normal[Series[q,{getSymbol[q],∞,n}]]];
series0[q_,n_]:=Simplify[Normal[Series[q,{getSymbol[q],0,n}]]];
cs={Cos[t], Sin[t]};
(* correction=(1/(13.73 ((((t/(2 π))))^2)+5));   not really helpful *) 

F[1]=(Sqrt[(t/(2π))]) cs;  (* get rid of this *) 

F[t_]:=(Sqrt[(t/(2π))]){ Cos[t],Sin[t]};
FFront[t_]:=Evaluate[Simplify[F[t]+(((∂_t)F[t])/(Sqrt[(∂_t)F[t] . (∂_t)F[t]]))]];


FDerUnit[1]=Simplify[unit[(∂_t)F[1]]];

F[t_]:=(Sqrt[(t/(2 π))]) {Cos[t],Sin[t]};
FDer[t_]:={((Cos[t])/(2 (Sqrt[2 π]) (Sqrt[t])))-(((Sqrt[t]) Sin[t])/(Sqrt[2 π])),(((Sqrt[t]) Cos[t])/(Sqrt[2 π]))+((Sin[t])/(2 (Sqrt[2 π]) (Sqrt[t])))};
FFront[t_]:={(((2+(Sqrt[(2/π)]) t (Sqrt[(1/t)+4 t])) Cos[t]-4 t Sin[t])/(2 (Sqrt[t]) (Sqrt[(1/t)+4 t]))),((4 t Cos[t]+(2+(Sqrt[(2/π)]) t (Sqrt[(1/t)+4 t])) Sin[t])/(2 (Sqrt[t]) (Sqrt[(1/t)+4 t])))};





atn[x_] := If[x=={0,0},0, Mod[ArcTan@@x, 2π]];

(*F0[x_]:=If[x>0,(E^(-1/x)),0]; F[x_]:=((F0[1-x])/(F0[1-x] +F0[x]));
Plot[F[x],{x,-1,2}, Frame->True] ; (* nonanalytic spline connector *)  *) 
           
(* bikeErrorOld[f_, unitfDer_, {t_,tVal_}, offset_:1]:= With[{front =f+unitfDer /.  t->tVal},
     αAngle =(offset + Quotient[tVal,2 π]) 2 π+atn[front];
     objWithOffset[s_?NumericQ] := With[{q = front - (f /.  t->s)},(((q.q))^0.5)];
      Quiet[First@FindMinimum[objWithOffset[sss],{sss,αAngle}, PrecisionGoal->10]]];  *) 
 
 
 
 bikeError[f_, unitfDer_, {t_,tVal_}, offset_:1]:= With[{front =f+unitfDer /.  t->tVal},
     αAngle =(offset + Quotient[tVal, 2 π]+If[atn@front< atn [f/. t-> tVal], 1,0]) 2 π + atn[front];
     objWithOffset[s_?NumericQ] := With[{q = front - (f /.  t->s)},(q.q)];
      ans=Quiet[First[FindMinimum[objWithOffset[sss],{sss,αAngle},PrecisionGoal->10]]^(1/2)];
      If[Abs[tVal-(1+Quotient[tVal, 2 π] )2π ] < 10^-6 || Abs[Mod[tVal,2π] ] < 10^-6,
    Min[ans, 
      Quiet[First[FindMinimum[objWithOffset[sss],{sss,αAngle+2π},PrecisionGoal->10]]^(1/2)],
      Quiet[First[FindMinimum[objWithOffset[sss],{sss,αAngle-2π},PrecisionGoal->10]]^(1/2)]],  ans]
      
      ]; 
      (* this has the square distance as the objective and that is a lot faster *)
     
      
(*bikeErrorBetter[f_, unitfDer_, {t_,tVal_}, offset_:1]:= With[
        {front =f+unitfDer /.  t->tVal},
        αAngles=offset + (Quotient[tVal, 2 π]+{-1,0,1}) 2π + atn[front];
        αAngle=SortBy[αAngles, ED[front, (f /.  t->#)&]] [[1]];    
   objWithOffset[s_?NumericQ] := With[{q = front - (f /.  t->s)},(q.q)];
      Quiet[First[FindMinimum[objWithOffset[sss],{sss,αAngle},PrecisionGoal->10]]^(1/2)]
      ]; 
      (* this has the square distance as the objective and that is a lot faster *)
  *)   
     
     
bikeErrorBetter//Clear; 
sss//Clear;objForFindRoot//Clear;
Options[bikeErrorBetter]= {WorkingPrecision->MachinePrecision, PrecisionGoal -> Automatic, Offset-> 1, Output->Error, InitialLoops->{-1,0,1}};
bikeErrorBetter[f_,fDer_,{t_,tVal_}, opts___]:= Module[{sssVal,sss, αAngles, αAngle, objForFindRoot,wp, pg, offset, output},
With[
        {front = (f /. t-> tVal)  + unit[fDer /.  t->tVal]},
        {wp, pg, offset, output,initLoops} = {WorkingPrecision, PrecisionGoal, Offset, Output,InitialLoops}/. {opts}/. Options[bikeErrorBetter];
        
        αAngles=(offset + Quotient[tVal, 2 π]+initLoops) 2π + atn[front];     
        αAngle = SortBy[αAngles, N[ ED[front, (f /.  t->#)],wp] &] [[1]]; 
        objForFindRoot[s_?NumericQ] := (fDer /. t -> s) . (front - (f /. t -> s));  
        Quiet[sssVal = sss /. (FindRoot[objForFindRoot[sss],{sss, αAngle},PrecisionGoal->pg, WorkingPrecision->wp])];
      (* this has the dot product as the objective and that is a lot faster *) 
      If[output === Error, qq = (f /. t-> sssVal) - front; (qq.qq)^(1/2), sssVal]  ]];

initialLoops is a brute force way to find the closest loop. But it requires more of the function.

bikeErrorBetterFInt[n_,tVal_, opts___]:= Module[{sssVal,sss, αAngles, αAngle, objForFindRoot,wp, pg, offset, output,f=FInt[n], fDer=FIntDer[n]},
With[
        {front = (f /. t-> tVal)  + unit[fDer /.  t->tVal]},
        {wp, pg, offset, output} = {WorkingPrecision, PrecisionGoal, Offset, Output}/. {opts}/. Options[bikeErrorBetter];
        
        αAngles=(offset + Quotient[tVal, 2 π]+{0}) 2 π + atn[front];     
        αAngle = SortBy[αAngles, N[ ED[front, (f /.  t->#)],wp] &] [[1]]; 
        objForFindRoot[s_?NumericQ] := (fDer /. t -> s) . (front - (f /. t -> s));  
        Quiet[sssVal = sss /. (FindRoot[objForFindRoot[sss],{sss, αAngle},PrecisionGoal->pg, WorkingPrecision->wp])];
      (* this has the dot product as the objective and that is a lot faster *) 
      If[output === Error, qq = (f /. t-> sssVal) - front; (qq.qq)^(1/2), sssVal]  ]]; 
      
  
       
           
  (Options[bikeErrorPolarSquareRoot]={WorkingPrecision->MachinePrecision,PrecisionGoal->Automatic,Offset->1,Output->Error};)
   (* this is twice as fast as the general function *)  
 bikeErrorPolarSquareRoot[tVal_, opts___] := Module[{ sss,sssVal,objForFindRoot,wp, pg, offset, output},
     Module[{fDer={((Cos[tVal]-2 tVal Sin[tVal])/(2 (Sqrt[2 π]) (Sqrt[tVal]))),((2 tVal Cos[tVal]+Sin[tVal])/(2 (Sqrt[2 π]) (Sqrt[tVal])))} ,
      front = {(((Sqrt[tVal])/(Sqrt[2 π]))+(1/(Sqrt[1+4 (tVal^2)]))) Cos[tVal]-((2 tVal Sin[tVal])/(Sqrt[1+4 (tVal^2)])),((2 tVal Cos[tVal])/(Sqrt[1+4 (tVal^2)]))+(((Sqrt[tVal])/(Sqrt[2 π]))+(1/(Sqrt[1+4 (tVal^2)]))) Sin[tVal]}},
     
        
        {wp, pg, offset, output} = {WorkingPrecision, PrecisionGoal, Offset, Output}/. {opts}/. Options[bikeErrorPolarSquareRoot];
        objForFindRoot[s_?NumericQ] := {((Cos[s])/(2 (Sqrt[2 π]) (Sqrt[s])))-(((Sqrt[s]) Sin[s])/(Sqrt[2 π])), (((Sqrt[s]) Cos[s])/(Sqrt[2 π]))+((Sin[s])/(2 (Sqrt[2 π]) (Sqrt[s])))} . (front - (Sqrt[(s/(2 π))]){Cos[s],Sin[s]});  
       Quiet[sssVal = sss /. (FindRoot[objForFindRoot[sss], {sss, tVal+2 π + ArcCot[(Sqrt[(tVal/(2 π))])]-(1/(tVal^2))(π + (1/(8(Sqrt[1+(tVal/(2π))]))))},PrecisionGoal->pg, WorkingPrecision->wp])];
        
        If[output === Error, qq = ((Sqrt[(sssVal/(2 π))]) {Cos[sssVal],Sin[sssVal]}) - front; (Sqrt[(qq . qq)]), sssVal]  ]]; 

     
       
 (*bikeErrorLocation[f_, unitfDer_, {t_,tVal_}, offset_:1]:= With[{front =f+unitfDer /.  t->tVal},
     αAngle =(offset + Quotient[tVal,2 π]+If[atn@front< atn [f/. t-> tVal], 1,0]) 2 π+atn[front];
     objWithOffset[s_?NumericQ] := With[{q = front - (f /.  t->s)},(q.q)];
      Quiet[sss/.Last@FindMinimum[objWithOffset[sss],{sss,αAngle}, PrecisionGoal->10]]]; *)
      
NLim = NumericalMath`NSequenceLimit;
bikeErrorList//Clear;
(* this uses the radial function to make a smaller interpolating function on the required domain *) 

Options[bikeErrorList] = {IntResolution -> 30000};
bikeErrorList[F_, {t_, range_}, opts___]:= With[{
FShort = Interpolation[Table[{t,Norm[F] cs},
 {t, range[[1]], range[[-1]]+8π, (range[[-1]]+8π)/( IntResolution /. {opts} /. Options[bikeErrorList])}], InterpolationOrder->4][t]},

FShortDer = D[ FShort, t];
Table[{τ, bikeErrorBetter[FShort, FShortDer, {t, τ}]}, {τ, range}]];

bikeErrorListSlow[F_, FD_, {t_, range_}]:= Table[{τ, bikeError[F, FD, {t, τ}]}, {τ, range}];  (* can be more reliable than radial method *) 


Options[initialize]= {InitialRadialFunction->(Sqrt[(t/(2 π))]),WorkingPrecision->MachinePrecision,PrecisionGoal->Automatic,Resolution->1200};
initialize//Clear;

initialize[mxVal_,opts___] := (Clear[FDerUnit,getNextInitialValue, objFR, start,mx];
{wp, pg, initFunc, ρ}= {WorkingPrecision, PrecisionGoal, InitialRadialFunction, Resolution}/. {opts} /. Options[initialize];


mx[1] = mxVal;   (* not far enough for 214, and results get ragged. go farther *)  
F[1] = initFunc cs;
(*F[1] =((Sqrt[(t/(2π))])+((t-2π)(-(1/3))(1/(t^2)) ))cs;*) 
FDerUnit[1]=Simplify[unit[(∂_t)F[1]]];
FDer[1]=Simplify[(∂_t)F[1]];
FInt[1] = Interpolation[Table[{t,F[1]}, {t, 2π,mx[1],N[2, wp]π/ρ}]][t];
FIntFunc[1][t_] := Evaluate[FInt[1]];
FIntDer[n_] :=D[FInt[n],t];
FIntDerUnit[n_] :=unit[FIntDer[n]];
objFR[n_,t0_][t1_?NumericQ] := EDSquaredDot[FInt[n]/. t->t1,FInt[n]/. t->t0];
start[n_,t0_, wp_, pg_] := Module[{t1}, t1/.FindRoot[objFR[n, t0][t1]==1, {t1,t0+2π+1/2,t0+2π+9/10}, WorkingPrecision->wp, PrecisionGoal->pg]];

getNextInitialValue[n_,t0_]:=FInt[n] /. t->start[n,t0];

front[1][t_?NumericQ]:=Evaluate[F[1]];
frontDer[1][t_?NumericQ]:=Evaluate[(∂_t)F[1]];
frontDerUnit[1][t_?NumericQ]:=Evaluate[unit[(∂_t)F[1]]];
BikeVector[1][t_?NumericQ] := Evaluate[{FInt[1],FInt[1]+FIntDerUnit[1]}]);

Init

Options[GetFIntSequence] ={TraceQ->True, Resolution->1200, WorkingPrecision->MachinePrecision, PrecisionGoal->6, FunctionsToKeep->{}, ErrorValuesToStore->{2π}, FunctionPointsToKeep->{}, InitialRadialFunction ->(Sqrt[t/(2π)]) ,MaxSteps-> 3450000};

GetFIntSequence[nMax_, tMax_, opts___] := 
Module[{tStart, ms, init, mx, backSol, backSolPts},
{trQ,res, wp, pg, fcnKeep, errorKeep,fcnPtsKeep, initFunc,ms}=
{TraceQ,Resolution, WorkingPrecision, PrecisionGoal, 
FunctionsToKeep,ErrorValuesToStore,FunctionPointsToKeep, InitialRadialFunction,MaxSteps}/. {opts}/.Options[GetFIntSequence];

initialize[tMax,Resolution-> res, WorkingPrecision-> wp, PrecisionGoal->pg,InitialRadialFunction-> initFunc] ;

Do[errorData[t0] ={  {1,Log10@bikeErrorBetter[FInt[1],  FIntDer[1],{t, t0}, WorkingPrecision->wp, PrecisionGoal->pg]}},
{t0, errorKeep}];

Do[points2keep[t0]={}, {t0,fcnPtsKeep}] ;

Do[tStart=SetPrecision[start[n-1,2π,wp+3,If[pg===Automatic, Floor[wp/2],pg+5]], wp];
mPrint[Precision[tStart]];

init[n]={1,0};mx[n]=intDomMax[FInt[n-1]];

backSol=NDSolveValue[{b'[t]==frontDer[n-1][t].(b[t]-front[n-1][t]) (b[t]-front[n-1][t]),b[tStart]==init[n]},b[t],{t,tStart,mx[n]},
MaxSteps->ms, WorkingPrecision->wp,PrecisionGoal->pg];
backSolPts=Table[backSol,{t,tStart,intDomMax[backSol],2π/ρ}];
backSolPts=SetPrecision[backSolPts, wp];

FInt[n]=(old=atn[init[n]];loops=If[backSolPts[[1,2]]≥0, 1,0];
Interpolation[Table[new=atn[backSolPts[[i]]];If[new<old,loops++];old=new;
{atn[backSolPts[[i]]]+2 π loops,backSolPts[[i]]},{i,Length[backSolPts]}]][t]);
If[trQ&&n==2, mPrint[initFunc]];
Do[AppendTo[errorData[t0], 
 {n,Log10[bikeErrorBetter[FInt[n], FIntDer[n],{t, t0}, WorkingPrecision->wp, PrecisionGoal->pg]]}], {t0, errorKeep}];

Do[AppendTo[points2keep[t0],  {n,FInt[n]/. t-> t0}], {t0, fcnPtsKeep}];

If[trQ, Print[Row[{n, "; DE for FInt[",n, "] solved. ", intCount[backSol],"; ",  mx[n], "; ",errorData[2π][[-1,2]],"; ", 10^errorData[2π][[-1,2]]}]]];
Clear[backSolPts];
If[n≥4 && !MemberQ[fcnKeep,n-2], Unset[FInt[n-2]]];
front[n][t_?NumericQ]:=Evaluate[FInt[n]];
frontDer[n][t_?NumericQ]:=Evaluate[(∂_t)FInt[n]];
frontDerUnit[n][t_?NumericQ]:=Evaluate[unit@(∂_t)FInt[n]];
BikeVector[n][t_?NumericQ] := Evaluate[{FInt[n],FInt[n]+FIntDerUnit[n]}], {n,2,nMax}]]

errorF240={{6.28318530717958647692528676655900576836`19.735683541462205,-5.282837786124634`},{6.42953949403826835791288919201973720905`19.735683541462205,-5.288434725750334`},{6.57930270784170556362966587481183782396`19.735683541462205,-5.291309548840874`},{6.73255435502820716269380627044214906621`19.735683541462205,-5.292696911376423`},{6.8893756916496367990379313621601866048`19.735683541462205,-5.295301617845482`},{7.04984986645444694101010225025398170948`19.735683541462205,-5.299105536495965`},{7.21406196497424591467060438160494535446`19.735683541462205,-5.3011468149850955`},{7.38209905463727300268026332264715489033`19.735683541462205,-5.304426926918669`},{7.55405023093270137042515977677673929698`19.735683541462205,-5.308915715021129`},{7.73000666465024574384596292935207962439`19.735683541462205,-5.311525900459883`},{7.91006165022012190425279603244118728129`19.735683541462205,-5.315135637234124`},{8.09431065517898848650479201402736823848`19.735683541462205,-5.316783713032114`},{8.28285137078809857765487859255991817208`19.735683541462205,-5.31928360223306`},{8.47578376383049953004783874521645514138`19.735683541462205,-5.322934373545603`},{8.67321012961474454982954361388085991173`19.735683541462205,-5.3274091015013365`},{8.87523514621321933033864925014529555619`19.735683541462205,-5.3296595842394945`},{9.08196592996384160909090734448284845831`19.735683541462205,-5.3326601217171445`},{9.29351209226456138411304647563273183315`19.735683541462205,-5.336493890675407`},{9.50998579769077498540500727715711504358`19.735683541462205,-5.337423790422557`},{9.73150182346646762374613895767523191382`19.735683541462205,-5.342934272505896`},{9.95817762032061680381836711005528487137`19.735683541462205,-5.34537135704344`},{10.19013337476112347226569881738497104694`19.735683541462205,-5.348375990885638`},{10.42749207279928936327769204822644024193`19.735683541462205,-5.351988189782119`},{10.67037956515862810307869229826654824989`19.735683541462205,-5.356084145116292`},{10.91892463400258464811109127813204647254`19.735683541462205,-5.35664946169256`},{11.17325906121654297700451562258254958418`19.735683541462205,-5.36145183866753`},{11.43351769828032606062844810674455180086`19.735683541462205,-5.366585345773902`},{11.69983853776823543458427781544742420777`19.735683541462205,-5.367740069460849`},{11.97236278651454064153302253144664483277`19.735683541462205,-5.373220643808605`},{12.25123494048321185432078862771703428781`19.735683541462205,-5.374378030560048`},{12.53660286138159260315681803029646896162`19.735683541462205,-5.379755057587243`},{12.82861785505863419022591671826422705964`19.735683541462205,-5.380671722637226`},{13.12743475172925957334520337623474157324`19.735683541462205,-5.385128855545845`},{13.43321198806739273829128518843818394026`19.735683541462194,-5.389578916308507`},{13.7461116912111803706007726797717790118`19.735683541462205,-5.393317286744049`},{14.06629976472494650731508358173526890428`19.735683541462205,-5.396388207088415`},{14.3939459765634583348668750752701488341`19.735683541462205,-5.398762977519272`},{14.72922404908514295117884931691194676496`19.735683541462205,-5.404349757203179`},{15.0723117511619812909679747927947277777`19.735683541462205,-5.409074923445821`},{15.42339099243491709923290072061523003591`19.735683541462205,-5.4125394775314035`},{15.78264791976475641838013744258495003421`19.735683541462205,-5.414507225302954`},{16.15027301592969713257565950768948436494`19.735683541462205,-5.419777865629339`},{16.52646120062181930590259342547281702677`19.735683541462205,-5.423240179443562`},{16.91141193379608599034887161094227089267`19.735683541462205,-5.424985961083327`},{17.30532932142665151183899519699034331094`19.735683541462205,-5.42928305538087`},{17.70842222372655062881338883679851502163`19.735683541462205,-5.436382361382662`},{18.12090436588814807500415593217599042848`19.735683541462205,-5.4408723720714836`},{18.54299445140306453857795186042274250786`19.735683541462205,-5.4429722067166075`},{18.97491627802166280236666165355977961178`19.735683541462205,-5.447094431804443`},{19.41689885641357729964268648949259609537`19.735683541462194,-5.45311217158168`},{19.86917653159220246886710134064070224572`19.735683541462205,-5.456260588282878`},{20.33198910716752077837948132675333974389`19.735683541462205,-5.460835406087984`},{20.80558197249315091512869874761655660413`19.735683541462205,-5.461516713992543`},{21.29020623277503118538450008035097868955`19.735683541462205,-5.469097209937982`},{21.78611884221072347356094087334803322006`19.735683541462194,-5.472327232061459`},{22.29358274022992998041767281901738307952`19.735683541462205,-5.476391961171411`},{22.81286699090845926597010737993953977286`19.735683541462194,-5.48113588145366`},{23.34424692562956072726640439402018025583`19.735683541462205,-5.486069364700422`},{23.88800428906826843891740323680580310776`19.735683541462205,-5.485899828780697`},{24.44442738857615718781497453575364091238`19.735683541462194,-5.490935955594975`},{25.013811247045716477029405533371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Questions:  Can one approximate r(t) in some better way?

