Parabolic case

(Subscript[B, 0])[u_]=(((1-u))^2);(Subscript[B, 1])[u_]=2(1-u)u;(Subscript[B, 2])[u_]=(u^2);
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={d,0};
(Subscript[b, 2])={Cos[φ],Sin[φ]};
b20[t_,φ_,d_]=Sum[(Subscript[b, j])(Subscript[B, j])[(t+1)/2],{j,0,2}];
e20[t_,φ_,d_]=((b20[t,φ,d][[1]])^2)+((b20[t,φ,d][[2]])^2)-1//FullSimplify;

(Subscript[ψ, 2])[t_]:=e20[t,φ,d];
(Subscript[ϕ, 2])[t_]:=(Sqrt[e20[t,φ,d]+1])-1;

Extrema of (Subscript[ϕ, 2]) (and (Subscript[ψ, 2]))

extrema2=Solve[D[(Subscript[ψ, 2])[t],t]==0,t]//Simplify

Assuming[d>Cos[φ],(t/.extrema2[[3]])-((Sqrt[(d^2)+(c^2)-2])/(d-c))/.{c->Cos[φ],s->Sin[φ]}//Simplify]

Function f

f[d_]=(d^4)-8(d^3)+2((c^2)+4c+6)(d^2)-8c(4-c)d+(c^4)-8(c^3)+12(c^2)+4;

(Subscript[ϕ, 2])[0]+(Subscript[ϕ, 2])[t/.extrema2[[3]]]//Simplify

4((((2(d-Cos[φ])-(1/2) (d-Cos[φ])(d+Cos[φ])))^2)-(-1+(d^2)) ((Sin[φ])^2))-f[d]/.{c->Cos[φ],s->Sin[φ]}//Simplify

Value (Subscript[ϕ, 2])(0)

(Subscript[ϕ, 2])[0]//Simplify

Lemma 3.1

f[3-3c+(c^2)]-(-(((1-c))^4) (22(1-(c^2))+(1-(c^4))+8c(1+(c^2))))//Simplify

f[-x-c+2]-(4(((1-c))^4) +x(8(((1-c))^2)c+4(((2-2c-x))^2)+x((((2-2c-x))^2)+4(c^2))))//Simplify

f'[(1-2c+(c^2))x+2-c]-(-4(((1-c))^2)(3(x^2)(((1-c))^2)c+4(1-x (c^2))+(4-(x^2)(((1-c))^3))x(1-c)+2(c+x)))//Simplify

Manipulate[Plot[(d^4)-8(d^3)+2((c^2)+4c+6)(d^2)-8c(4-c)d+(c^4)-8(c^3)+12(c^2)+4,{d,0,3},GridLines->{{2-c,(c^2)-3c+3},{}}],{c,0,1}]

Numerical computations

dr=Select[d/.NSolve[f[d]==0/.c->Cos[(π/2)],d,Reals,WorkingPrecision->20],#>2-Cos[(π/2)]&&#<((Cos[(π/2)])^2)-3Cos[(π/2)]+3&][[1]];
ds=Select[d/.NSolve[-4+(c^4)+16 c d-4 (d^2)+(d^4)-2 (c^2) (2+3 (d^2))==0/.c->Cos[(π/2)],d,Reals,WorkingPrecision->20],#≥1&&#≤1/2(5-3Cos[(π/2)])&][[1]];
Plot[{(Sqrt[e20[t,(π/2),dr]+1])-1,(Sqrt[e20[t,(π/2),ds]+1])-1,e20[t,(π/2),dr]},{t,-1,1},PlotStyle->{{Black,Thick},{Black,Dashed},Gray}]