A Spiral Bicycle Track that Can Be Made by a Unicycle

Stan Wagon, Macalester College, St. Paul, MN; November 2023  wagon@macalester.edu

Summary.  Starting with the polar square root curve and iterating a simple construction using a differential equation apparently leads to a perfect unibike spiral with no self-intersections. A unibike curve is a track that can be made by either a bicycle or a unicycle. More precisely, for a unibike curve, the end of a unit tangent vector at any point on the curve always lies on the curve (so the bike’s rear wheel always lies on the track made by the front wheel). The outstanding question here is: Can one prove that the limit of the iteration described here exists and has unibike error 0? All computations indicate that this is true. Using an extrapolation idea as in Romberg integration allows for error-reduction that yields a spiral curve whose unibike error is everywhere under 5·(10^(-7)).

References

Anything else in ...TMI?

1.  The First Unibike

A bicycle has two wheels and when the bike rolls the wheels generally make two different tracks; Figure  shows a typical case. Because the rear wheel always points to the front wheel (we ignore the fact that the front fork is angled, which makes this only approximately true), the tangent to the path of the rear wheel at time t strikes the front wheel path at the point where the front wheel is at time t. The distance between the two wheels for all bicycles here is one unit.

DynamicModule[{locQ$224421=False,met$224421="Bezier",p$224421={{0,1},{0.6`,1.2`},{1.112`,0.8600000000000001`},{1.5499999999999998`,1.7349999999999999`},{2.51`,1.835`},{4.196`,0.505`},{5.5`,2}},randir=1,tanQ$224421=True,tantt$224421=0.388`,xmin$224421=1.06`,xrange$224421=3.9`,ymin$224421=0.28`,yrange$224421=1.64`},Module[{ranPts$,t$,backTrack$,frontTrack$},backTrack$=Which[met$224421==="Bezier",BezierFunction[p$224421,SplineDegree->Length[p$224421]-1][t$],met$224421==="Spline",Interpolation[Transpose[{Range[0,1,(1/(Length[p$224421]-1))],p$224421}],Method->"Spline"][t$]];frontTrack$=backTrack$+((randir (∂_t$)backTrack$)/(Norm[(∂_t$)backTrack$]));im=Show[ParametricPlot[Evaluate[{frontTrack$,backTrack$}],{t$,0.01`,1},PlotStyle->{{Thick,Red},{Thick,Blue}},PlotRange->All],Graphics[{If[locQ$224421,{EdgeForm[Thickness[0.0025`]],FaceForm[LightYellow],(Disk[#1,0.13`]&)/@p$224421,MapIndexed[Text[Style[#2[[1]],9],#1]&,p$224421]},{}],If[tanQ$224421,{Thickness[0.008`],PointSize[0.02`],Point[backTrack$],Arrow[{backTrack$,frontTrack$}]}/. t$->If[randir==1,tantt$224421,1-tantt$224421],{}]}],Frame->True,FrameTicks->False,ImageSize-> 400,Axes->False,PlotRange->{{xmin$224421,xmin$224421+xrange$224421},{ymin$224421+.4,ymin$224421+yrange$224421}}]]]

A natural question is whether there can be a path P for the rear wheel that is not a straight line and is such that the corresponding front path agrees with P.

Definition.  Let f(t) be a differentiable curve. The corresponding front track, Φ(f)(t), is f(t)+((f'(t))/(Abs[(Abs[f'(t)])])). Let s(t) be the parameter such that f(s(t)) is the closest point to Φ(f)(t) (ties will not arise). Then the unibike error of f at t, denoted E(f)(t), is (Abs[(Abs[Φ(f)(t)-f (s t)])]). A unibike curve is a curve that is not a straight line and has zero unibike error at all points; i.e., each point Φ(f)(t) lies on the curve f(t).   XXX later take f to be front and define Ω(f)(t) to be rear.

The unibike terminology refers to the fact that the curve is a single track that can be made by either a unicycle, or by both wheels of a bicycle. While the motion might go from –∞ to a finite point, we generally care only about paths that start at a point and then have infinite length from that point, or perhaps are infinite in both directions. In 2002 David Finn [] used a nonanalytic function to construct a unibike path (Fig. ); the red curve over [0,1] that starts his process is infinitely flat at its two ends: f(t)=4(e^((-1)/(t(1-t)))); the 4 is inessential but clarifies the curve. Starting with f, one looks at Φ(f), Φ(Φ(f)), and so on; that curve is shown at right in Figure . In [, Figs. 1.7, 1.8], we showed how Finn’s curve can be extended infinitely in the reverse direction as well, giving a path g(t) that is defined on (-∞,∞) and is such that Φ(g)=g (Fig. ). For more on Finn’s construction, including an animation, see [].

unitVec[v_]:=(v/(Sqrt[v.v]));CC=4;
f[t_]:={t,CC Exp[-(1/(t (1-t)))]}
fInt[1]=Interpolation[Table[{t,f[t]}, {t, 0, 1, 1./1000}], InterpolationOrder->5][t];//Quiet

Do[ff=fInt[n-1]+unitVec[(∂_t)fInt[n-1]];
fInt[n]=Interpolation[ttt=Table[{t,ff},{t, 0, 1, 1/1000}]][t], {n,2,5}];

im0=Quiet@ParametricPlot[{t,CC Exp[((-1)/(( t) (1-t)))]},{t,0.001,1},Frame->True, Axes->False,PlotRange->{{-.1,1.1}, {-1/50,1.6/20}}, PlotStyle->{at[2],Red},BaseStyle->FontFamily->"Times", GridLines->{{},{0}}, GridLinesStyle->{at@0.82,Black}, AspectRatio-> 1, FrameTicks-> {{Range[-.02,.08,.02]// Chop,{}},{Range[0,1,.2]/. 0. -> 0/. 1.-> 1,{}}}];

cols={Red,Blue,Green,Yellow,Cyan,Orange};
nmax=5;xmin=-0.2;xmax=nmax+0.2;ymin=-3;ymax=2.9;
im1=Show[
Graphics[{GrayLevel[.4],Rectangle[{xmin,ymin},{xmax,ymax}]}],
ParametricPlot[fInt[1], {t, 0, 1}, PlotStyle->cols[[1]]]//Quiet,
ParametricPlot[fInt[2], {t, 0, 1}, PlotStyle->cols[[2]]]//Quiet,
ParametricPlot[fInt[3], {t, 0, 1}, PlotStyle->cols[[3]]]//Quiet,
ParametricPlot[fInt[4], {t, 0, 1}, PlotStyle->cols[[4]]]//Quiet,
ParametricPlot[fInt[5], {t, 0, 1}, PlotStyle->cols[[5]]]//Quiet,
Graphics[{{Black,AbsolutePointSize[7],Point[{#,0}]&/@Range[0,5]},White,AbsoluteThickness@1,Arrowheads[.02],arrows={Arrow[{fInt[1], fInt[2]}/. t-> .3],Arrow[{fInt[2], fInt[3]}/. t-> .92],Arrow[{fInt[3], fInt[4]}/. t-> .75],
Arrow[{fInt[3], fInt[4]}/. t-> .865],Arrow[{fInt[4], fInt[5]}/. t-> .83],
Arrow[{fInt[1], fInt[2]}/. t-> .67]},
AbsolutePointSize[4], Point[First/@ First/@ arrows]}],
PlotRange->{{xmin,xmax},{ymin,ymax}},FrameTicks->{Range[0,nmax],Range[-4,4],None,None},ImageSize->500,BaseStyle->FontFamily->"Times", PlotRangePadding->0];
Row[{Show[im0,Graphics[{ap[5], Point[{{1,0}, {0,0}}], Text["4 (E^((-1)/(t (1 - t))))", {.12, .04}]}],ImageSize-> 200],Spacer[30], Show[im1, ImageSize->300]}]

Clear[f];
f[t_]:=If[t==0||t==1,0,4 Exp[-(1/(t (1-t)))]];
eqns=Thread[Simplify[c'[t]==c'[t-1].(c[t]-c[t-1]) (c[t]-c[t-1])/. {c[t_]->{x[t],y[t]},c'[t_]->{x'[t],y'[t]}}]];
solExpr=NDSolveValue[eqns~Join~{x[t/;t<=0]==t,y[t/;t<=0]==f[t+1]},{x[t],y[t]},{t,-1,10},WorkingPrecision->30, PrecisionGoal->15, MaxSteps->200000];
solExprDer=D[solExpr,t]/Sqrt[D[solExpr,t].D[solExpr,t]];
solFcn[t_] := solExpr//Evaluate;
solFcnDer[t_] := solExprDer//Evaluate;

Show[ParametricPlot[solExpr/.{x_,y_}:>{-x,y},{t,0,12},PlotStyle->{Darker@Green,at@1.8}],
Plot[f[t],{t,0,1},PlotStyle->{at@1, Red}],
ParametricPlot[Evaluate[{t,f[t]}+unit[D[{t,f[t]},t]]],{t,0,.7},PlotStyle->{at@1, Blue}],
Graphics[{Black,
u=0.5;Arrowheads[.02],at@1,  Arrow[{solFcn[u],solFcn[u-1]}/.{x_?NumericQ,y_}:>{-x,y}],

 ap@4, Point[{{0,0}, solFcn[u]{-1,1},{1,0}}]}],
PlotRange->{{-2, 1.3}, {-.2, .4}},
FrameTicks-> {{Range[-.2, .4, .2], {}}, {Automatic, {}}},Axes->False,Frame->True,PlotRangePadding->0, ImageSize->Large,AspectRatio->Automatic, BaseStyle->FontFamily-> Times]//Quiet

2.  A Spiral Pseudo-Unibike

Questions:

What is the profile for F[2]? Also (Subscript[F, 240]). I think these are like (π/3)(t^(-2)), but they are a little ragged
Define Φ(F) to be the front wheel for F. 
Define Ω(F) to be the rear from F. So my work is (lim_(n->∞))(Ω^((n)))((Subscript[F, 1]))
What is the profile for Φ(F)+Ω(F)?  Or Ω(Φ(F)

Why is it that working backwards causes an improvement each time at 2π: ??? 
Can one make a Manipulate where one adjust the curve dynamically, until the front curve matches?

A very natural idea is to seek a spiral-shaped unibike path with no self-intersections. In [], we observed that the polar square root r=(Sqrt[θ]) comes close. Here we use the polar form (Subscript[F, 1])(t)=(Sqrt[(t/(2π))])(cos t,sin t), which passes through (1,0). This function is not analytic as t rises to ∞. Figure  shows the polar square root curve (blue) and the corresponding front wheel path (yellow), which starts at (1,0). Note that E((Subscript[F, 1])(0))=0. I found this curve a few years ago just by fooling around. But below will be a proof that E((Subscript[F, 1]))(t)∼(π/3)(t^(-2)) (∼ means asymptotic). If one varies the exponent in (Subscript[F, 1]) from 1/2 to a little larger or smaller, computations indicate that the error is not quadratically small.

Show[pr=1.06{{-1.6,1.6}, {-1.6,1.75}};
Graphics[{GrayLevel[.768], Rectangle[First/@ pr, Last/@ pr]}],
ParametricPlot[ F[t], {t, 0, 10π}, PlotStyle-> {Blue, at[5], CapForm[None]}],
ParametricPlot[FFront[t],{t, 0,8 π-.49},PlotStyle->{CapForm[None],at[1.2],Yellow}],Graphics[{ Red, ap[5],
EdgeForm[Black], FaceForm[{Yellow, Opacity[0.3]}],Arrowheads[.04],
Table[{Red, Point[F[τ]]}, {τ, .001, 2π, π/3}], Table[Arrow[{F[τ], FFront[τ]}], {τ, 0.001, 5π, π/3}]}],  Frame->True, BaseStyle->FontFamily->Times,PlotRangePadding->0, PlotRangeClipping->True, PlotRange-> pr, ImageSize-> 400]

Clear[b];

Manipulate[f=((Sqrt[(t/(2 π))])+ (t-2π)((a/(t^(3/2)))) +b ArcTan[t-2π]/(π/2))cs;
Show[ParametricPlot[f,{t,2π,16 π},PlotPoints-> 100,PlotStyle->{at[6],Lighter@Blue}],ParametricPlot[Evaluate[f+unit[(∂_t)f]],{t,2π,16 π},PlotStyle->{at[2],Red},PlotPoints-> 100], GridLines-> {{1},{0}},PlotRange->1 {{-3,3},{-3,3}},

PlotLabel-> Max[Log10@Table[bikeErrorBetter[f,(∂_t)f,{t,τ}],{τ,2 π,100 π,π/2}]]],{{a,.134042; .5},0,10}, {{b,0},-5,5}]

f=((Sqrt[(t/(2 π))])+ (t-2π)(((0.135 )/(t^(3/2)))) )cs;
ListLinePlot[Log10@Table[{τ,bikeErrorBetter[f,(∂_t)f,{t,τ}]},{τ,2 π,100 π,π/2}], PlotRange->{-6, -1.5}]

Code to show varying the exponent makes things much worse.

For some of these error exponents my error computation might be imperfect.

f[t_]:=((((t/(2 π))))^(50001/100000)) cs;
dL=Table[Log10[{τ,bikeErrorBetter[f[t],(∂_t)f[t],{t,τ},WorkingPrecision->40, PrecisionGoal-> 25]}],{τ,2 π,200000 π,2000π}];
ListPlot[dL,Frame->True]
fit=Fit[Take[dL, -25],{1,L},L]
(t^(fit[[2,1]]))

Figure  shows the unibike error E((Subscript[F, 1]))(2π) by the location of the first small circle, centered at the front wheel’s position when the rear is at (Subscript[F, 1])(2π), or (1,0). The circle is not centered on the blue (Subscript[F, 1]) curve, though its distance from (Subscript[F, 1]) is small, about 0.014. As t increases, the loops get closer together; this means that, because the error is bounded by the distance between the loops, E((Subscript[F, 1]))(t)=O((t^(-1/2))) as t->∞. But in fact E((Subscript[F, 1]))(t) is asymptotic to (π/3) (t^(-2)), as will be proved in the next section.

Define s(t) to be the parameter value giving the closest point on the curve f(t) to the front wheel; precisely, (Abs[(Abs[Φ(f)(t)-f(s)])]) is minimized when s=s(t). Regarding the error computation, note that the minimal distance from a point P to a curve f(t) can be quickly computed using the fact that when Q is the closest point on the curve f, the line PQ is perpendicular to the tangent to the curve. Therefore s(t), whether for the polar square root spiral (Subscript[F, 1]) or another curve, can be obtained by root-finding on the dot product (Φ(f)(t)-f(t))·f'(t). This is fast but there are many solutions and therefore it is important to start with a value close to the optimal value.

Note that one can look at the polar square root starting from a very large parameter value (Subscript[t, 0]); the error from that point on will be nicely small; and as (Subscript[t, 0]) increases this gives curves with maximum unibike error approaching 0. But this process of deleting initial segments leads to no limiting curve, so is useless. Similarly, the polar curve r=(θ^(1/m)) as m rises has loops that are so close together that the maximum unibike error approaches 0; but again there is no limiting curve.

Show[pr={{-2,2.1}, {-2,1.9}};
Graphics[{GrayLevel[.768], Rectangle[First/@ pr, Last/@ pr]}],
ParametricPlot[ F[t], {t, 2π, 10π}, PlotStyle-> {Blue, at[2]}],Graphics[{{Text[Style["(Subscript[F, 1])(t) = (Sqrt[(t/(2  π))])(cos t, sin t)",16], {-0.3,0.45}], Text[Style["(Subscript[F, 1])(2 π)",12], {.88,-.1}]}, Red, ap[5],
EdgeForm[Black], FaceForm[{Yellow, Opacity[0.3]}],Arrowheads[.02],
Table[
{Disk[FFront[τ],.033], Red, Point[F[τ]]}, {τ, 2π, 5.8π, π/3}], Table[Arrow[{F[τ], FFront[τ]}], {τ, 2π, 5.8π, π/3}],

{Black ,Text[Style["Visible error",12, Bold], {1.55,1.75}], White, AbsoluteThickness[4],Arrowheads[.06], Arrow[{{2,1.8}, {1.15,1.05}}]}}],  Frame->True, BaseStyle->FontFamily->Times,PlotRangePadding->0, PlotRangeClipping->True, PlotRange-> pr, ImageSize-> 500]

The error is always positive. Proof?

Figure  shows that (π/3)(t^(-2)) is an asymptotically perfect fit to E((Subscript[F, 1]))(t) as t->∞. We can use a closed-form approximation to the closest point on the path to prove that the error is asymptotically bounded above by (π/3)(t^(-2)), and then a geometric argument to show it is asymptotic to this bound.