angles={(π/2),(π/3),(π/4),(π/6),(π/8),(π/12)};
sez2={{"φ","(Subscript[d, r])","(Subscript[ψ, 2, 0])","(Subscript[d, s])","(Subscript[ϕ, 2, 0])"}};For[i=1,i≤Length[angles],i++,φ0=angles[[i]];dr=Select[d/.NSolve[f[d]==0/.c->Cos[φ0],d,Reals,WorkingPrecision->20],#>2-Cos[φ0]&&#<((Cos[φ0])^2)-3Cos[φ0]+3&][[1]];
ds=Select[d/.NSolve[-4+(c^4)+16 c d-4 (d^2)+(d^4)-2 (c^2) (2+3 (d^2))==0/.c->Cos[φ0],d,Reals,WorkingPrecision->20],#≥1&&#≤1/2(5-3Cos[φ0])&][[1]];AppendTo[sez2,{φ,NumberForm[dr,{6,5}],ScientificForm[N[(Sqrt[e20[0,φ0,dr]+1])-1],6],NumberForm[ds,{6,5}],ScientificForm[N[1-(Sqrt[e20[((Sqrt[-3+2 (ds^2)+Cos[2 φ0]])/((Sqrt[2]) (Sqrt[(((ds-Cos[φ0]))^2)]))),φ0,ds]+1])],6]}/.φ->angles[[i]]]];

Grid[sez2,Frame->All]

The best interpolant for an arbitrary angle

φ0=(π/2); (Subscript[b, 0])={Cos[φ],-Sin[φ]}; (Subscript[b, 1])={d,0}; (Subscript[b, 2])={Cos[φ],Sin[φ]}; dr=Select[d/.NSolve[f[d]==0/.c->Cos[φ0],d,Reals,WorkingPrecision->20],#>2-Cos[φ0]&&#<((Cos[φ0])^2)-3Cos[φ0]+3&][[1]];
ds=Select[d/.NSolve[-4+(c^4)+16 c d-4 (d^2)+(d^4)-2 (c^2) (2+3 (d^2))==0/.c->Cos[φ0],d,Reals,WorkingPrecision->20],#≥1&&#≤1/2(5-3Cos[φ0])&][[1]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b20[t,φ0,dr],{t,-1,1},PlotStyle->Red],
ParametricPlot[b20[t,φ0,ds],{t,-1,1},PlotStyle->Green],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2])}]}],AspectRatio->Automatic,PlotRange->All],Plot[{e20[t,φ0,dr],e20[t,φ0,ds]},{t,-1,1},PlotStyle->{Red,Green}]}]/.{φ->φ0,d->ds}

Parabolic case (revisited)

(Subscript[B, 0])[u_]=(((1-u))^2);(Subscript[B, 1])[u_]=2(1-u)u;(Subscript[B, 2])[u_]=(u^2);
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={d,0};
(Subscript[b, 2])={Cos[φ],Sin[φ]};
b20[t_,φ_,d_]=Sum[(Subscript[b, j])(Subscript[B, j])[(t+1)/2],{j,0,2}];
(Subscript[ψ, 2])[t_,d_]=((b20[t,φ,d][[1]])^2)+((b20[t,φ,d][[2]])^2)-1//FullSimplify;
(Subscript[ϕ, 2])[t_,d_]=(Sqrt[(Subscript[ψ, 2])[t,d]+1])-1;

(Subscript[q, 2])[t_,d_]:=(1/4) ((((d-c))^2)(t^2)-(((d+c))^2)+4);

(Subscript[ψ, 2])[t,d]-((t^2)-1)(Subscript[q, 2])[t,d]/.{c->Cos[φ]}//Simplify

Lemma 5.1.

(Subscript[q, 2])[τ,1]-(1/4)(1-c)(3+c+(1-c)(τ^2))//Simplify

CoefficientList[(Subscript[q, 2])[τ,d],d][[-1]]-(-(1/4) (1-(τ^2)))//Simplify

Lemma 5.2.