Theorem.  E((Subscript[F, 1])(t))∼(π/3)(t^(-2)) as t->∞.

Experiment gives approximation as (π/3)(t^(-2))+(1/2)(t^(-5/2))

sqrtError=Table[{N@τ, bikeErrorPolarSquareRoot[τ, WorkingPrecision->20, PrecisionGoal-> 10]},  {τ, 2 π,  25001,20 }];

im0=Show[Plot[π/3(t^(-2)),{t,2 π,2000 },PlotStyle->{at@8,Blue}, PlotRange->All],
ListPlot[sqrtError[[1;;-1]],Frame->True,PlotStyle->{ap@4,RGBColor[1,1,0]}, PlotRange->All],PlotRange->{{0,210}, {0, .03}}, Frame->True, GridLines->{{}, {}},
FrameTicks->{{ { .03, .02, .01, .001},{}},{{2π,  50, 100,150,  200},{}}},
BaseStyle-> {FontFamily->"Times", 10}];
im1=Show[Plot[Log10[(π/3)]-2L,{L,Log10@(2π),Log10@(25000)},PlotStyle->{at@10,Blue}],ListPlot[sqrtError/. {x_, y_}:> {Log10@x, Log10@y},Mesh->All, PlotStyle->{ap@3,Yellow}],PlotRange->All,Frame->True,Axes->False, 
BaseStyle-> {FontFamily->"Times", 10},FrameTicks->{{{#,(("10")^(ToString@#))}& /@ {-2,-4,-6,-8},{}},{{#,10^#}& /@ Log10[{1,2,2π, 30,  100, 1000,25000}],{}}}];
GraphicsRow[{im0,Spacer@25, im1}]

Better:   (π/3)(t^(-2))+(2/π)(t^(-5/2))

sqrtError=Table[{N@τ, bikeErrorPolarSquareRoot[τ, WorkingPrecision->20, PrecisionGoal-> 10]},  {τ, 2 π,  25001,20 }];


im1=Show[Plot[Log10[2/π]-5/2 L,{L,Log10@(2π),Log10@(25000)},PlotStyle->{at@10,Lighter@Blue}], ListLinePlot[dL=sqrtError/. {x_,y_}:>{Log10[x],Log10[Abs[y-(π/3)(x^(-2))]]},Mesh->All,PlotRange->All,PlotStyle->{ap[3],Red}],PlotRange->All,Frame->True,Axes->False, 
BaseStyle-> {FontFamily->"Times", 10},FrameTicks->{{{#,(("10")^(ToString@#))}& /@ {-2,-4,-6,-8},{}},{{#,10^#}& /@ Log10[{1,2,2π, 30,  100, 1000,25000}],{}}}]

Proof.  We will get both upper and lower bounds on E((Subscript[F, 1]))(t). Let W=Φ(f)(t). For an asymptotic upper bound we simply have to find a good approximation to s(t). A first try is α(t)=t+2π+(cot^(-1))(Sqrt[(t/(2π))]). This is a reasonable first approximation to s(t): adding 2π moves to the next loop and the (cot^(-1)) term approximates the angle made by the two ends of the bike with the origin. In Fig. , the angle made by the tangent line at (1,0) is not perpendicular to the line from the origin to the point (the x-axis in this case), but this angle approaches a right angle as t increases. If we assume perpendicularity, then the angle made by W at the origin is (cot^(-1))(Sqrt[(t/(2π))]). Using the parameter α(t) leads to the black point in Figure , which visibly overshoots the value of s(t). (And this approximation always overshoots: this follows from the fact that the relevant dot product is asymptotic to -(1/(2t))). Note that the distance from (Subscript[F, 1])(α(t)) to W is an upper bound on the unibike error at (Subscript[F, 1])(t). This distance is asymptotic to (Sqrt[(π/2)])(t^(-3/2)), which is inadequate.

τ=2π;trueLoc=F@bikeErrorPolarSquareRoot[2π, Output->Location];
im0=Show[Graphics[{ap@5, Point[{0,0}],Line[{F[t],{0,0}, F[t]+unit@FDer[t]}/. t-> τ], 
Circle[{0,0}, 0.05, {0, .67}], 
Text["(cot^(-1))(Sqrt[(t/(2  π))])", {.2, .06}],Text["1", {1.03, .6}],Text["(Subscript[F, 1])(2 π)", {.97, .028}, {1,0}]}],
ParametricPlot[F[1], {t, 2  π, 10π}, PlotStyle-> {at@2, Blue}],
 Graphics[{ap@6, Point[F[1]/. t-> τ+2π+ArcCot[(Sqrt[τ/(2π)])]],Green,{Black, Point[F[2π]]}, Red,Arrow[{F[1],  F[1]+FDerUnit[1]}/. t-> τ]}], Axes->False, PlotRangeClipping->True,Frame->True, PlotRange->{{-.02,1.13}, {-.02, 1.06}}, PlotRangePadding-> None, BaseStyle-> {FontFamily->Times, 10}, FrameTicks->{{Automatic,{}}{Automatic,{}}}]

τ=6π;
true=bikeErrorPolarSquareRoot[τ, Output->Loc];
α[t_] := t+2 π+ArcCot[(Sqrt[(t/(2 π))])]; σ[t_] := α[t]-(1/(4π))((((t/(2π))))^(-2))- ((1/(32(π^2))))((((t/(2π))))^(-5/2));
trueMin=F[true];
frontWheel=FFront[τ];
arccotApprox=F[α[τ]];
Show[ParametricPlot[F[t], {t, 4π, 11π}, PlotStyle-> {at@2, Blue}],
Graphics[{ 
{Black, ap@6,Point[F[α[τ]]]},
{Red,ap@5,Point[F[σ[τ]]]},
Circle[F[σ[τ]], .003],  {Black,ap@5, Point[trueMin]},{Red, ap@4, Point[FFront[τ]]}}], Frame->True, PlotRange->{{1.68,1.8}, {.96, 1.05}}, PlotRangePadding->{}]

Studying the error in this first approximation leads to the improvement α(t)-(1/(4π))((((t/(2π))))^(-2)). This leads to an upper bound of (Sqrt[((π^2)/9)+(1/64)])(t^(-2)), which is a nice upper bound, but the constant is a little too large. Studying the error of the improvement leads to further improvement to σ(t)=α(t)-(1/(4π))((((t/(2π))))^(-2))-((1/(32(π^2))))((((t/(2π))))^(-5/2)). One can go farther, but σ(t) suffices for the proof. Let δ be the distance from (Subscript[F, 1])(h(t)) to (Subscript[F, 1])(t)+((F'(t))/(|F'(t)|)). Then δ bounds the error and one can examine the product (t^4)(δ^2), which is:

-((t^(3/2))/(8 (Sqrt[2 π])))+((t^5)/π)+2 (t^4)+(((t^4) (cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))))/(2 π))-((t^2)/2)+(((Sqrt[(2/π)]) (t^(9/2)))/(Sqrt[4 (t^2)+1]))-(((t^(9/2)) (Sqrt[-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π])))+16 π]) cos((1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))))))/(2 (Sqrt[2]) π))+(((t^5) (Sqrt[-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π])))+16 π]) sin((1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))))))/(Sqrt[4 π (t^2)+π]))-(((t^4) (Sqrt[-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π])))+16 π]) cos((1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))))))/(2 (Sqrt[π]) (Sqrt[4 (t^2)+1]))).

Mathematica computes this expression to be ((π^2)/9)+O((1/t)), which gives for the desired upper bound. To do this by hand......   use approximations for all the terms, recursive on operations.   replace terms is sums by things that have a limit-difference of 0. 
An alternative to using σ(t) is to use the intersection point of the curve with the polar line from the origin to W. 

For the lower bound, let P be the point on the curve determined by α(t) from the upper bound proof, let Q be the point on the tangent line L at P that is closest to W, and let R be the closest point on the curve to W (Fig. ). There is a right angle at Q and the distance from W to Q is less than the length of the hypotenuse of the right triangle determined Q, W, and the line WR. That hypotenuse is shorter than WR, so we have the lower bound WQ. Using the standard formula for the distance from a point to the line, this bound is

(((Sqrt[2]) (Sqrt[4 (t^2)+1]) (t^2)+(Sqrt[4 π (t^2)+π]) (Sqrt[t])+(Sqrt[2]) π (2 t ((Sqrt[4 (t^2)+1])+2 t)+1)+(cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))) ((Sqrt[2]) (Sqrt[4 (t^2)+1]) (t-(Sqrt[t+2 π]) (Sqrt[t+(cot^(-1))(((Sqrt[t])/(Sqrt[2 π])))+2 π]))+4 (Sqrt[2]) π t+2 (Sqrt[π]) (Sqrt[t]))-(Sqrt[2]) (((t+2 π))^(3/2)) (Sqrt[4 (t^2)+1]) (Sqrt[t+(cot^(-1))(((Sqrt[t])/(Sqrt[2 π])))+2 π])+8 (Sqrt[2]) (π^2) t+4 (π^(3/2)) (Sqrt[t]))/((Sqrt[π]) (Sqrt[t+2 π]) (Sqrt[4 (t^2)+1]) (Sqrt[4 (((t+2 π))^2)+4 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))) (2 t+(cot^(-1))(((Sqrt[t])/(Sqrt[2 π])))+4 π)+1])))

and this (using Mathematica) is (π/3)(t^(-2))+O((t^(-5/2))). So we have the desired lower bound.  Now, there are some geometrical points to check: (1) the error RW is less than the distance from W to the closest outer loop; (2) the tangent L strikes  WR between W and R. Surely these can be done. I did the second using the “LeftOf” formula for point and line. That is: W is left of PQ. We also have convexity of the curve. And a recursive approach to asymptotics shows the (t^(-2)) formula.  □

Corollary ((E((Subscript[F, 1]))(t))/(E((Subscript[F, 1]))(t+2π)))∼1+((4π)/t).   Also  1+((4π)/t)+((4(π^2))/(t^2))

Show[ListPlot[Table[Log10@{τ,((bikeErrorPolarSquareRoot[τ])/(bikeErrorPolarSquareRoot[τ+2 π]))-1},{τ,2 π,50 π,π}], PlotRange->All],
ListLinePlot[Table[Log10@{τ,((4π)/τ)},{τ,2 π,50 π,π}],PlotStyle->Red, PlotRange->All], Frame->True, PlotRange->All]

((π/3)(t^(-2)))/((π/3)(((t+2π))^(-2)))//Apart

τ=2π;
α[t_] := t+2 π+ArcCot[(Sqrt[(t/(2 π))])];
t0=bikeErrorPolarSquareRoot[2π, Output->Loc];
trueMin=F[t0];
frontWheel=FFront[τ];
arccotApprox=F[α[τ]];
tangent=-FDer[α[τ]]//N;
new=(arccotApprox+s tangent)/. FindRoot[((tangent).(frontWheel-(arccotApprox+s tangent))), {s,1}];
S=RegionIntersection[{Line[{arccotApprox, new}], Line[{frontWheel, trueMin}]}][[1]];
im0=Show[Graphics[{ap@4,Line[{F[1],{0,0},frontWheel}];  
Text["(cot^(-1))(Sqrt[(t/(2  π))])", {.2, .06}],Text["1", {1.03, .6}], Text["P", arccotApprox+{.002,.001}], tW=Text["W", frontWheel+{.0001,.0027}],tQ=Text["Q", new+{.003,-.0003}], tR=Text["R", trueMin+{-.0022,-.001}]}],
ParametricPlot[F[1], {t, 2  π, 10π}, PlotStyle-> {at@2, Blue}],
 Graphics[{ap@6,Point[arccotApprox],Green, Point[new],  Red,Arrow[{F[τ],  FFront[τ]}/. t-> τ],
  Line[{F[α[τ]],F[α[τ]]-FDer[α[τ]]}],
ap@8,Point[qq=FFront[τ]],
{Black,at@1, Darker@Green, Line[{trueMin,frontWheel,new}], {Black, ap[4],Point[arccotApprox], Darker@Black,Point[new],Black, Point[trueMin], Point[S]}}

}], Axes->False, PlotRangeClipping->True,FrameTicks->{{Automatic, {}},{Automatic, {}}},Frame->True, PlotRange->{{1.025,1.084}, {.98, 1.035}}, PlotRangePadding-> None, BaseStyle-> {FontFamily->Times, 10}];
im1=Show[im0/.tQ->Text["Q", new+{-0.0001,-.0005}]/. tR->Text["R", trueMin+{-.0003,-.0003}]/.
tW->Text["W", frontWheel+{-.00045,.0002}],  Graphics[{at@.7, Line[{Q1=new+.00035 unit@ (frontWheel-new),Q2=Q1+.00035 unit @(new-arccotApprox), Q2-.00035 unit@ (frontWheel-new)}]}], PlotRange->{{1.065,1.08}, {.987, .9978}}];
GraphicsRow[{im0, im1}]

distancePointLine[{{x1_, y1_},{x2_, y2_}}, {x0_,y0_}]:= (((x2-x1)(y1-y0)-(x1-x0)(y2-y1))/(Sqrt[(((x2-x1))^2)+(((y2-y1))^2)]));

(* lower bound proof *) raw=FullSimplify[PowerExpand[Simplify[Simplify[TrigExpand[distancePointLine[{F[α[t]],F[α[t]]+FDer[α[t]]},FFront[t]]]]/. (Sqrt[(1/t)+4 t])->((Sqrt[1+4 (t^2)])/(Sqrt[t]))/. (Sqrt[1+((2 π)/t)])->((Sqrt[t+2 π])/(Sqrt[t]))/. (1/(Sqrt[(1/t)+4 t]))->((Sqrt[t])/(Sqrt[1+4 (t^2)]))]]]/. 1/(Sqrt[1+((2 π)/t)])-> (Sqrt[t])/(Sqrt[t+2 π])/. (Sqrt[2+8 (t^2)])-> (Sqrt[2])(Sqrt[(1+4(t^2))])
N[raw/. t->2 π]
Series[raw,{t,∞,2}]

FullSimplify[(Sqrt[frontWheel.frontWheel])-(Sqrt[arccotApprox.arccotApprox]), τ>2π]

Simplify[(Sqrt[FFront[t].FFront[t]])-(Sqrt[arccotApprox.arccotApprox]), τ>2π]

Code for the good approximation

(* FFrontSer[t_]:=Evaluate[series[FFront[t],1]]; *) 
(* h[t_]:=Evaluate[Identity[t+2 π+ArcCos[(((a^2)+(b^2)-(c^2))/(2 a b))]/. c->1/. b->(Sqrt[(t/(2 π))])/. a->(Sqrt[FFront[t].FFront[t]])]]; *)

σ[t_]:=t+2 π+ArcCot[(Sqrt[(t/(2 π))])]-(1/(t^2))(π+(1/8)(Sqrt[2π])(1/(Sqrt[t])));
δSquare=Simplify[(F[σ[t]]-FFront[t]).(F[σ[t]]-FFront[t])/. {(Sqrt[4 t+(1/t)])->((Sqrt[4 (t^2)+1])/(Sqrt[t])),(1/(Sqrt[4 t+(1/t)]))->((Sqrt[t])/(Sqrt[4 (t^2)+1]))}/. (((a_+b_))^2):>(a^2)+(b^2)+2 a b/. ((Sin[z_])^2):>1-((Cos[z])^2)]
goodPolarSquareRootErrorApproximation=δSquare^(1/2);
series[δSquare,4]

The preceding establishes the asymptotic behavior. So we multiply by (t^4) and examine the limit.

δProd=Expand[δSquare (t^4)];
N[{δProd/. t->(10^10), (π^2)/9},20]
δProd//TraditionalForm
limit∞[δProd]

This below gives an error bound of 1/(Sqrt[t]) so constitutes a proof!!

δProd1=Expand[δProd];
Series[δProd1, {t,∞,0}]//Simplify

This breaks down into a sum of 9 terms.

tryList=List@@δProd1
tryList//Length

We replace each term by its asymptotic value and add them up: the result is (π^2)/9. This is legal because in each case approximation - summand -> 0.

ser = series[tryList, 0]
Total[ser]//Expand

A Proof by Hand

The last one:

-(5/64)-((19 (π^2))/3)+(1/2) (Sqrt[(π/2)]) (Sqrt[t])-(92/15) (Sqrt[2]) (π^(5/2)) (Sqrt[t])+((5 π t)/2)+(8/3) (Sqrt[2]) (π^(3/2)) (t^(3/2))+((t^2)/2)-2 (t^4) - tryList[[9]]//limit∞

-(5/64)-((19 (π^2))/3)+(1/2) (Sqrt[(π/2)]) (Sqrt[t])-(92/15) (Sqrt[2]) (π^(5/2)) (Sqrt[t])+((5 π t)/2)+(8/3) (Sqrt[2]) (π^(3/2)) (t^(3/2))+((t^2)/2)-2 (t^4) - tryList[[9]]//Expand

(* limit∞[try]//Timing *) 
series[try,0]//Timing
tryList=List@@try
tryList//Length

ser = series[#, 0]& /@ tryList
Total[ser]//Expand

Limit[Table[Together[Expand[ser[[i]]-tryList[[i]]]],{i,9}], t-> ∞]

Great. But the problem is I really do not understand the algorithm unerlying Series, so I need to check these 9.  First 5 are triviual.  ANd one can do the rset using approximations.

Code for the two weak approximations

σ0[t_]:=t+2 π+ArcCot[(Sqrt[(t/(2 π))])];
δSquare=Simplify[(F[σ0[t]]-FFront[t]).(F[σ0[t]]-FFront[t])/. (((a_+b_))^2):>(a^2)+(b^2)+2 a b/. ((Sin[z_])^2):>1-((Cos[z])^2)];
δ=(Sqrt[δSquare]);
Asymptotic[δSquare,{t,∞,3}]
δProd=Expand[Expand[δSquare (t^3)]/. ((Sin[z_])^2):>1-((Cos[z])^2)]/. (Sqrt[1+((2 π)/t)])->((Sqrt[t+2 π])/(Sqrt[t]))/. (1/(Sqrt[(1/t)+4 t]))->((Sqrt[t])/(Sqrt[1+4 (t^2)]))
N20[δProd]

Logic of proof.  We wish to prove the limit of the sum of the 10 terms is π/2. We can replace each summand A above by B, provided A-B->0 without changing the limit of the sum. The following does this and the resulting expression clearly converges to π/2.

δProd1=Apart[(∑_i^10)If[i≤3,δProd[[i]],series[δProd[[i]],0]]]

term #4

i=4;
try=Together[δProd[[i]]-series[δProd[[i]],0]]
limit∞[num=((Numerator[try])/((Sqrt[2π])(t^2)))]
limit∞[Simplify[((Denominator[try])/((Sqrt[2π])(t^2)))]]

The numerator limit above can be proved to vanish by conjugation. Then the denominator of the result -> ∞ and numerator to a constant. Easier:  Use 0th approximation, as it is a sum.