(Subscript[q, 2])[0,d]-(1/4) (4-(((c+d))^2))//Simplify

(Subscript[q, 2])[1,d]- (1-c d)//Simplify

The algorithm for the radial error

φ0=(π/2); ϵ=(10^(-5)); (Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={d,0};(Subscript[b, 2])={Cos[φ],Sin[φ]};
τl=0;τr=1;
While[τr-τl>ϵ,τ0=(1/2) (τr+τl);
d0=Select[d/.NSolve[(Subscript[ψ, 2])[τ0,d]==0/.{φ->φ0},d,Reals,WorkingPrecision->20],#>1&][[1]];
If[(Subscript[ϕ, 2])[0,d0]+(Subscript[ϕ, 2])[((Sqrt[(d0^2)+((Cos[φ0])^2)-2])/( (d0-Cos[φ0]))),d0]>0/.{φ->φ0},τr=τ0,τl=τ0]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b20[t,φ0,d0],{t,-1,1},PlotStyle->Red],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2])}]}],AspectRatio->Automatic,PlotRange->All],Plot[(Subscript[ϕ, 2])[t,d0]/.{φ->φ0},{t,-1,1},GridLines->{{},{-(Subscript[ϕ, 2])[0,d0],(Subscript[ϕ, 2])[0,d0]}/.{φ->φ0}}]}]/.{φ->φ0,d->d0}
N[b20[t,φ0,d0]]//Simplify

The algorithm for the simplified radial error

φ0=(π/2); ϵ=(10^(-5)); (Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={d,0};(Subscript[b, 2])={Cos[φ],Sin[φ]};
τl=0;τr=1;
While[τr-τl>ϵ,τ0=(1/2) (τr+τl);
d0=Select[d/.NSolve[(Subscript[ψ, 2])[τ0,d]==0/.{φ->φ0},d,Reals,WorkingPrecision->20],#>1&][[1]];
If[(Subscript[ψ, 2])[0,d0]+(Subscript[ψ, 2])[((Sqrt[(d0^2)+((Cos[φ0])^2)-2])/( (d0-Cos[φ0]))),d0]>0/.{φ->φ0},τr=τ0,τl=τ0]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b20[t,φ0,d0],{t,-1,1},PlotStyle->Red],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2])}]}],AspectRatio->Automatic,PlotRange->All],Plot[(Subscript[ϕ, 2])[t,d0]/.{φ->φ0},{t,-1,1},GridLines->{{},{-(Subscript[ϕ, 2])[0,d0],(Subscript[ϕ, 2])[0,d0]}/.{φ->φ0}}]}]/.{φ->φ0,d->d0}
N[b20[t,φ0,d0]]//Simplify

Cubic case

(Subscript[B, 0])[u_]=(((1-u))^3);(Subscript[B, 1])[u_]=3(((1-u))^2)u;(Subscript[B, 2])[u_]=3(1-u)(u^2);(Subscript[B, 3])[u_]=(u^3);
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+d{Sin[φ],Cos[φ]};(Subscript[b, 2])={Cos[φ],Sin[φ]}+d{Sin[φ],-Cos[φ]};(Subscript[b, 3])={Cos[φ],Sin[φ]};
b31[t_,φ_,d_]=Sum[(Subscript[b, j])(Subscript[B, j])[(t+1)/2],{j,0,3}];
(Subscript[ψ, 3])[t_,d_]=((b31[t,φ,d][[1]])^2)+((b31[t,φ,d][[2]])^2)-1//FullSimplify;
(Subscript[ϕ, 3])[t_,d_]=(Sqrt[(Subscript[ψ, 3])[t,d]+1])-1;

(Subscript[q, 3])[t_,d_]=(1/16)((((3 d c-2s))^2)(t^2)+(((3d s+4c))^2)-16);

(Subscript[ψ, 3])[t,d]-((((t^2)-1))^2)(Subscript[q, 3])[t,d]/.{c->Cos[φ],s->Sin[φ]}//Simplify

Lemma 6.1.

CoefficientList[(Subscript[q, 3])[t,d],d][[-1]]//Simplify

(Subscript[q, 3])[τ,0]-(-(1/4)(s^2)(4-(τ^2)))/.{c->Cos[φ],s->Sin[φ]}//Simplify

Lemma 6.2.