Apart[num]

series[#,0]& /@ List@@Apart[num]

((Sqrt[t]) (Sqrt[1+4 (t^2)]))-(series[(Sqrt[t]) (Sqrt[1+4 (t^2)]),0])//conjugate//Simplify//conjugate

term #5

i=5;
try=Together[δProd[[i]]-series[δProd[[i]],0]]
limit∞[num=((Numerator[try])/(16 (Sqrt[2]) π (t^2)))]
limit∞[Simplify[((Denominator[try])/(16 (Sqrt[2]) π (t^2)))]]

So the quotient -> 0. Proof:

series[((Denominator[try])/(16 (Sqrt[2]) π (t^2))),0]
series[((Numerator[try])/(16 (Sqrt[2]) π (t^2))),0]

The numerator limit above can be proved to vanish by conjugation. Then the denominator of the result -> ∞ and numerator to a constant. Easier:  Use 0th approximation, as it is a sum.

Apart[num]

series[#,0]& /@ List@@Apart[num]//Total//Together

series[#,0]& /@ List@@Apart[num]

conjugate[#-series[#,0]]&/@ List@@Rest[Apart[num]]//Simplify//conjugate

Clearly all summands go to 0 when made asymptotic.

term #6

i=6;
try=Together[δProd[[i]]-series[δProd[[i]],0]]
limit∞[num=Numerator[try]/(2 π+t)]
limit∞[Simplify[Denominator[try]/(2 π+t)]]

So the quotient -> 0. Proof:

((Denominator[try])/(2 π+t))

series[((Numerator[try])/(2π+t)),0]

Apart[num]

limit∞[ArcCot[(Sqrt[t])]-1/(Sqrt[t])]
limit∞[t ArcCot[(Sqrt[t])]-(Sqrt[t])]
limit∞[(t^2) ArcCot[(Sqrt[t])]-((t^(3/2))-((Sqrt[t])/3))]

We need to prove the three above and they allow the whole thing to simplify to 0. 

Lemma 1. lim (cot^(-1))(t)-(1/t)=0. Proof. Same  as (cot^(-1))((1/t))-t->0 as t->0. Easy by derivative/Taylor
Lemma 2. lim  (t^2)(cot^(-1))(t)-t=0. Proof. Same  as (1/(t^2))(cot^(-1))((1/t))-(1/t)->0 as t->0. Easy by l’Hopital
Lemma 3. lim  (t^4)(cot^(-1))(t)-((t^3)-(t/3))=0. Proof. Same  as ....   as t->0. Easy by l’Hopital.

limit∞[(t^4) ArcCot[t]-((t^3)-(t/3))]
Simplify[(∂_({t,4}))Numerator[Together[Together[(t^4) ArcCot[t]-((t^3)-(t/3))]/. t->(1/t)]]]//Expand

limit0[((ArcCot[(1/t)])/(t^4))-(1/t)]

D[((ArcCot[(1/t)])/(t^2))-(1/t)//Together//Numerator, {t,2}]//Simplify

Proof. Take cot.   t-cot(1/t), or t-tan t. t-sin(t)/cos(t). But t-sin t -> 0 and cost t -> 1

(t^2)ArcCot[t] - t//limit∞
(t^2)ArcCot[t] - t//limit∞

limit∞[t ArcCot[t]-1]
limit∞[(t^2) ArcCot[t]-t]
limit∞[(t^(-1)) ArcCot[t]-1/(t^2)]
limit∞[(((2π+t))^(-1)) ArcCot[t]-1/(t^2)]

num/.ArcCot[((Sqrt[t])/(Sqrt[2 π]))]-> series[ArcCot[((Sqrt[t])/(Sqrt[2 π]))],2]//N20

num/. Table[(t^i) ArcCot[((Sqrt[t])/(Sqrt[2 π]))]->(((Sqrt[2 π]) )/(Sqrt[t]))(t^i),{i,0,0}]//N20

series[#,0]& /@ List@@Apart[num]
series[#,0]& /@ List@@Apart[num]//Total//Expand

series[ArcCot[(Sqrt[t])],2]

limit∞[series[ArcCot[(Sqrt[t])],2]-ArcCot[(Sqrt[t])]]

Together[Simplify[(∂_({t,2}))Numerator[Together[Expand[((Together[ArcCot[(Sqrt[t])]-((-1+3 t)/(3 (t^(3/2))))])^2)]]]]]//N20

List @@Expand[((Together[ArcCot[(Sqrt[t])]-((-1+3 t)/(3 (t^(3/2))))])^2)]

All these -> 0. So the series subst. is legal.

Therefore  (cot^(-1))((Sqrt[t]))-1/(Sqrt[t]) -> 0

limit∞[ArcCot[t]-1/t]
limit∞[t ArcCot[t]-1]
limit∞[(t^2) ArcCot[t]-t]
limit∞[(t^(-1)) ArcCot[t]-1/(t^2)]
limit∞[(((2π+t))^(-1)) ArcCot[t]-1/(t^2)]

series[#,0]& /@ List@@Apart[num]

conjugate[#-series[#,0]]&/@ List@@Rest[Apart[num]]//Simplify//conjugate

Clearly all summands go to 0 when made asymptotic.

δProd1=Together[δProd/. (((t^(i_. 1)) ArcCot[((Sqrt[t])/(Sqrt[2 π]))])/(w_)):>series[(((t^i) ArcCot[((Sqrt[t])/(Sqrt[2 π]))])/w),0]/. a_ (Sqrt[2 π+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]):>series[a (Sqrt[2 π+t+ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]),0]]/. (Sqrt[1+4 (t^2)])-> series[(Sqrt[1+4 (t^2)]),1]/. 1/(Sqrt[1+4 (t^2)])-> series[1/(Sqrt[1+4 (t^2)]),1]//Expand
limit∞[List@@ δProd1]

To be certain we want to understand why each limit substitution is legal.

σ0[t_]:=t+2 π+ArcCot[(Sqrt[(t/(2 π))])]-(1/(4π))((((t/(2π))))^(-2));
δSquare=Simplify[(F[σ0[t]]-FFront[t]).(F[σ0[t]]-FFront[t])/. (((a_+b_))^2):>(a^2)+(b^2)+2 a b/. ((Sin[z_])^2):>1-((Cos[z])^2)];
δProd=Expand[Expand[δSquare (t^4)]/. ((Sin[z_])^2):>1-((Cos[z])^2)];
(Sqrt[N20[δProd]])
Simplify[series[δProd, 0], t>0]
N20[δSquare-(((1/64)+((π^2)/9))/(t^4))]
(Sqrt[δSquare])-(Sqrt[((1/64)+((π^2)/9))])(t^(-2))//limit∞

tryList[[1;;5]]
ser[[1;;5]];
Expand[Expand[FullSimplify[Together[tryList[[5]]-ser[[5]]],t>0]]/. (((Sqrt[(2/π)]) (t^(9/2)))/(Sqrt[1+4 (t^2)]))->series[(((Sqrt[(2/π)]) (t^(9/2)))/(Sqrt[1+4 (t^2)])),0]]

tryList[[6]]-ser[[6]]
Numerator[Together[tryList[[6]]-ser[[6]]]]/. t^4ArcCot[z_]:>t^4 series[ArcCot[z], 4]//Expand

lHopital does it

ee=Simplify[(t^4) ArcCot[((Sqrt[t])/(Sqrt[2 π]))]-t^4 series[ArcCot[((Sqrt[t])/(Sqrt[2 π]))],4], t>2π]
Simplify[(∂_({t,4}))Numerator[Together[Simplify[Together[ee],0<t]]]]/. t-> 0

needed=(Expand[Simplify[Expand[Simplify[Expand[Simplify[try[[{7,8,9}]]-series[try[[{7,8,9}]],0]]]/. ArcCot[((Sqrt[t])/(Sqrt[2 π]))]->series[ArcCot[((Sqrt[t])/(Sqrt[2 π]))],4]/. Cos[z_]:>series[Cos[z],5]/. Sin[z_]:>series[Sin[z],5]/. (Sqrt[16 π-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+((8 (Sqrt[2 π]) (-120 (π^3)+84 (π^2) t-70 π (t^2)+105 (t^3)))/(105 (t^(7/2))))])->series[(Sqrt[16 π-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+((8 (Sqrt[2 π]) (-120 (π^3)+84 (π^2) t-70 π (t^2)+105 (t^3)))/(105 (t^(7/2))))]),4]/. (1/(Sqrt[1+4 (t^2)]))->series[(Sqrt[1+4 (t^2)]),0]]]/. (1/(Sqrt[1+4 (t^2)]))->series[(1/(Sqrt[1+4 (t^2)])),3]]]/. t-> 1/t//Simplify[#, t>0]&)/. t-> 0
(N[#1/. t->(10^200),20]&)[needed]
limit∞[List@@needed]

So all things work with more uses of series in legal ways.

Cases[+((121329 (π^(11/2)))/(8960 (Sqrt[2]) (t^(11/2)) (Sqrt[1+4 (t^2)]))), 1/(Sqrt[z_]), ∞]
+((70943 (π^3))/(1260 (Sqrt[1+4 (t^2)])))/. 1/(Sqrt[1+4t^2])-> 999

;

(List@@needed)[[7;;-1]]

tryList[[7]]-ser[[7]]/.- (((t^(9/2)) (Sqrt[16 π-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]) Cos[(1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 ArcCot[((Sqrt[t])/(Sqrt[2 π]))])])/(2 (Sqrt[2]) π))-> series[((-(t^(9/2)) (Sqrt[16 π-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]) Cos[(1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 ArcCot[((Sqrt[t])/(Sqrt[2 π]))])])/(2 (Sqrt[2]) π)),0]

needed=- (((t^(9/2)) (Sqrt[16 π-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]) Cos[(1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 ArcCot[((Sqrt[t])/(Sqrt[2 π]))])])/(2 (Sqrt[2]) π))-series[- (((t^(9/2)) (Sqrt[16 π-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 ArcCot[((Sqrt[t])/(Sqrt[2 π]))]]) Cos[(1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 ArcCot[((Sqrt[t])/(Sqrt[2 π]))])])/(2 (Sqrt[2]) π)),0];
needed//limit∞

needed

-

Numerator[Together[tryList[[6]]-ser[[6]]]]/. t^4ArcCot[z_]:>t^4 series[ArcCot[z], 4]//Expand

&

series[ArcCot[(Sqrt[t])/(Sqrt[2π])], 2]

try1=tryτ/. ArcCot[z_]:>series[ArcCot[z],3]//Simplify
try1=try1/. {Cos[z_]:>series[Cos[z],5],Sin[z_]:>series[Sin[z],5]};
ddlimit∞[try1];
N20[(Sqrt[try1 9]), 10000000000]

2+(t/π)+(((cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))))/(2 π))-(1/(8 (Sqrt[2 π]) (t^(5/2))))+(((Sqrt[(2/π)]) (Sqrt[t]))/(Sqrt[4 (t^2)+1]))-(1/(2 (t^2)))-((((Sqrt[2]) (Sqrt[t]) (Sqrt[4 (t^2)+1])+2 (Sqrt[π])) (Sqrt[-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π])))+16 π]) cos((1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))))))/(4 π (Sqrt[4 (t^2)+1])))+((t (Sqrt[-((Sqrt[2 π])/(t^(5/2)))-((8 π)/(t^2))+8 t+8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π])))+16 π]) sin((1/8) (((Sqrt[2 π])/(t^(5/2)))+((8 π)/(t^2))-8 (cot^(-1))(((Sqrt[t])/(Sqrt[2 π]))))))/(Sqrt[4 π (t^2)+π])).

Mathematica can evaluate the limit of this as t->∞ and it is ((π^2)/9).  User Series. 
One can confirm this because it is a sum with 9 terms etc. , with a lot of algebraic work, confirm that this limit is correct. Having (t^4)(δ^2)->((π^2)/9) means that δ is asymptotically bounded by (π/3)(t^(-2)).

Lower bound: Use tangent. See lower bound file.

3.  Computing the Error

The minimal distance from a point P to a curve f(t) can be quickly computed using the fact that when Q is the closest point on the curve, the line PQ makes a right-angle with the tangent to the curve. Therefore the optimal value of the parameter t can be obtained by root-finding on the dot product equation (P-f(t))·f'(t)=0. This is fast, but there are many solutions and therefore it is critical to start with a value close to the optimal value.

4.  The Rear Track

Given the small error for the polar square root, it is natural to try to improve the curve in some way so as to reduce the error. The hope is to find an iterative process that starts with (Subscript[F, 1]) and gets curves (Subscript[F, n]), with domain [2π,∞), for which the unibike error approaches 0 and the curves approach a limiting curve, (Subscript[F, ∞]). Then (Subscript[F, ∞]) will be a unibike curve and, one hopes, in the shape of a spiral with no self-intersection). We describe such a process here. It is not clear that there is a closed form for such a zero-error curve, but our work here is aimed at seeing if a spiral unibike curve exists. This work was inspired by some recent attempts in the same direction by Bob Bixler.

It is easy to get the front track from the rear track: just add the unit tangent. Figure  shows the front curve defined from the polar square root. Iterating this process is not useful because a loop is lost at each iteration and there is no limiting curve. But it is of interest to look at the error: it gets worse. For example, the error for the polar square root when t corresponds to the point closest to the first point on the front curve is 0.0022. The error for the front curve at its first point is 0.0038, about 70% larger. So moving forward does not yield anything useful. (One can try other ideas to make use of the front curve, such as rotating it and translating it so  that it starts at (1,0), but nothing useful arises.)

Show[ParametricPlot[F[t],{t,2 π,10 π},PlotStyle->{at[6],Blue,CapForm[None]}],ParametricPlot[FFront[t],{t,2 π,8 π-.49},PlotStyle->{CapForm[None],at[1.5],Yellow}],Graphics[{Red,Arrow[{F[2 π],FFront[2 π]}]}],Frame->True,FrameTicks->{{Automatic,{}},{Automatic,{}}},Axes->None,BaseStyle->FontFamily->Times, PlotRangePadding->None, PlotRange-> {{-2.3,2.39}, {-2.35,2.26}}]

Code for front error; needed for previous figure

Error for polar root at t = 19.45

G=Simplify[F[1]+FDerUnit[1]];
dot=Numerator[Together[FullSimplify[(∂_t)F[1].((G/. t->4 π)-F[1])]]];
tVal=t/. FindRoot[0==dot,{t,19}]
dot1=Numerator[Together[FullSimplify[(∂_t)F[1].((G/. t->tVal)-F[1])]]];
tVal1=t/. FindRoot[0==dot1,{t,19+7}];
Norm[(G/. t-> tVal)-( F[1]/.t-> tVal1)]

Error for front curve at 4π

GFront=G+unit@D[G,t];
dot=Simplify[((∂_t)G).((GFront/. t->4 π)-G)];
tVal=t/. FindRoot[0==dot,{t,6π}]
Norm[(GFront/. t->4 π)-(G/. t-> tVal)]

To move backward we need to solve a differential equation. The equation is quite simple (see [, chap. 18, or ]; it was first derived by Dunbar et al []. Suppose F(t) is the front curve and we seek the rear curve R(t). Note that this is the physically natural direction to consider because a bicyclist steers the front wheel and the rear wheel follows behind according to geometry. This idea applies also to cars, trucks, and busses where again the steering is at the front and the rear follows. The rear wheel’s velocity vector always points to the location of the front wheel. It follows that the speed of the rear wheel is the magnitude of the velocity vector of the front wheel in the direction of the line connecting the two wheels. That leads to this differential equation:

R'(t)=F'(t)·(R(t)-F(t)) [R(t)-F(t)]

I could insert a diagram explaining the DE (from VDS?)

t0=t/.FindRoot[(F[t]-{1,0}).(F[t]-{1,0})==1, {t, 4π+ArcCot[1]}];
Show[ParametricPlot[ F[t], {t, 2π, 4.5π}, PlotStyle->{Blue, at[1]}],
Graphics[{{Dashed, Circle[{1,0},1, {1,2.2}]},
Text["(Subscript[F, 1])(2π)", {.93, .03}],Text["(Subscript[F, 1])((Subscript[t, 0]))", {.937,1.035}],
{Arrowheads[.025],  at@1.1,Arrow[{{.981,1.033},{1.043,1.007}}]},
Red, ap[5], Point[{1,0}],Arrow[{{1,0},FFront[2π]}],
EdgeForm[Blue], FaceForm[Red], Disk[FFront[2π], 0.01],
 EdgeForm[Black], FaceForm[RGBColor[0.5,0.5,1]], Disk[F[t0], .01]}],  Frame->True,PlotRangePadding-> 0, PlotRange-> {{0, 1.2}, {-.1, 1.2}},ImageSize-> 400, GridLines->{{}, {0}},BaseStyle->FontFamily->Times, GridLinesStyle->Directive@{Black, AbsoluteThickness@.6}]

We can solve the equation numerically over a large t-domain and because we want our tracks to always start at (1,0) we will use the initial condition R((Subscript[t, 0]))=(1,0). Mathematica’s methodology for solving differential equations numerically presents the solution as an interpolating function over the requested domain, interpolating by piecewise cubics with matching derivatives at the junctions (Hermite interpolation). We will want to solve this up to a large value, (Subscript[t, max]). To find the initial parameter (Subscript[t, 0]), we need (Subscript[t, 0]) such that the distance from F((Subscript[t, 0])) to (1,0) is one unit exactly (see Fig. ). For (Subscript[F, 1]) this is easily found by root-finding and it is (Subscript[t, 0])=13.3217. Note that the initial value for this root-finding can be taken to be 4π+(cot^(-1))(1) (about 13.35); the 4π determines the loop on which (Subscript[t, 0]) lies and the (cot^(-1)) is an approximation based on the approximate right triangle determined by the (0,0), (Subscript[F, 1])(2π), and the front point Φ((Subscript[F, 1]))(2π). Here is complete code to get the rear track.

F1=(Sqrt[(t/(2 π))]) {Cos[t],Sin[t]}; F1Der=(∂_t)F1;
F1DerUnit=(F1Der/(Sqrt[F1Der.F1Der])); F1Front=F1+F1DerUnit;
t0=t/. FindRoot[1==(F1-{1,0}).(F1-{1,0}),{t,4 π+ArcCot[1]}]
R[t_]:={Rx[t],Ry[t]};
sol=NDSolveValue[{R'[t]==(F1Der.(R[t]-F1Front)) (R[t]-F1Front),R[t0]==(F1/. t->t0)},R[t],{t,2π,40}];
Show[ParametricPlot[F1, {t, t0, 8π}, PlotStyle->{AbsoluteThickness[6], Blue}],ParametricPlot[sol,{t,2π,8π}, PlotStyle->{AbsoluteThickness[2],Red}], Frame->True, Axes->False]

Fig XX Style

The solution, (Subscript[F, 2])(t) (and we will iterate the process to get {(Subscript[F, n])}), will be an interpolating function with domain [(Subscript[t, 0]),(Subscript[t, max])] and starting at (1,0). But we must reparametrize the solution because we want all our curves (Subscript[F, n])(t) to be such that (Subscript[F, n])(2π)=(1,0). And the reparametrization must be done in such a way that the t-value indicates the angle, including the number of loops. That aspect of the solution is critical to the task of computing the error: the distance from Φ((Subscript[F, n]))(t) to the nearest point on (Subscript[F, n])(t). But such a reparametrization is not a big problem. (If one cares only about the final curve, say (Subscript[F, 200]), the reparametrization step is not necessary.)

We will want to iterate this process, but let us first learn some critical properties of (Subscript[F, 2]). We cannot prove the validity of these observations, but they are computationally solid, have heuristic explanations (???), and form the basis for the work to come. Figure  shows that E((Subscript[F, 2])) is monotonically decreasing (as is E((Subscript[F, 1]))): its maximum occurs at t=2π. More important, the error for (Subscript[F, 2]) is significantly less than the error for (Subscript[F, 1]): the max errors are 0.0137 for (Subscript[F, 1]) and 0.0058 for (Subscript[F, 2]), a reduction by more than half. The asymptotic behavior appears to be the same: (π/3)(t^(-2)); this means the reduction ratio converges to 1. But the fact that the maximum error decreases so sharply is very promising.

dd=Table[{τ,bikeErrorBetter[FInt[2],FIntDer[2],{t,τ}]},{τ,2 π,150 π,(π/2)}];

dd1[[1,2]]/dd[[1,2]];
dd1=Table[{τ,bikeErrorPolarSquareRoot[τ]}, {τ, 2π, 150π, π/2}];
im1=Show[ListLinePlot[dd1, PlotStyle->{at@5,Lighter@Red}, Frame->True, PlotRange->All],
ListLinePlot[dd, Frame->True, PlotStyle->{at@2,Blue},PlotRange->All], BaseStyle->FontFamily->Times, FrameTicks->{  { Automatic,{}}, {{2π, 100π,50π,  150π},{}}},PlotRange->{-.0005,.015}];
im2=Show[ListLinePlot[ddL1=Log10@dd1, PlotStyle->{at@5,Lighter@Red}], 
ListLinePlot[ddL=Log10@dd,PlotStyle->{at@2,Blue}],FrameTicks->{  {{#, (("10")^(ToString@#))}& /@ {-4, -6, -5,-3, -2, -1},{}}, {{#, 10^#}& /@ Log10@{2π,10π,50π,  150π},{}}}, PlotRange-> {-5.38, -1.8},Frame->True,BaseStyle->FontFamily->Times];
GraphicsRow[{im1, im2}]
Fit[ddL[[-5;;-1]], {1,L}, L];

Clear[e];e[1]=(π/(3 ))(t^(-2))+ 0 (2/π)(t^(-5/2));  (* compare this way out there?  but not too important really *) 
e[n_]:=e[n]=(e[n-1]/. t->t+2 π) (1+(π/t));
n=2
e[2]

n=2; Show[ListLinePlot[Log10[errorTable[n,50]],PlotStyle->{at[6],Lighter[Blue]}],ParametricPlot[
Evaluate[Log10[{t,e[n]}]], {t, 2π, 102π}, PlotStyle->{at@2,Red}], Frame->True, PlotRange->{-5.1, -2}, Axes->False, FrameTicks->{  {{#, (("10")^(ToString@#))}& /@ {-4, -6, -5,-3, -2, -1},{}}, {{#, 10^#}& /@ Log10@{2π,5π, 10π,50 π, 100π},{}}}, BaseStyle->FontFamily -> Times]

It is a little jaggy at 100 π. Can I improve?

n=40;
Show[ListLinePlot[Log10[errorTable[n,50]],PlotStyle->{at[6],Lighter[Blue]}],ParametricPlot[
Evaluate[Log10[{t,e[n]}]], {t, 2π, 102π}, PlotStyle->{at@2,Red}], Frame->True, PlotRange->{-5.1, -2.1}, Axes->False];

ListLinePlot[dL=Table[{Log10[2 π j], Log10[Abs[e[n] - errorTable[n,50][[j,2]] /. t-> 2π j]]}, {j, 1, 10}],Mesh->All,PlotStyle->{at[2],Lighter[Blue]}, Frame->True, PlotStyle->All]
Fit[dL[[-3;;-1]], {1,L},L]

e[2]

So we have that E((Subscript[F, n]))(t)∼E((Subscript[F, n-1]))(t+2π) (1+(π/t)). The fit is excellent.

10^-.5

ρ=(((π/3) (t^(-2)))/((π/3)  (((t+2 π+ ArcCot[(Sqrt[(t/(2 π))])]))^(-2))));
ρ=(((π/3) (t^(-2)))/((π/3)  (((t+2 π+ 1/(Sqrt[(t/(2 π))])))^(-2))));
ρ=(((π/3) (t^(-2)))/((π/3)  (((t+2 π))^(-2))));
Apart[FullSimplify[ρ]]
series[ρ,2];

τ=10^3 2 π;
N[ρ/. t-> τ,20]
((bikeErrorPolarSquareRoot[τ,WorkingPrecision-> 40,PrecisionGoal->20])/(bikeErrorPolarSquareRoot[τ+2π,WorkingPrecision-> 40,PrecisionGoal->20]))
N[1+((4π)/τ), 20]

τ=100π;
((bikeErrorBetter[FInt[2], FIntDer[2], {t, τ},WorkingPrecision-> 40,PrecisionGoal->20])/(bikeErrorPolarSquareRoot[τ+2π,WorkingPrecision-> 40,PrecisionGoal->20](1+(π/τ))))
((bikeErrorBetter[FInt[1], FIntDer[1], {t, τ},WorkingPrecision-> 40,PrecisionGoal->20])/(bikeErrorPolarSquareRoot[τ+2π,WorkingPrecision-> 40,PrecisionGoal->20](1+((4π)/τ))))

Let us examine E((Subscript[F, 2]))(t) more closely and see how it relates to E((Subscript[F, 1]))(t). Figure XX shows the polar square root in blue, but starting only at (Subscript[t, 0])=13.3217. The red curve is (Subscript[F, 2]), which is Ω((Subscript[F, 1])). The lower arrow shows the bike for (Subscript[F, 2]) at the start, while the upper is the same for (Subscript[F, 1]) at 13.3217. If (Subscript[F, 1]) were a  unibike curve, (Subscript[F, 2]) would equal (Subscript[F, 1]). They are not equal but they are close. Suppose they were equal. What would the connection be between E((Subscript[F, 1]))(2π) and E((Subscript[F, 1]))((Subscript[t, 0]))?  Using Theorem XX the error ratio ρ is close to 

     (((π/3)(t^(-2)))/((π/3)(((t+2π+(cot^(-1))((Sqrt[(t/(2π))]))))^(-2))))   which is essentially the same as (((π/3)(t^(-2)))/((π/3)(((t+2π+(Sqrt[((2π)/t)])))^(-2)))), and ρ is asymptotic to 1+((4 )/t). So we suggest ((E((Subscript[F, 1]))(t))/(E((Subscript[F, 1]))(t+2π)))≈((E((Subscript[F, 2]))(t))/(E((Subscript[F, 1]))(t+2π)))≈1+((4π)/t)   or  E((Subscript[F, 2]))(t) = E((Subscript[F, 1]))(t+2π)) (1+((4π)/t)). 
     
     But there is a connection between error t and error t+2π:  ((E((Subscript[F, 1])(t)))/(E((Subscript[F, 1])(t+2π))))∼1+((4 π)/t). So then   E((Subscript[F, 2]))(t) = E((Subscript[F, 1])) (t) / (1/((1+((4 π)/t))))    (1+(π/t))

Show[ListPlot[Table[Log10@{τ,((bikeErrorPolarSquareRoot[τ])/(bikeErrorPolarSquareRoot[τ+2 π]))-1},{τ,2 π,50 π,π}], PlotRange->All],
ListLinePlot[Table[Log10@{τ,((4π)/τ)},{τ,2 π,50 π,π}],PlotStyle->Red, PlotRange->All], Frame->True, PlotRange->All]

ttt=Table[{τ,((bikeErrorBetter[FInt[2], FIntDer[2], {t, τ}])/(bikeErrorBetter[FInt[2], FIntDer[2], {t, τ+2π}]))},{τ,2 π,52 π,10 π}];

Show[ListPlot[Log10[ttt-1], PlotRange->All],
ListLinePlot[Log10@Table[{τ,((4π)/τ)+4((π^2)/(τ^2))},{τ,2 π,52 π,π}],PlotStyle->Red, PlotRange->All], Frame->True, PlotRange->All]

The error in using this approximation arises from: the fact that (Subscript[F, 2])≠(Subscript[F, 1]), and the other approximation to the point s, and the series trunction. 

But maybe this is no heuristic at all....   It says that 1+(C/t) governs the error as we move back 2π. But for all we know (Subscript[E, 2]) could go up\, not down. So this is not yet explaining in any way why (Subscript[F, 2]) < (Subscript[F, 1]). Now, possibly my frontward expansion idea is relevant...moving forward expands error. Again, why?

intDomMax[FInt[2]]
τ=380;
err1=N[bikeErrorBetter[FInt[2],FIntDer[2],{t,τ},WorkingPrecision->32,PrecisionGoal->15]]
err2=N[bikeErrorPolarSquareRoot[τ+2 π+(Sqrt[(τ/(2 π))]),WorkingPrecision->32,PrecisionGoal->15] /(1+((4 π)/τ)+0 (((2 (Sqrt[2]) π)/(τ^(3/2)))+((4 (π^2))/(τ^2))))]
(err1/err2)

While (Subscript[F, 2]) and (Subscript[F, 1]) are not identical (be

it appears that the differential equation extends (Subscript[F, 1]) backwards to get (Subscript[F, 2]). Now, forward movement along the spiral compresses the spiral, so backward movement expands the spiral. So on the one hand it appears that there should be a connection between the error for (Subscript[F, 2]) at t=2π and the error for (Subscript[F, 1]) at 13.3217.  [[Aside: similarly there should be a connection between the error at 2π to the arror at, say 5π]]/

ss=start[1, 2π, MachinePrecision,6];
nearest=bikeErrorBetter[FInt[2], FIntDer[2],{t, 2π}, Output->Location];
im1=Show[ParametricPlot[F[t], {t,ss, 8π}, PlotStyle->{Blue, at@8, CapForm[None]}],
ParametricPlot[F[t], {t,2π, ss}, PlotStyle->{Cyan, at@6, CapForm[None]}],ParametricPlot[FInt[2], {t,2π, 8π}, PlotStyle->Red],
Graphics[{Cyan, ap@5, Point[FInt[1]/. t-> ss],ap@5,Darker@Green, Point[FInt[2]/. t-> nearest],Black, Arrow[{{1,0}, {1,0}+(unit@D[FInt[2],t]/. t-> 2π)}],
Arrow[{FInt[2] , FInt[2]+unit@D[FInt[2],t]}/. t-> ss]}], Frame->True, Axes->False, BaseStyle->FontFamily->"Times"];
GraphicsRow[{im1, Show[im1, PlotRange-> {{.99, 1.09}, {.9, 1.04}}]}];

nearest2=FInt[2]/. t->bikeErrorBetter[FInt[2], FIntDer[2],{t, 2π}, Output->Location]
nearest1=FInt[1]/. t->bikeErrorPolarSquareRoot[ 2π, Output->Location]

im1=Show[ParametricPlot[F[t], {t,4π, 7π}, PlotStyle->{Blue, at@8, CapForm[None]}],
ParametricPlot[F[t], {t,2π, 4π}, PlotStyle->{Cyan, at@6, CapForm[None]}],
ParametricPlot[FInt[2], {t,2π, 7π}, PlotStyle->Red],
Graphics[{Cyan, ap@5,ap@5,Darker@Green, Point[ nearest];Black, 
Arrow[{FInt[1] , FInt[1]+unit@D[FInt[1],t]}/. t-> 4π],
Arrow[{FInt[2] , FInt[2]+unit@D[FInt[2],t]}/. t-> 2π], 
Arrow[{FInt[1] , FInt[1]+unit@D[FInt[1],t]}/. t-> 2π],
Line[{nearest1,  FInt[1]+unit@D[FInt[1],t]/. t-> 2π}],
Line[{nearest2,  FInt[2]+unit@D[FInt[2],t]/. t-> 2π}],
{Dashed,Circle[{1,0},1]}
}], 

Frame->True, Axes->False, BaseStyle->FontFamily->"Times"];
GraphicsRow[{im1, Show[im1/. at@8-> at@4/. Blue ->Lighter@Blue, PlotRange-> {{.99, 1.09}, {.9, 1.04}}]}]

Let (Subscript[E, n])(t) denote the unibike error for (Subscript[F, n]) at t. So the theorem is (Subscript[E, 1])(t)∼(π/3)(t^(-2)).

Consider the ratio (((Subscript[E, 2])(t))/((Subscript[E, 1])(t+2π+(tan^(-1))(Sqrt[((2π)/t)])))). We hope it is near 1. Plot (Subscript[E, 2])/(Subscript[E, 1])== (π/t)+1

From file:  Power3Over2try. But study this more closely for n=2 first. It seems that there should be a connnection between (Subscript[E, 2])(t) and (Subscript[E, 1])(t). Experiments show that (Subscript[E, 2])(t) is close to (Subscript[E, 1])(t+2π+(cot^(-1))). But how close 

(Subscript[e, 1])(t)=(π/3)(t^(-2)) (the proved asymptotic for the polar square root)
(Subscript[e, n])(t)=(Subscript[e, n-1])(t+2π)(1+(π/t))  (a translation of the error by 2π and a multiplicative error factor, with the power of t and the coefficient π found experimentally).

GetFIntSequence[nMax=40,800, WorkingPrecision->22,PrecisionGoal->10,FunctionsToKeep->Range[2, nMax]]//AbsoluteTiming

intDomMax[FInt[2]]/π

Do[errorTable[n, 50], {n, 2, 40}]//AbsoluteTiming

max=50 π; im1=Show[ListLinePlot[Table[Log10[{τ,bikeErrorPolarSquareRoot[τ,WorkingPrecision->22,PrecisionGoal->12]}],{τ,2 π,max, π}],PlotStyle->{at[1],Red},PlotRange->All],
ListLinePlot[Table[Log10[{τ,bikeErrorPolarSquareRoot[τ+2 π,WorkingPrecision->22,PrecisionGoal->12]}],{τ,2 π,max, π}],PlotStyle->{at[3],Cyan},PlotRange->All],ListLinePlot[Table[Log10[{τ,bikeErrorPolarSquareRoot[τ+2 π,WorkingPrecision->22,PrecisionGoal->12] (1+(π/τ))}],{τ,2 π,max, π}],PlotStyle->{at[2],Black},PlotRange->All],
ListPlot[Log10[Take[dd=errorTable[2,50], 20]],PlotStyle->{ap[6],Blue}],PlotRange->All,Frame->True,Axes->False, BaseStyle->FontFamily->"Times"];
GraphicsRow[{im1, Show[im1, PlotRange-> {{1.8, 2.2}, {-4.4, -3.4}}, PlotRangePadding->0]}]

Show[Plot[0, {x, 0, max}, PlotStyle->Blue], 
ListLinePlot[Table[{d[[1]],bikeErrorPolarSquareRoot[ d[[1]]]-d[[2]]}, {d, dd}],PlotStyle->Red, PlotRange->All],
ListLinePlot[Table[{d[[1]],bikeErrorPolarSquareRoot[d[[1]]+2 π]-d[[2]]}, {d, dd}],PlotStyle->Cyan, PlotRange->All],
ListLinePlot[Table[{d[[1]],(1+(π/(d[[1]])))bikeErrorPolarSquareRoot[d[[1]]+2 π]-d[[2]]}, {d, dd}],PlotStyle->Black, PlotRange->All], Frame->True, PlotRange->{-.00001, .00001}, BaseStyle->FontFamily->Times]

Show[ListLinePlot[Log10[Abs[Table[{d[[1]],bikeErrorPolarSquareRoot[ d[[1]]]-d[[2]]}, {d, dd}]]],PlotStyle->Red, PlotRange->All],
ListLinePlot[Log10[Abs[Table[{d[[1]],bikeErrorPolarSquareRoot[d[[1]]+2 π]-d[[2]]}, {d, dd}]]],PlotStyle->Cyan, PlotRange->All],
ListLinePlot[Log10[Abs@Table[{d[[1]],(1+(π/(d[[1]])))bikeErrorPolarSquareRoot[d[[1]]+2 π]-d[[2]]}, {d, dd}]],PlotStyle->Black, PlotRange->All], Frame->True, PlotRange->All, BaseStyle->FontFamily->Times]

ListLinePlot[Log10[Abs[Table[{d[[1]],(((bikeErrorPolarSquareRoot[d[[1]]]-d[[2]]))/(d[[2]]))},{d,dd}]]],PlotStyle->Red,PlotRange->All]

Show[ListLinePlot[Log10[Abs[Table[{d[[1]],(((bikeErrorPolarSquareRoot[d[[1]]]-d[[2]]))/(d[[2]]))},{d,dd}]]],PlotStyle->Red,PlotRange->All],ListLinePlot[Log10[Abs[Table[{d[[1]],(bikeErrorPolarSquareRoot[d[[1]]+2 π]-d[[2]])/d[[2]]},{d,dd}]]],PlotStyle->Cyan,PlotRange->All],ListLinePlot[dL=Log10[Abs[Table[{d[[1]],((((1+(π/(d[[1]]))+ 0   (( (π^2))/((d[[1]])^2))) bikeErrorPolarSquareRoot[d[[1]]+2 π]-d[[2]]))/(d[[2]]))},{d,dd}]]],PlotStyle->Black,PlotRange->All],Plot[Evaluate[{.75 - 1.6 L,1 - 1.6 L}[[1]]],  {L, Log10[2π], 2.5}, PlotStyle-> Dashed], Frame->True,PlotRange->All,BaseStyle->FontFamily->Times]

So adding the extra term helps, but it will make symbolics more difficult. So go with 1+(4/t) and observe that it is an UPPER BOUND.  Use htis to redo all symbolics in my attempt at faster functions. Or closed form etc.

Length[dL]
Fit[dL[[5;;50]], {1,L}, L]
10^1.33 /(π^2)

(t^(-5/4))

Figure XX  The fit of the heuristic formula using 1+(4/π) is phenomenal.  The above shows differences but also RELATIVE differences!!  Convergence for the latter is slower, but it is clearly -> 0.  And it appears to be an upper bound when I use the 1+(4/t) ratio.

τ = 50*Pi; 
bikeErrorBetter[FInt[2], FIntDer[2], {t, τ}]
(bikeErrorPolarSquareRoot[τ + 2*Pi + ArcCot[Sqrt[τ/(2*Pi)]]]*
   (1 + Pi/τ))

The red curve is E((Subscript[F, 1])) and it bounds E((Subscript[F, 2])). The green curve is E((Subscript[F, 1])(t+2π+(cot^(-1))(Sqrt[(t/(2π))]))). It nicely approximates E((Subscript[F, 2])) but is an underestimate.
The black curve is (1+(π/t))E((Subscript[F, 1])(t+2π+(cot^(-1))(Sqrt[(t/(2π))]))). I found 1+π/t by studying the ratio.