(Subscript[q, 3])[0,d]-(1/16) ((((3 d s+4 c))^2)-16)//Simplify

(Subscript[q, 3])[1,d]-(3/16) (3 (d^2)+4s c d -4 (s^2))/.{c->Cos[φ],s->Sin[φ]}//Simplify

The algorithm for the radial error

Solve[D[(Subscript[ψ, 3])[t,d],t]==0,t]//Simplify

φ0=(π/4); ϵ=(10^(-10)); 
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+d{Sin[φ],Cos[φ]};(Subscript[b, 2])={Cos[φ],Sin[φ]}+d{Sin[φ],-Cos[φ]};(Subscript[b, 3])={Cos[φ],Sin[φ]};
τl=0;τr=1;
While[τr-τl>ϵ,τ0=(1/2) (τr+τl);
d0=Select[d/.NSolve[(Subscript[ψ, 3])[τ0,d]==0/.{φ->φ0},d,Reals,WorkingPrecision->20],#>0&][[1]];
If[N[(Subscript[ϕ, 3])[0,d0]+(Subscript[ϕ, 3])[((Sqrt[12-3 (d0^2)+3 (-4+3 (d0^2)) Cos[2 φ0]-20 d0 Sin[2 φ0]])/((Sqrt[2]) (Sqrt[(((3 d0 Cos[φ0]-2 Sin[φ0]))^2)]))),d0]/.{φ->φ0},20]>0,τl=τ0,τr=τ0]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b31[t,φ0,d0],{t,-1,1},PlotStyle->Red],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])}]}],AspectRatio->Automatic,PlotRange->All],Plot[(Subscript[ϕ, 3])[t,d0]/.{φ->φ0},{t,-1,1},GridLines->{{},{-(Subscript[ϕ, 3])[0,d0],(Subscript[ϕ, 3])[0,d0]}/.{φ->φ0}}]}]/.{φ->φ0,d->d0}
N[b31[t,φ0,d0],10]//Simplify

The algorithm for the simplified radial error

φ0=(π/4); ϵ=(10^(-10));
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+d{Sin[φ],Cos[φ]};(Subscript[b, 2])={Cos[φ],Sin[φ]}+d{Sin[φ],-Cos[φ]};(Subscript[b, 3])={Cos[φ],Sin[φ]};
τl=0;τr=1;
While[τr-τl>ϵ,τ0=(1/2) (τr+τl);
d0=Select[d/.NSolve[(Subscript[ψ, 3])[τ0,d]==0/.{φ->φ0},d,Reals,WorkingPrecision->20],#>0&][[1]];
If[N[(Subscript[ψ, 3])[0,d0]+(Subscript[ψ, 3])[((Sqrt[12-3 (d0^2)+3 (-4+3 (d0^2)) Cos[2 φ0]-20 d0 Sin[2 φ0]])/((Sqrt[2]) (Sqrt[(((3 d0 Cos[φ0]-2 Sin[φ0]))^2)]))),d0]/.{φ->φ0},20]>0,τl=τ0,τr=τ0]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b31[t,φ0,d0],{t,-1,1},PlotStyle->Red],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])}]}],AspectRatio->Automatic,PlotRange->All],Plot[(Subscript[ϕ, 3])[t,d0]/.{φ->φ0},{t,-1,1},GridLines->{{},{-(Subscript[ϕ, 3])[0,d0],(Subscript[ϕ, 3])[0,d0]}/.{φ->φ0}}]}]/.{φ->φ0,d->d0}
N[b31[t,φ0,d0],10]//Simplify

Quartic case

(Subscript[B, 0])[u_]=(((1-u))^4);(Subscript[B, 1])[u_]=4(((1-u))^3)u;(Subscript[B, 2])[u_]=6(((1-u))^2)(u^2);(Subscript[B, 3])[u_]=4(1-u)(u^3);(Subscript[B, 4])[u_]=(u^4);
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+((Sqrt[3])/2)(Sqrt[1- Cos[φ] d]){Sin[φ],Cos[φ]};
(Subscript[b, 2])={d,0};(Subscript[b, 3])={Cos[φ],Sin[φ]}+((Sqrt[3])/2)(Sqrt[1- Cos[φ] d]){Sin[φ],-Cos[φ]};(Subscript[b, 4])={Cos[φ],Sin[φ]};
b42[t_,φ_,d_]=Sum[(Subscript[b, j])(Subscript[B, j])[(t+1)/2],{j,0,4}];//Simplify;
(Subscript[ψ, 4])[t_,d_]=((b42[t,φ,d][[1]])^2)+((b42[t,φ,d][[2]])^2)-1//Simplify;
(Subscript[ϕ, 4])[t_,d_]=(Sqrt[(Subscript[ψ, 4])[t,d]+1])-1;