M<aybe make one showing the differences? Use log τ to get the data.

Show[ListLinePlot[Table[Log10[{τ,bikeErrorPolarSquareRoot[τ,WorkingPrecision->22,PrecisionGoal->12]}],{τ,2 π,50 π, π}],PlotStyle->{at[1],Red},PlotRange->All],
ListLinePlot[Table[Log10[{τ,bikeErrorPolarSquareRoot[τ+2 π+ArcCot[(Sqrt[(τ/(2 π))])],WorkingPrecision->22,PrecisionGoal->12]}],{τ,2 π,20 π, π}],PlotStyle->{at[1],Green},PlotRange->All],ListLinePlot[Table[Log10[{τ,bikeErrorPolarSquareRoot[τ+2 π+ArcCot[(Sqrt[(τ/(2 π))])],WorkingPrecision->22,PrecisionGoal->12](1+π/τ)}],{τ,2 π,50 π, π}],PlotStyle->{at[1],Black},PlotRange->All],
ListPlot[Log10[Take[dd, 25]],PlotStyle->{ap[6],Blue}],PlotRange->All,Frame->True,Axes->False]

Show[ListPlot[Log10[dCompare]],Plot[-L+Log10[π],{L,.7,2.5},Frame->True]]

dCompare=Table[{Log10[τ=d[[1]]],Log10[((d[[2]])/(bikeErrorPolarSquareRoot[τ+2 π+ArcCot[(Sqrt[(τ/(2 π))])],WorkingPrecision->22,PrecisionGoal->12]))-1]},{d,Take[dd,15]}];
Show[ListPlot[dCompare],ListPlot[Table[{Log10@τ,Log10[ ((π/τ))]},{τ,2,30 π,.5}], PlotStyle->Red, PlotRange->All], PlotRange->All,Frame->True,Axes->False]

dd[[15]]

6

Fit[dL[[4;; 10]], {1,L},L]
10^-1.1

The cyan line is this, with t set  to 2π (but  general t is needed in the recurrence b/c of the translation).

Clear[e];
e[1]= (π/3)(t^(-2));
e[n_] := e[n]= (e[n-1] /. t-> t+2π) (1+((4 )/t))
e[6]//Together

n=13;
Together[e[n]];
Together[e[n]]/. t-> 2.π
(π/3)(1/((((2 n-2) π+t))^2))(( (∏_(k=0)^(n-2))(4+2 k π+t))/( (∏_(k=0)^(n-2))(2 k π+t)))/. t-> 2.π
n=.;
(π/3)(1/((((2 n-2) π+t))^2))(( (∏_(k=0)^(n-2))(4+2 k π+t))/( (∏_(k=0)^(n-2))(2 k π+t)))

PochFunc=(π/3)((4+t)/t)((Pochhammer[1+((4+t)/(2 π)),-2+n])/( (((2 (n-1) π+t))^2) Pochhammer[1+(t/(2 π)),-2+n]))
PochFunc/.n-> 13/. t-> 2.π
Normal[Series[PochFunc,{n,∞,2}]]/. t-> 2π//Simplify

-2+2/π//N

((4+2 π)/(2 π))//Apart

There is in fact a closed form for (Subscript[e, n]) even for general t:   (π/3)(( Pochhammer[1+(2/π)+(t/(2π)),n-1])/( (((2 (n-1) π+t))^2) Pochhammer[1+(t/(2 π)),n-1]))

And the asymptotics of this are (1/(9(π^(3/2)) ))(n^(-3/2)), better written as, or (1/50.115)(n^(-3/2)). So my (π^(-3/2)) guess was right, but then it gets multiplied not by (2^(3/2))π (which is 8.8), but simply by 9. Nice. 
This is the red line.

ddNew=Table[{τ, bikeErrorBetter[FInt[2], FIntDer[2], {t, τ}, WorkingPrecision->20, PrecisionGoal->12]}, {τ, 2π, 182π, 10π}];

GetFIntSequence[nMax=2,2000,Resolution->1200, FunctionsToKeep-> Range[nMax], ErrorValuesToStore-> {}, 
MaxSteps-> 2500000, WorkingPrecision->32, PrecisionGoal->20]//AbsoluteTiming

intDomMax[FInt@2]/π

τ=200π;ss=bikeErrorBetter[FInt[1], FIntDer[1], {t,τ}, Output->loc];
{bikeErrorPolarSquareRoot[ss], bikeErrorPolarSquareRoot[ss]( 1+(1/τ))}
(*{bikeErrorPolarSquareRoot[τ+2π+ArcCot[(Sqrt[2π/τ])]],bikeErrorPolarSquareRoot[τ+3π]}*)
bikeErrorBetter[FInt[2], FIntDer[2], {t, τ}]

This is not literally true, but if (Subscript[F, 1]) were a true unibike curve, then this would be literally true. Because (Subscript[F, 1]) is close to a unibike curve, let us assume it is true and see what follows. Well, if it were literally true the error from the lower arrow would ethis backward

bikeErrorPolarSquareRoot[4π+ArcTan[1]]

ddL=

Repeat the process (11 hours) to get (Subscript[F, 1]),…,(Subscript[F, 224]). Details: Start with domain [2π,2350) use 1100 points on each loop when reinterpolating. WP 22, PG 15.  Do not save any except (Subscript[F, 224]) and (Subscript[F, 112]).
THen (Subscript[F, 224]) has domain [2π, 297), or about 148 loops.  

There is a reemarkable exact fit at (1/(2 π+3)) (1/(((π m))^(3/2))). The coeff might be wrong, but the 3/2 exponent is clearly right.
(Subscript[F, 224]) is a piecewise cubic interpolation with 162306 cubics.

(* 8+ hours *) GetFIntSequence[224,2350,Resolution->1100,InitialRadialFunction->(Sqrt[(t/(2 π))]), FunctionsToKeep-> {224, 112,56}, ErrorValuesToStore-> {2π,20π}, FunctionPointsToKeep-> {50 π/2, 100π/2},WorkingPrecision->22, PrecisionGoal->15]//AbsoluteTiming

AbsoluteTiming[GetFIntSequence[240,2500,Resolution->1000,InitialRadialFunction->(Sqrt[(t/(2 π))]),FunctionsToKeep->{240,120},ErrorValuesToStore->{2 π,300 π},FunctionPointsToKeep->{((50 π)/2),((100 π)/2)},WorkingPrecision->22,PrecisionGoal->15]]

41757/3600.

points2keep[50π/2][[13,2]]

ListPlot[Table[Log10[{points2keep[((50 π)/2)][[i,1]],points2keep[((50 π)/2)][[i,2,1]]- points2keep[((50 π)/2)][[i-1,2,1]]}],{i,2,Length[points2keep[((50 π)/2)]]}]]

ListPlot[errorData[2π]/. {x_, y_} :> {Log10@x, y}, Frame->True, GridLinesStyle->Thick, GridLines->{{Log10@600},{-5, -6}}, PlotRange-> {{0,Log10@650},{-6.3, -1}}]

FInt240=FInt[240];
FInt120=FInt[120];

FInt224=FInt[224];
FInt112=FInt[112];
FInt56=FInt[56];

FInt2=FInt[2];

Save["/Users/wagon/Desktop/exportFInt2.nb",FInt2];//Timing

Save["/Users/wagon/Desktop/exportFInt240.nb",FInt240];//Timing
Save["/Users/wagon/Desktop/exportFInt120.nb",FInt120];//Timing

THis will restore it to the  sumbol  name FInt[224], etc,

Get["/Users/wagon/Desktop/exportFInt240.nb"];  (* move files *) 
Get["/Users/wagon/Desktop/exportFInt120.nb"];

Get["/Users/wagon/Desktop/exportFInt240.nb"];  (* move files *) 
Get["/Users/wagon/Desktop/exportFInt120.nb"];

FInt[240]=FInt240;
FInt[120]=FInt120;
FIntDer[240]=D[FInt[240], t];
FIntDer[120]=D[FInt[120], t];

The error data at

intDomMax[FInt[240]]/π

be240=bikeErrorBetter[FInt[240], FIntDer[240], {t, 2π}, WorkingPrecision->20, PrecisionGoal->10]
be120=bikeErrorBetter[FInt[120], FIntDer[120], {t, 2π}, WorkingPrecision->20, PrecisionGoal->10]

Log2[(be120/be240)]

Show[ParametricPlot[FInt[240], {t,2π, 24π}, PlotStyle->{Blue,at@1}], 
Graphics[{Red, Arrow[{FInt[240], FInt[240]+unit[FIntDer[240]]}/. t-> 2π],
at@.7, Black, Circle[ FInt[240]+unit[FIntDer[240]]/. t-> 2π, .075],
ap@3, Point[{{1,0}, FInt[240]+unit[FIntDer[240]]/. t-> 2π}]}
], Frame->True, Axes->False, BaseStyle->FontFamily->Times, PlotRangePadding->0, PlotRange-> {{-2.2, 2.2}, {-2.2, 2.2}}]

domain=(Floor[((intDomMax[FInt[240]])/π)]-3) π

AbsoluteTiming[error240=Table[{τ,bikeErrorBetter[FInt[240],FIntDer[240],{t,τ}]},{τ,2 π,(Floor[((intDomMax[FInt[240]])/π)]-3) π,π}];]
AbsoluteTiming[error120=Table[{τ,bikeErrorBetter[FInt[120],FIntDer[120],{t,τ}]},{τ,2 π,(Floor[((intDomMax[FInt[240]])/π)]-3) π,π}];]

ListPlot[error240/. {x_, y_}:> {Log10@x, y}, Frame->True]

ListPlot[errorData[2π]/. {x_, y_}:> {Log10@x, y}, Frame->True, PlotRange]

GraphicsRow[{Show[ListPlot[error240],ListPlot[error120],Frame->True],
Show[ListPlot[Table[{error240[[i,1]],((error120[[i,2]])/(error240[[i,2]]))},{i,Length[error240]}]],Frame->True]}]

errorData(2π) and errorData(20π) cached

errorData[2π]={{1,-1.86463569105005507289862979210605113248`18.664765995366704},{2,-2.23657321341079073944526803193656286421`17.393181705467903},{3,-2.4729407365978776690860562998577475474`17.2026625003283},{4,-2.64644497029829954593239721590453259192`17.06086628766573},{5,-2.78362543978665461743583406370139499057`16.947770614637687},{6,-2.89708722806984323966494116426229915943`16.85360947995908},{7,-2.99383547924442292594128168157140539402`16.772885589279714},{8,-3.07817394197392551883866003835647682891`16.702194058895905},{9,-3.15291744877870745123246731180876387725`16.639296104944474},{10,-3.2200254706511737500875407831933655093`16.58262555686578},{11,-3.28091507379054507228909137562612288835`16.53104511092279},{12,-3.33663929067920594064660933133890841069`16.48370687208243},{13,-3.3880073424212439333485872551357272986`16.43995698996782},{14,-3.43564551768238914448837657396214588648`16.399288402268745},{15,-3.48006400932272505693872779021251297958`16.36128622641691},{16,-3.52166423727073621530700061062620566424`16.32562394939683},{17,-3.56078999845803318015333320741762630535`16.29202038511049},{18,-3.597707518594551684372484706094311311`16.26025846709155},{19,-3.63266310854478344368360262954209522076`16.23013552390262},{20,-3.66584972788496251268814016137581298027`16.201493372005697},{21,-3.6974359634170506018220232642726737163`16.174193330798154},{22,-3.72757118178375631799130240999909394513`16.148112092649455},{23,-3.75638406690679814321578931999130345454`16.123143277448424},{24,-3.78398276794696261406072610071595436158`16.099197591007805},{25,-3.81046813390483766834719641952112162458`16.076191290035837},{26,-3.83592035202854227882611430108221460362`16.054058180274616},{27,-3.86042104953906126378381507780301270516`16.032730163908038},{28,-3.8840459970278557853678281762182034656`16.01214376821288},{29,-3.90684071408960936703066311338384700897`15.992261920055824},{30,-3.92888348799223487522820977906821509907`15.973018028675899},{31,-3.95018327056907832550874658925049511017`15.95440684761969},{32,-3.97083763316357494713725351892221803314`15.93634392014465},{33,-3.9908409708614704002611618205048449169`15.918836441333802},{34,-4.01025961830025287080783586014181463363`15.901827262626195},{35,-4.0292157972505059061882097949750561231`15.88520908197124},{36,-4.04745422832773242752677890888288950135`15.86921138378965},{37,-4.06526597838198023312039800282021760496`15.85357594012351},{38,-4.08263192378525315838241743685915785062`15.838321085252238},{39,-4.09954848312471709575383388062295645353`15.823451239255098},{40,-4.11602261351461939182650938248391008571`15.808961335232622},{41,-4.13211200370941818387742677586413286692`15.794800827690619},{42,-4.14779685393050021013797900051540175128`15.78098839862443},{43,-4.1631288412331445479332876714193732393`15.767478681402544},{44,-4.17809524426726856018246645286659837503`15.754283902532398},{45,-4.19273511928009250423415514588696664062`15.741369834774872},{46,-4.20705930702585922601068267186207307568`15.728727468176627},{47,-4.22106437346380500669822802005558173956`15.716360556974516},{48,-4.23476897418324512346765003585303107396`15.704252986399695},{49,-4.2482256567426425622041940301721480198`15.692358161130974},{50,-4.2613828798526869509957027975369618117`15.680722892748395},{51,-4.27429254212928729922415904603988031664`15.66930108496608},{52,-4.28693816775795889174073790710829836114`15.65810801916639},{53,-4.29935209526998328744709131505655081047`15.647115073176536},{54,-4.31154801039318204247196941706134576756`15.636310342916659},{55,-4.32350546172398783141891225373169369372`15.625712627938572},{56,-4.33524159054833775837772437000966630346`15.615306879036106},{57,-4.34678485001320763873078107235330236683`15.605067841121269},{58,-4.35812668728000407064037036583630602353`15.595003531538202},{59,-4.36924874003931869628670299749609444226`15.585130801811799},{60,-4.38022574934872032672265619453900502507`15.575382636670584},{61,-4.39099715890013116685372985712061323385`15.565813841868238},{62,-4.40161014271404074515113413947146318791`15.55638219149195},{63,-4.41202270098562915835364323819590357469`15.547125772905094},{64,-4.4223025948604587583054502479362693371`15.53798375437605},{65,-4.43238903987976256147428826030009270728`15.529011167808848},{66,-4.44235893477520608365737128270579068031`15.520138854591188},{67,-4.45218470896223405370250314659440373352`15.511391878543646},{68,-4.46182937900925485216620557749767446175`15.502803819499684},{69,-4.47136792222002879795282907149687200682`15.49430718033623},{70,-4.48069820588882811290894400514883551527`15.485994368147045},{71,-4.48996417858454017602371963930836837633`15.477735548690115},{72,-4.49910319481290166871695892004423951039`15.469587439837328},{73,-4.50808581598490569391167853502342384098`15.46157680550771},{74,-4.51693155527748676400602252140941420309`15.453686155645954},{75,-4.52573233262349092711183614318301774368`15.445832549423576},{76,-4.53435971173689164220118303534739625896`15.438132208454855},{77,-4.5428721675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errorData[2π]
errorData[20π]

Show[
ParametricPlot[{Log10[m],Log10[(1/((2π+3)(((π m))^(3/2))))]}, {m,1, 250}, PlotRange->All,PlotStyle-> ({CapForm@None,at@9, RGBColor[.95,.95,0];Blue})],
ParametricPlot[{Log10[m],Log10[(1/(1.28 (((10 π m))^1.43)))]}, {m,1, 250}, PlotRange->All,PlotStyle-> ({CapForm@None,at@3, RGBColor[.95,.95,0];Green})],
ListLinePlot[ww=errorData[2π]/. {x_, y_}:> {Log10@x, y}, PlotStyle-> {at@3.5,Blue;Yellow}],
ListLinePlot[ww=errorData[2π]/. {x_, y_}:> {Log10@x, y}, PlotStyle-> {at@1,Blue;Yellow; Black}],
ListLinePlot[ww20=errorData[20π]/. {x_, y_}:> {Log10@x, y}, PlotStyle->Red], FrameTicks->{  {{#, (("10")^(ToString@#))}& /@ {-4, -6, -5,-3, -2, -1},{}}, {{#, 10^#}& /@ Log10@{1, 10,50,  100, 240},{}}}, BaseStyle->{FontFamily-> Times, 12},
 PlotRange-> {-6, -1.5}, AspectRatio-> 0.6, Frame->True, Axes->False, GridLines->{{},{-5,-6}}, GridLinesStyle->Black]
Fit[ww[[160;;240]], {1,L}, L]
Fit[ww20[[160;;240]], {1,L}, L]

Solve[1/(2π+3) (π nnn)^-(3/2)==(10^(-6))]//N

n=.;
Solve[10^-6==(1/(2π+3))(((π n))^(-3/2)), n]//N

σ=(2^(3/2));
GG[n_]:=((σ FInt[n]-FInt[(n/2)])/((σ-1)));
GGDer[n_]:=((σ FIntDer[n]-FIntDer[(n/2)])/(σ-1));
 bikeErrorBetter[GG[240], GGDer[240], {t, 2π}]

bike error down to 3.44·(10^(-7))

loc=bikeErrorBetter[GG[240], GGDer[240], {t, 2π}, Output->Location]

δ=(1/(2 (10^5)));
rad=(1/(10^6));
Show[ParametricPlot[GG[240],{t,4 π,4.5 π},PlotStyle->{at[1.5],Red},PlotPoints->4000],Graphics[{{EdgeForm[Black],FaceForm[{Opacity[.75],RGBColor[1,1,.6]}],Disk[GG[240]/. t->loc,rad]},
{ap[5],Point[pp=GG[240]/. t->loc]}, 
{ EdgeForm[Black],FaceForm[White], Disk[GG[240]+unit[GGDer[240]]/. t->2π, 10^-7]},{ap[1.6], Blue,Point[GG[240]+unit[GGDer[240]]/. t->2π]},
Arrow[{GG[240],GG[240]+unit[GGDer[240]]}]/. t->2π

}],Frame->True,PlotRange->{{pp[[1]]-δ,pp[[1]]+δ},{pp[[2]]-δ,pp[[2]]+δ}},PlotLabel->"Error at (Subscript[G, 240]) (average by Romberg) at t = 2 π is less than 4 · (10^(-7))",BaseStyle->FontFamily->Times]

δ=(1/(2 (10^5)));
rad=(1/(10^6));
loc = bikeErrorBetter[try, D[try,t], {t, 2π}, Output-> lo]
error = bikeErrorBetter[try, D[try,t], {t, 2π}]

tryDerUnit = unit@D[try, t];

Show[pp=try/. t->loc;
ParametricPlot[try,{t,13.341063327038338-.005,13.341063327038338+.005 },PlotStyle->{at[1.5],Red},PlotPoints->4000/10],
Graphics[{{EdgeForm[Black],FaceForm[{Opacity[.75],RGBColor[1,1,.6]}],Disk[pp,rad]},
{ap[5],Point[pp]}, 
{ EdgeForm[Black],FaceForm[White], Disk[try+tryDerUnit/. t->2.π, 10^-7]},
{ap[1.6], Blue,Point[try+tryDerUnit/. t->2π]},
Arrow[{try,try+tryDerUnit}/. t->2 π]}],Frame->True,PlotRange->{{pp[[1]]-δ,pp[[1]]+δ},{pp[[2]]-δ,pp[[2]]+δ}},BaseStyle->FontFamily->Times]

ParametricPlot[GG@240, {t, 2π, 300π}, PlotPoints->500, Frame->True, PlotStyle-> {at@.5}, Axes->False, ImageSize->600]

ParametricPlot[{t,Norm[GG[240]]},{t,2 π,300 π},PlotPoints->500,Frame->True,PlotStyle->{at[2], Black},Axes->False,ImageSize->600,AspectRatio->.6]

ParametricPlot[Log10[{t,(((Sqrt[t])/(Sqrt[2 π])))-Norm[GG[240]]}],{t,2 π,300 π},PlotPoints->500,Frame->True,PlotStyle->{at[2],Black},Axes->False,ImageSize->600,AspectRatio->.6]

Cached

errorF224={{6.28318530717958647692528676655900576836`19.735683541462205,-5.238634930765438`},{6.42953949403826835791288919201973720905`19.735683541462205,-5.242839650142165`},{6.57930270784170556362966587481183782396`19.735683541462205,-5.24505746181207`},{6.73255435502820716269380627044214906621`19.735683541462205,-5.250161632856029`},{6.8893756916496367990379313621601866048`19.735683541462205,-5.252002918924572`},{7.04984986645444694101010225025398170948`19.735683541462205,-5.25464199701263`},{7.21406196497424591467060438160494535446`19.735683541462205,-5.257868679564094`},{7.38209905463727300268026332264715489033`19.735683541462205,-5.259871226123369`},{7.55405023093270137042515977677673929698`19.735683541462205,-5.262437726644677`},{7.73000666465024574384596292935207962439`19.735683541462205,-5.265802090153381`},{7.91006165022012190425279603244118728129`19.735683541462205,-5.269704655814311`},{8.09431065517898848650479201402736823848`19.735683541462205,-5.2722010163954005`},{8.28285137078809857765487859255991817208`19.735683541462205,-5.275099900441532`},{8.47578376383049953004783874521645514138`19.735683541462205,-5.278621344685677`},{8.67321012961474454982954361388085991173`19.735683541462205,-5.280237154112712`},{8.87523514621321933033864925014529555619`19.735683541462205,-5.284774394078849`},{9.08196592996384160909090734448284845831`19.735683541462205,-5.287296116467747`},{9.29351209226456138411304647563273183315`19.735683541462205,-5.290174515807158`},{9.50998579769077498540500727715711504358`19.735683541462205,-5.293479736967072`},{9.73150182346646762374613895767523191382`19.735683541462205,-5.29728395032242`},{9.95817762032061680381836711005528487137`19.735683541462205,-5.301189997727421`},{10.19013337476112347226569881738497104694`19.735683541462205,-5.302736345209208`},{10.42749207279928936327769204822644024193`19.735683541462205,-5.307085822030593`},{10.67037956515862810307869229826654824989`19.735683541462205,-5.311378553706476`},{10.91892463400258464811109127813204647254`19.735683541462205,-5.313297334164229`},{11.17325906121654297700451562258254958418`19.735683541462205,-5.317896034922459`},{11.43351769828032606062844810674455180086`19.735683541462205,-5.319562761088133`},{11.69983853776823543458427781544742420777`19.735683541462205,-5.323816439411153`},{11.97236278651454064153302253144664483277`19.735683541462205,-5.327977990179924`},{12.25123494048321185432078862771703428781`19.735683541462205,-5.331778047367404`},{12.53660286138159260315681803029646896162`19.735683541462205,-5.335116637717773`},{12.82861785505863419022591671826422705964`19.735683541462205,-5.338172146883507`},{13.12743475172925957334520337623474157324`19.735683541462205,-5.340542065827972`},{13.43321198806739273829128518843818394026`19.735683541462194,-5.345324980711081`},{13.7461116912111803706007726797717790118`19.735683541462205,-5.34936677500163`},{14.06629976472494650731508358173526890428`19.735683541462205,-5.352567668331728`},{14.3939459765634583348668750752701488341`19.735683541462205,-5.354842246111635`},{14.72922404908514295117884931691194676496`19.735683541462205,-5.359116411266512`},{15.0723117511619812909679747927947277777`19.735683541462205,-5.362340313661663`},{15.42339099243491709923290072061523003591`19.735683541462205,-5.367342647499347`},{15.78264791976475641838013744258495003421`19.735683541462205,-5.370931083724749`},{16.15027301592969713257565950768948436494`19.735683541462205,-5.373649549044703`},{16.52646120062181930590259342547281702677`19.735683541462205,-5.377353308999473`},{16.91141193379608599034887161094227089267`19.735683541462205,-5.3826030768458475`},{17.30532932142665151183899519699034331094`19.735683541462205,-5.3861420392316495`},{17.70842222372655062881338883679851502163`19.735683541462205,-5.391177219828835`},{18.12090436588814807500415593217599042848`19.735683541462205,-5.393902076055799`},{18.54299445140306453857795186042274250786`19.735683541462205,-5.397791005317497`},{18.97491627802166280236666165355977961178`19.735683541462205,-5.402640627515628`},{19.41689885641357729964268648949259609537`19.735683541462194,-5.404887203123426`},{19.86917653159220246886710134064070224572`19.735683541462205,-5.411207367305899`},{20.33198910716752077837948132675333974389`19.735683541462205,-5.414548662140091`},{20.80558197249315091512869874761655660413`19.735683541462205,-5.418347881726244`},{21.29020623277503118538450008035097868955`19.735683541462205,-5.4224659610668455`},{21.78611884221072347356094087334803322006`19.735683541462194,-5.426617905629559`},{22.29358274022992998041767281901738307952`19.735683541462205,-5.431613055940034`},{22.81286699090845926597010737993953977286`19.735683541462194,-5.434403596135121`},{23.34424692562956072726640439402018025583`19.735683541462205,-5.437872774607301`},{23.88800428906826843891740323680580310776`19.735683541462205,-5.444490014503224`},{24.44442738857615718781497453575364091238`19.735683541462194,-5.443053533112054`},{25.013811247045716477029405533371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intDomMax[FInt[240]]/π

errorG240=errorG224;

dommax=300 π;
n=240;(errorG240=Table[{τ1, Log10[bikeErrorBetter[GG[n], GGDer[n], {t, τ1}, WorkingPrecision->MachinePrecision, PrecisionGoal-> 6]]}, {τ1, 10^Range[N[Log10[2π], 20],1. Log10[ dommax], (1/200)]}])//Timing//Quiet

Cached

(errorF240=Table[{τ1, Log10@bikeErrorBetter[FInt[n], FIntDer[n], {t, τ1}, WorkingPrecision->MachinePrecision, PrecisionGoal-> 6]}, {τ1, 10^Range[N[Log10[2π], 20],Log10[dommax], (1/100)]}])//Timing//Quiet

(errorF1=Table[{τ1, Log10@bikeErrorBetter[F[1], FDer[1], {t, τ1}, WorkingPrecision->MachinePrecision, PrecisionGoal-> 6]}, {τ1, 10^Range[N[Log10[2π], 20],Log10[dommax], (1/100)]}])//Timing//Quiet;

τ1

Show[ListLinePlot[errorF1/. {x_, y_}:> {Log10@x, y}, PlotStyle-> {at@1,Black}],
ListLinePlot[errorF240/. {x_, y_}:> {Log10@x, y}, PlotStyle-> {at@1,Red}],
ListLinePlot[errorG240/. {x_, y_}:> {Log10@x, y}, PlotStyle-> {at@1.5,Blue}, PlotRange->All], Frame->True, 
FrameTicks->{  {{#, (("10")^(ToString@#))}& /@ {-8,-7,-4, -6, -5,-3, -2, -1},{{Log10[4 10^-7],"4·(10^(-7))"}}}, {{#, 10^#}& /@ Log10@{2π, 3π, 10 π,20π, 50π, 100 π, 300π},{}}}, PlotRange-> {{.75, 3.1}, {-8.68,-1.5}}, PlotRangePadding->0, BaseStyle->FontFamily->Times, GridLines->{{},{-6,Log10[5 10^-7]}}, GridLinesStyle->Directive@{Black, at@.5}]

Caption.  The polar square root ((Subscript[F, 1])) error (black), (Subscript[F, 240]) error (red), and (Subscript[G, 240]) error (blue). The largest error for (Subscript[G, 240]) is about  3.75·(10^(-7)) at t=2π. The blue spike at left is where the front curve crosses (Subscript[G, 240]), giving 0 unibike error. Perhaps the asymptotic error for (Subscript[G, 240]) matches that of (Subscript[F, 240]) as t->∞, but the error for the initial part of the curve is vastly reduced.

SortBy[errorG224, Last][[-5;;-1]]

Here a zoom plot shows what happens at the first upward spike. It is similar at a spike near 61.3. THere are 162000 cubics that make up these functions.

FInt[224]//intCount

Max[Last/@ errorG224Zoom]

(10^(-6.33401))

ListLinePlot[errorG224Zoom, PlotRange->All, Frame->True, GridLines-> {{6.702, 6.704},{-6.33}}]

(errorG224Zoom=Table[{τ1, Log10@bikeErrorBetter[GG[n], GGDer[n], {t, τ1}, WorkingPrecision->MachinePrecision, PrecisionGoal-> 6]}, {τ1, 6.695, 6.707, .012/50}])//Timing//Quiet

(errorG224Zoom=Table[{τ1, Log10@bikeErrorBetter[GG[n], GGDer[n], {t, τ1}, WorkingPrecision->MachinePrecision, PrecisionGoal-> 6]}, {τ1, 6.68, 6.72, .04/100}])//Timing//Quiet

Max[Last/@ errorG224Zoom]

errorG224Zoom

τ1

Position[errorG224, SortBy[errorG224, Last][[-1]]]

ListLinePlot[errorG224[[96;;104]], PlotRange->All]

Redo around 61.

σ=(2^(3/2))  ;n=224;  
G[n]=((σ FInt[n]-FInt[n/2])/(σ-1));
GDer[n]=(∂_t)G[n];

{5306.887893`,}

err200=bikeErrorList[G[n],GDer@n,{t,(10^(Range[Log10[2 π],Log10@1000,.015]))}]

Max[Last/@ err160]

Plot[radF[160], {t, 2π, 100π}]

THis F2 stuff is out of order.

data=bikeErrorList[FInt[2], FIntDerUnit[2],  {t, Range[2π, 100π, 2π]}];

τ

data

dataL=Log10[data];
Fit[dataL, {1,L}, L]

ListLogLogPlot[data, PlotRange->All, Frame->True, PlotLabel -> ""]

bikeError[FInt[2], FIntDerUnit[2],{t, 52π}]

bikeError[FInt[1], FIntDerUnit[1], {t, 2π}]//Timing
bikeError[FInt[2], FIntDerUnit[2], {t, 2π}]

rTry=Interpolation[Table[{t,Norm[FInt[1]]}, {t, 2π, 100π, 100π/10000}], InterpolationOrder->3][t];
FShort[1] = rTry cs;
FShortDer[1] =unit[D[ FShort[1], t]];
bikeError[FShort[1], FShortDer[1], {t, 2π}]//AbsoluteTiming

errorData[1]=Table[bikeError[FInt[1], FIntDerUnit[1], {t, τ}], {τ, 2π, 20π, π/3}]

A question that occupied me for some years is: Can one improve (Subscript[F, 1]), the polar square root, by a deterministic process to get (Subscript[F, 2]), (Subscript[F, 3]), … such that the error decreases to 0 and the curves converge? That is, the limiting curve (Subscript[F, ∞]) will be a unibike curve with no self-intersection. I tried many things over several years with no success, but finally I have something that works. This was inspired by some recent attempts in the same direction by Bob Bixler.

The idea is simple. Take (Subscript[F, 1]) to be the front track of a bike and numerically solve the DE that gives the rear track; call that (Subscript[R, 1]). To set this up one needs a starting parameter (Subscript[t, 0]) and a starting location for the rear curve, (Subscript[R, 1])((Subscript[t, 0])). This location must be at unit distance from (Subscript[F, 1])((Subscript[t, 0])). I will use (1,0) for this point and then find (Subscript[t, 0]) that works. This forces the solution curve will pass through (1,0); and we will be interested in this curve only from (1,0) outwards. Simple root-finding finds that (Subscript[F, 1])(13.3217) is at distance 1 from (1,0) and is on the appropriate loop (the next loop), so (Subscript[t, 0])=13.3217….

Figure 3 shows the story. The unit tangent at (1,0) ends at the cyan point, not on the curve. The dashed arc is a radius-1 circle. So the t-value giving the red disk is what will be used to start the DE for the rear track: (Subscript[t, 0])=13.321697163362538`… .

n=1;
Show[ParametricPlot[ FInt[n], {t, 2π, 4.5π}, PlotStyle->{Blue, AbsoluteThickness[1]}],
Graphics[{{Dashed, Circle[{1,0},1, {1,2}]},
Text["(Subscript[F, 1])(2π)", {.93, .03}],Text["(Subscript[F, 1])((Subscript[t, 0]))", {1.01,0.96}],
Red, AbsolutePointSize[5], Point[{1,0}],Arrow[BikeVector[n][2π]],
EdgeForm[Blue], FaceForm[Cyan], Disk[BikeVector[n][2π][[2]], .01],
 EdgeForm[Black], FaceForm[Red], Disk[getNextInitialValue[n,2π], .01]}],  Frame->True,PlotRangePadding-> 0, PlotRange-> {{0, 1.2}, {-.1, 1.2}},ImageSize-> 400, GridLines->{{}, {0}},BaseStyle->FontFamily->Times, GridLinesStyle->Directive@{Black, AbsoluteThickness@.6}]

Figure 3.  The initial condition for the DE is set by the t-parameter at the red dot.

The DE is quite simple. See VisualDSolve (Slavik & Wagon), Chap. 18, or Bicycle or Unicycle?. Here is the general DE where F(t) is the given front curve and we seek the rear curve R(t):

R'(t)=F'(t)·(R(t)-F(t)) [R(t)-F(t)]  with initial value R((Subscript[t, 0]))=(1,0).

This equation arises from the observation that the speed of the rear wheel is the magnitude of the velocity vector of the front wheel in the direction of the line connecting the two wheels. We can solve this numerically over a large t-domain. The solution will be an interpolating function with domain [(Subscript[t, 0]),(Subscript[t, max])] and starting at (1,0). But we must reparametrize the solution because we want all our curves (Subscript[F, n])(t) to be such that (Subscript[F, n])(2π)=(1,0). And the reparametrization must be done in such a way that the t-value indicates the angle, including the number of loops. That aspect of the solution is critical to the job of computing the error: the distance to the nearest point on (Subscript[F, 1])(t). But such a reparametrization is not a big problem. (If one cares only about the final curve, say (Subscript[F, 200]), the reparametrization step is not necessary.)

Important points: We will (almost) always reparametrize the DE solutions so that the domain starts at 2π (giving the point (1,0)) and the parameter t is the angle of the point, including a count of the loops. This is critical for the root-finding process that is used to compute the error, as root-finding requires a good starting value. Another critical point is that the domain will shrink. In the passage from (Subscript[F, 1]) to (Subscript[F, 2]) we might start with the  domain [(Subscript[t, 0]),(Subscript[t, 1])] where (Subscript[t, 0]) is about 13. After reparametrization the upper end of the domain will be reduced by a little more than 2π.

Of course once we have obtained (Subscript[F, 2]) with domain [2π,(Subscript[M, 2])] from (Subscript[F, 1]), we want to iterate the process a hundred or more times. What happens turns out to be quite nice. The computation — numerical soluton of a DE over a large domain—can be a little slow, but the process appears to be quite stable. Most important, the error decreases in a very predictable way! That is the key to gathering evidence that (Subscript[F, ∞])=(lim_(n->∞))(Subscript[F, n]) exists and is a unibike curve.

This is what happens: I can get a sequence of 160 (or more) curves. Each is defined by an interpolating function on many points. I start with the polar square root on an interval [2π,(Subscript[M, 1])] where (Subscript[M, 1]) is 1000 or 1500, but each iteration loses about 2π. For fixed n, the error in (Subscript[F, n])(t) does not decrease as t rises, but it almost does. It appears that the max error occurs near t=100 (true for Romberg; check for (Subscript[F, 160]) etc); the error at t=2π is a good indicator of the order of magnitude of the max error of (Subscript[F, n]).

This error for (Subscript[F, 1]) is 0.0137. But a log-log plot of the error at 2π as a function of n is very close to linear (Fig. 4,  right), and so the error at 2π (what about 100) appears to follow this rule: (Subscript[error, max])∼(1/50) (n^(-1.5)). So the error decrease is not super-fast, but it is steady. Figure 4 shows two plots of the error at 2π as n rises. For (Subscript[F, 132]) the error is just under (10^(-5)). There are some jaggies for large n. They are likely due to error accumulation: after all, (Subscript[F, 132]) is a numerical solution to a DE based on 131 previous numerically solved DEs. Using more interpolation points did not eliminate them.

GraphicsRow[{
ListLinePlot[data2Pi, Frame->True, PlotStyle-> {AbsoluteThickness[4], Darker@Green}, Mesh->All, PlotRange-> {{-.3, 136}, {-.001, .015}}, MeshStyle->{AbsolutePointSize[3],Red},BaseStyle->{FontFamily -> "Times", 12}, FrameTicks->{{{.005,.01, .0001}, {0}},{{1,10,25,50, 75, 120},{}}}],
Spacer[30],
ListLinePlot[dL=data2Pi/. {x_, y_}:> {Log10@x, Log10@y},  PlotStyle-> {AbsoluteThickness[4],Darker@Green},Mesh->All, MeshStyle->{AbsolutePointSize[3],Red}, PlotRange-> {{-.06,Log10[132]+.06},{-5.2,-1}},FrameTicks-> {{{#, (("10")^(ToString@#))}& /@ {-1,-2,-3,-4, -5},{}},{{Log10[#], #}& /@ {1,2, 8,4, 16,32,64, 128},{}}},
Frame->True,
FrameTicks-> {{{0.01, 0.002},{}},{ {1,  10,25,50},{}}},BaseStyle->{FontFamily -> "Times", 12}, PlotRange-> All]}]

Figure 4.  Unibike error at (Subscript[F, n])(2π) (error at left, log-log plot at right). The max error for (Subscript[F, 132]) is under (10^(-5)).

ListLinePlot[dL=data2Pi[[50;;132]]/. {x_, y_}:> {Log10@x, Log10@y},  PlotStyle-> {AbsoluteThickness[1],Darker@Green},Mesh->None, MeshStyle->{AbsolutePointSize[3],Red}, PlotRange-> {All,{-5.25,-4.3}},FrameTicks-> {{{#, (("10")^(ToString@#))}& /@ {-1,-2,-3,-4, -5},{}},{{Log10[#], #}& /@ {1,2, 8,4, 16,32,64, 128},{}}},
Frame->True,
FrameTicks-> {{{0.01, 0.002},{}},{ {1,  10,25,50},{}}},BaseStyle->{FontFamily -> "Times", 12}, PlotRange-> All]

Figure 5.  Some numerical error shows up as n increases. This is a loglog plot of the error for (Subscript[F, 50]) to (Subscript[F, 132]).

The linearity in the loglog plot is very striking, and indicates that the error reduction is well behaved. Each time n is doubled the error gets cut by a factor of (2^1.63), or about 3. Figures 6 and 7 show (Subscript[F, 132]). Of course it looks little different from (Subscript[F, 1]), except that now there is no visible error. Going to (Subscript[F, ∞]) would look hardly different.

n=132;Show[pr={{-2,2.1}, {-2,1.9}};
Graphics[{GrayLevel[.768], Rectangle[First/@ pr, Last/@ pr]}],
ParametricPlot[ FInt[n], {t, 2π, 10π}, PlotStyle-> {Blue, AbsoluteThickness[3.5]}],Graphics[{Red, AbsolutePointSize[5], Point[front[n][start[3]]];
EdgeForm[Black], FaceForm[{Yellow, Opacity[.3]}],Arrowheads[.02],
Table[
{Disk[front[n][τ]+frontDerUnit[n][τ], .033], Red, Point[front[n][τ]]}, {τ, 2π, 5.8π, π/6}], Table[Arrow[BikeVector[n][τ]], {τ, 2π, 5.8π, π/6}]}],  Frame->True, BaseStyle->FontFamily->Times,PlotRangePadding->0, PlotRangeClipping->True, PlotRange-> pr]

Figure 6.  (Subscript[F, 132]) is nicely close to a perfect unibike curve, with max error under (10^(-5)).

intDomMax[FInt[132]];
ParametricPlot[FInt[132], {t, 2π, 75}, Frame->True, BaseStyle->FontFamily-> "Times", PlotStyle->{Blue, AbsoluteThickness[.75]}]

Figure 7.  (Subscript[F, 132]) shown as t goes from 2π to 75.

We also need to worry about convergence of the curves. Each (Subscript[F, n])(2π)=(1,0), so we must look at other values. Consider t=50. The distance between (Subscript[F, n])(50) and (Subscript[F, n+1])(50) gets smaller as n rises and appears headed to 0; Figure 8 shows a loglog plot of this difference. The log-linear behavior indicates a definite pattern, and this means that the sequence {(Subscript[F, n])(50)} is a Cauchy sequence. Therefore it converges.

dataCon=Table[{n,ED[FInt[n],FInt[n+1]]}/. t->50,{n,1, 131, 5}];
ListLogLogPlot[dataCon, Frame->True]

Figure 8.  A loglog plot of (Abs[(Abs[(Subscript[F, n+1])(50)-(Subscript[F, n])(50)])]).

More: One can adjust the polar square root to get a curve with smaller error. I found that the polar function (Sqrt[(t/(2 π))])-(t^(-(Sqrt[2]))) works well. And (Sqrt[(t/(2 π))])-(92/100)(t^(-1402/1000)) is even better. For this adjusted polar square root the error is invisible to the eye without doing any DE work. The next diagram compares the two. Blue is the well-behaved polar square root. Red shows the error with the adjustment term and that is never larger than 1/1000. It is not as well behaved, but it is an improvement for all t. To repeat, we can get very small error by just using the interval [(10^(-100)),∞), but that defeats the purpose which is to find a method that converges to a zero-error function.

The dip is a place where the error is going to 0 and so there is  a crossing of the curve with the derived front curve: 0 error at such a point. The nice blue curve means there is no such crossing for the polar square root.

This next shows that the adjusted version has no visible error. The error in the image is at most one quarter of the error in the pure square root curve.

References

Finn
Velleman Wagon
Slavik, Wagon, Schwalbe

The Method:
In 200X Finn....    His construction is clever, but leaves open the question of whether a nicer unibike curve exists. Define. It is very natural to consider a spiral. 
Start with the polar square root spiral given by (Subscript[F, 1])=r(cos t, sin t) where r=(Sqrt[(t/(2π))]). This spiral starts at (1,0) and is nicely close to being a unibike curve.

ParametricPlot[(Sqrt[(t/(2 π))]) cs,{t,2 π,10 π}, Frame->True]

But the error is visible: 1/62 at 2π

bikeError[FInt[1], FIntDer[1], {t, 2π}]

1/%

The error decreases as t->∞ because, after all, the loops get closer and closer. But our goal is to modify the polar square root to yield a spiral starting at (1,0) that looks like the polar square root but has zero unibike error; that is, we seek a spiral that is a unibike curve. The method is to define (Subscript[F, n]) from (Subscript[F, n-1]) in such a way that the error decreases.

Definition.  Given f, a curve defined on [2π,M], let g be the path of the rear tire as the front tire drives along f. Ignore the early part of g,  so that the domain of g will be [2π,M'] where M'< M-2π.

So the algorithm is: 
Set a resolution: ρ is number of points  per loop in the interpolating function that will be defined from the interpolating function that NDSolve produces so that the parameter t is angle. ρ=1000 seems fine.
Set a maximum to start. We want roughly (Subscript[F, 200]), or maybe only (Subscript[F, 160]). For that a maximum value of (Subscript[mx, 1])=2π 400 is more than adequate.
Use NDSolve on the DE to get the back track. Reparametrize so that t is the angle. Is there a way of setting up the DE so this happens automatically?
Then use a Romberg step on the pair ((Subscript[F, 80]),(Subscript[F, 160])). That gets error below 1 part in a million.

Question: Can extrapolation on (Subscript[F, 1]),...,(Subscript[F, 160]) yield a good value of the limiting function (Subscript[F, ∞])?

Improvements: Can I just store the r-values? That should be a simpler int fcn.
Take FInt and form r at 1000 points. Then recover FInt and see how close it is. This would save a huge amt. of memory. 
Maybe take r get F from it and compute the error?

Yes, this works

intDomMax[FInt[160]]
2π 400 1.

The next version below deletes old functions. It could be set up with FOld and FNew.

nMax = 240;
initialize[750 π, 1200, 22]  (* the use of 600 π gets me only 100 at the end of 200 iterations  700 π got me to 935 *)  
(* this sets wp value which is used below  *)

(100+160 (2π + .7))

nMax = 200;
initialize[1550;2200, 1200, 22]  (* the use of 600 π gets me only 100 at the end of 200 iterations  700 π got me to 935 *)  
(* this sets wp value which is used below  *)

errorData2π={  {1,Log10@bikeErrorBetter[FInt[1],  FIntDer[1],{t, 2π}]}};

points2={};

Do[tStart=SetPrecision[start[n-1,2π],wp];
init[n]={1,0};mx[n]=intDomMax[FInt[n-1]];
mPrint[{n, DEStart,{t,tStart,mx[n], intDomMin[FInt[n-1]],intDomMax[FInt[n-1]]} }];
backSol=NDSolveValue[{b'[t]==frontDer[n-1][t].(b[t]-front[n-1][t]) (b[t]-front[n-1][t]),b[tStart]==init[n]},b[t],{t,tStart,mx[n]},MaxSteps->3 450000, WorkingPrecision->wp,PrecisionGoal->(10)(*upgraded as seems necessary *)];
mPrint[{n, DESolved, intCount[backSol], mx[n]}];
backSolPts=Table[backSol,{t,tStart,intDomMax[backSol],2π/ρ}];
FInt[n]=(old=atn[init[n]];loops=If[backSolPts[[1,2]]≥0, 1,0];
Interpolation[Table[new=atn[backSolPts[[i]]];If[new<old,loops++];old=new;
{atn[backSolPts[[i]]]+2 π loops,backSolPts[[i]]},{i,Length[backSolPts]}]][t]);
AppendTo[errorData2π, {n,Log10@bikeErrorBetter[FInt[n], FIntDer[n],{t, 2π}]}];
Print[{n, DESolved, intCount[backSol], mx[n], errorData2π[[-1]]}];
Clear[backSolPts];
(*If[errs[[1]]<-6,Print[errs]; Break[]];*)
If[n≥4 && !MemberQ[{ 80, 160,25,50,100, 200, 110,220, 90, 180, 120, 240},n-2], Unset[FInt[n-2]]];
front[n][t_?NumericQ]:=Evaluate[FInt[n]];
frontDer[n][t_?NumericQ]:=Evaluate[(∂_t)FInt[n]];
frontDerUnit[n][t_?NumericQ]:=Evaluate[unit@(∂_t)FInt[n]];
BikeVector[n][t_?NumericQ] := Evaluate[{FInt[n],FInt[n]+FIntDerUnit[n]}], {n,2,nMax}]//AbsoluteTiming

σ=(2^(3/2))//N;
n=200;
G[n]=(((σ FInt[n]-FInt[n/2]))/(σ-1));
GDer[n]=(∂_t)G[n];
intDomMax[FInt[200]]
intDomMin[FInt[200]]

error200Short=bikeErrorList[G[200], {t, Range[2π,100, 1/6]}];

Max[Last/@ error200Short]

errorData2π

Redo a little farther than just 100?

Max[Last/@ error160Short]

ListLinePlot[error160Short/. {x_, y_}:> {x, Log10@y}, Frame->True, PlotRange-> {-8.5, -5.5}, GridLines-> {{},{-6, -6.1}}, GridLinesStyle->Thick]

error200=bikeErrorList[FInt[200], FIntDerUnit[200], {t, Range[2π, 100,4]}]

τ

errorData22π

Fit[errorData2π[[150;;200]], {1,L}, L]

Show[Plot[-1.71277-3/2L, {L, 1,3}, PlotStyle->{at@9, Black}],ListPlot[errorData2π, PlotStyle->{ap@3,Red}], ListPlot[errorData22π,PlotStyle->{ap@3,Green}], ListPlot[errorData100π,PlotStyle->{ap@3,Blue}], PlotRange->{{0, 2.8}, {-6, -1.8}}, Frame->True, AspectRatio->( 3/2; 1/GoldenRatio)  , GridLines->{{},{-6}}, GridLinesStyle->Black]

Figure XX.  A loglog plot of the unibike error at t=2π (red), t=22π (green), and t=100π (blue) for (Subscript[F, 1]),(Subscript[F, 2]),…,(Subscript[F, 240]). The red curve is consistent with slope (3/2), and the other two seem to approach the same slope.  A fit to the error for (Subscript[F, 150]),...,(Subscript[F, 200]) gives slope 1.50014, so for no reason we can discern, the error descends like c (t^(-3/2))The pattern here allows one to take a weighted average of  Maybe 200&100 is best?   Still 160/80?

ParametricPlot[FInt[160], {t, 2π, 368},PlotStyle -> at[.7], Frame->True]

im=Show[ParametricPlot[FInt[160], {t, 2π, 10π},PlotStyle -> {at[1.5], Red}],Graphics[{Table[Arrow[{FInt[160],FInt[160]+ FIntDerUnit[160]}]/. t-> τ, {τ, Range[2π, 6π, π/3]}]}], Frame->True]

Try plotting the error? Use PlotPoints-> 10 or so.

loc=bikeErrorLocation[G[160], GDer[160], {t,2π}]

loc=bikeErrorLocation[FInt[160], FIntDerUnit[160], {t,2π}]
bikeError[FInt[160], FIntDerUnit[160], {t,2π}]

(FInt[160]+ FIntDerUnit[160]/. t-> 2π)-
(FInt[160]/. t-> loc)//Norm

Show[ParametricPlot[FInt[160], {t, 4π, 4.5π},PlotStyle -> {at[1.5], Red}, PlotPoints->4000],Graphics[{Circle[FInt[160]/. t-> 13.3399952986, .00001],Table[Arrow[{FInt[160],FInt[160]+ FIntDerUnit[160]}]/. t-> τ, {τ, Range[2π, 2π, π/3]}]}], Frame->True, PlotRange->{{1.023, 1.024}, {.9995, 1.0}}]

Show[ParametricPlot[G[160], {t, 4π, 4.5π},PlotStyle -> {at[1.5], Red}, PlotPoints->4000],Graphics[{Circle[G[160]/. t->loc, .000001],Table[Arrow[{G[160],G[160]+ GDer[160]}]/. t-> τ, {τ, Range[2π, 2π, π/3]}]}], Frame->True, PlotRange->{{1.02257, 1.0226}, {.99973, .99976}}]

tt2=errorData2π//N;
tt5=errorData5π//N;
fit=Fit[tt2[[-35;;-20;;1]], {1,L}, L]

Show[
Plot[-1.6993923224616805-1.5L, {L, 1.6, 2.9}, PlotStyle->{at@10,Green}] ,
ListLinePlot[tt2, PlotStyle->{at@1,Blue},Mesh->All, MeshStyle->ap[3]],
 Frame->True, PlotRange-> {{0.27;2 , Log10@700},{-6.3;-5.5,-2;-4.65 }}, PlotLabel->Row[{n, "; Slope = ", fit[[2,1]]}], Axes->False, GridLines->{{Log10@160,Log10@170,  Log10@200},{-6}}, GridLinesStyle->Thick, PlotRangePadding->0]

intDom[FInt[160]]

error160Romberg=Table[{τ, bikeError[G@160, GDer@160, {t,τ}]}, {τ, 2.π, 100, 3}]

bikeError[G@160, GDer@160, {t,200}]

τ

error160=Table[{τ, bikeError[FInt[160], FIntDerUnit[160], {t,τ}]}, {τ, 2.π, 100, 4}]

τ

ListPlot[error160Romberg, Frame->True, PlotRange->All]
ListPlot[error160, Frame->True, PlotRange->All]

Convergence to a Limit

ListPlot[First/@ Last/@points2]
ListLogLogPlot[Differences[First/@ Last/@points2]]

Radius Model: r=(Sqrt[(t/(2π))])-(SuperMinus[t])

Plot[Norm[G[160]],{t, 2π, 1000}, Frame->True]

LogLogPlot[((Sqrt[t])/(Sqrt[2π]))-Norm[G[160]],{t, 2π, 1000}, Frame->True]

Romberg on F(160)

Try other than 1.5?

σ=(2^1.50)//N
n=180;
G[n]=(1/(σ-1))(σ FInt[n]-FInt[n/2]);
GDer[n]=unit[(∂_t)G[n]];
ParametricPlot[G[n], {t, 2π, 20π}];
LogLogPlot[-Norm@G[n]+(Sqrt[t/(2π)]), {t, 2π, intDomMax[FInt[n]]}, Frame->True]
intDomMax[FInt[n]]

Table[{τ, Log10@bikeError[G[180], GDer[180], {t, τ}]}, {τ, 22π,22π, π/4}]

err=bikeErrorList[G[180], GDer[180], {t, Range[2π, intDomMax[FInt[180]]-3π, 15]}]  (* the -5 is important *)

im180=ListLinePlot[err/. {x_, y_}:> {x, Log10@y},GridLines->{{},{-6}},Frame->True,PlotRange->{-8.5,-5.5}, GridLines-> {{},{-6}}, GridLinesStyle->Thick]

σ=(2^(3/2))//N
σ=2^1.51//N
indices = {5;10,20,40,80,160};
Do[G[n]=(((σ FInt[n]-FInt[n/2]))/(σ-1)), {n, indices}]
Do[GDer[n]=unit[(∂_t)G[n]], {n, indices}];
intDomMax[FInt[160]]

rTry=Interpolation[Table[{t,Norm[G[160]]}, {t, 2π, 450π, 450π/20000}], InterpolationOrder->4][t];
GShort[160] = rTry cs;
GShortDer[160] =unit[D[ rTry cs, t]];

bikeError[GShort[160], GShortDer[160], {t, 2π}]

errTryr=Table[{τ, Log10@bikeError[GShort[160], GShortDer[160], {t, τ}]}, {τ, 2π,440π, π/4}];
ListLinePlot[errTryr, PlotRange->{-7.5,-5}, Frame->True, GridLines->{{},{-6}}]

Can I smooth out the curve by....averaging?

errorRomberg = Table[{n, bikeError[G[n],GDer[n], {t, 2π}]}, {n, indices}]

errorG160Large=Table[{τ,bikeError[G[160],GDer[n], {t, τ}]}, {τ,2π, 422π;2π, 40π}]

ListLinePlot[eL=errorG160Large/. {x_, y_}:> {Log10@x, Log10@y},Frame->True, Mesh->All, PlotRange->All]

errorG160=Table[{τ,bikeError[G[160],GDer[n], {t, τ}]}, {τ,2π, 30π, π/2}]

ListLinePlot[eL=errorG160/. {x_, y_}:> {Log10@x, Log10@y},Frame->True, Mesh->All, GridLines->{{},{-6}}, GridLinesStyle->Thick]

Fit[eL[[-3;;-1]], {1,L},L]

σ1=2^.8483

n=20;Monitor[errorG[n]=Table[{τ, bikeError[G[n], GDer[n], {t,τ}]}, {τ, 2.π, 5π, π}], τ]

ListPlot[errorG[20]]

σ1=2^.8483
n=160;G1[n]=((σ1 G[n]-G[n/2])/(σ1-1));
G1Der[n]=unit[(∂_t)G1[n]];
bikeError[G1[160],G1Der[160], {t, 2π}]

ParametricPlot[G1[160],{t, 2π, 30π}]

bikeError[G[160],GDer[160], {t, 2π}]

Monitor[error211=Table[{τ, bikeError[FInt[211], FIntDerUnit[211], {t,τ}]}, {τ, 2.π, 100, 4}];, τ]

ListLinePlot[error211/. {x_, y_}:> {x, Log10@y}, Frame->True, PlotRange-> {-8, -4},Mesh-> All]

Notes: 

1. The error is solidly going down like (1/50)(n^(-(3/2))).
2. The error at 5π is staying under the error at 2π, though some numerical error is creeping in once n≥160.

NExt time set this up to Rombergize with 1.5 and 0.5. Keep track of what you need going forward in the iteration. Can I do that? Or just do it for the end?  Keep just a couple pairs k,2k.  Use 1.5 and .5

Keep pairs
(50,100)  certainly
(60,120)
(80,160)  beyond 160 error is creeping in

ttDiff=Table[{errorData2π[[i,1]],  errorData2π[[i,2]]/errorData5π[[i,2]]}, {i,Length@errorData2π}]//N;
ListPlot[ttDiff/. {x_, y_}:> {Log10@x, Log10@y}, Frame->True]

Show[
Plot[fit, {L, 0.3, 2.9}, PlotStyle->{at@2,Dashed,Green}] ,ListLinePlot[tt2, Mesh->All, MeshStyle->ap[4]],
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ListPlot[Last/@ First/@points2]
ListPlot[First/@ Last/@points2, Frame->True]

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