(Subscript[q, 4])[t_,d_]=(1/64)((12-3(c^2)-30 c d +12 (c^3) d+9(d^2)+12(Sqrt[3]) c s(Sqrt[1- c d])-12(Sqrt[3]) s d(Sqrt[1- c d]))(t^2)+
52-13(c^2)-12(c^3)d-9(d^2)-12(Sqrt[3]) s d(Sqrt[1- c d])-2c(9d+10(Sqrt[3]) s(Sqrt[1- c d])));

(Subscript[ψ, 4])[t,d]-((((t^2)-1))^3)(Subscript[q, 4])[t,d]/.{c->Cos[φ],s->Sin[φ]}//Simplify

a4=((Sqrt[3])/3)s((c^2)+(Sqrt[3])(1-c));b4=((Sqrt[3])/3)s((Sqrt[3+(c^2)])-c);

Plot[{a4/.{s-> (Sqrt[1-(c^2)])},b4/.{s-> (Sqrt[1-(c^2)])}},{c,0,1}]

Limit[((1-(a4^2))/c)/.{s-> (Sqrt[1-(c^2)])},c->0]

Limit[((1-(b4^2))/c)/.{s-> (Sqrt[1-(c^2)])},c->0]

(Subscript[r, 4])[t_,x_]=Assuming[x>0,(Subscript[q, 4])[t,((1-(x^2))/c)]//Simplify];

Lemma 7.1.

CoefficientList[(Subscript[r, 4])[t,x],x][[-1]]-(-((9 )/(64 (c^2)))(1-(t^2)))//Simplify

Manipulate[Plot[(Subscript[r, 4])[t,x]/.{c->Cos[φ],s->Sin[φ]},{x,-1.5,2},GridLines->{{-1,a4/.{c->Cos[φ],s->Sin[φ]},b4/.{c->Cos[φ],s->Sin[φ]},1},{}}],{{φ,(π/3)},0,(π/2)},{t,0,1}]

(Subscript[r, 4])[t,-1]-(1/64) (39+8 (Sqrt[3]) c s+9 (t^2)+12(Sqrt[3]) c s (1- (t^2))+(1-(c^2)) (13+3 (t^2)))//Simplify

(Subscript[r, 4])[t,b4]-(1/4)(1-(c^2))(((c(Sqrt[3+(c^2)])-1-(c^2)))^2)/.{c->Cos[φ],s->Sin[φ] }//Simplify

(Subscript[r, 4])[t,1]-(1/64) (52+12(t^2)+4 (Sqrt[3]) c s(-5+3(t^2))-(c^2)(13+3(t^2)))//Simplify

(Subscript[s, 4])[t_]=-16+24 (Sqrt[3])+20 (-1+(Sqrt[3])) c-15 (c^2)-16 (Sqrt[3]) (c^2)+34 (c^3)+8 (Sqrt[3]) (c^3)-16 (c^4)-8 (Sqrt[3]) (c^4)+2 (c^5)+4 (Sqrt[3]) (c^5)-(c^6)+(24 (-2+(Sqrt[3]))+12 (-5+3 (Sqrt[3])) c-(-15+8 (Sqrt[3])) (c^2)-(-54+32 (Sqrt[3])) (c^3)-16 (-2+(Sqrt[3])) (c^4)-(-6+4 (Sqrt[3])) (c^5)+(c^6)) (t^2);

(Subscript[r, 4])[t,a4]-(-(1/64) (((1-c))^3) (1+c) (Subscript[s, 4])[t])/.s->(Sqrt[1-(c^2)])//Simplify

(Subscript[s, 4])[0]-((((4-2(1+(Sqrt[3])) (c^2)))^2)+12 (2+(Sqrt[3]))(c^3)(1-c)+20 ((Sqrt[3])-1)c(1-(c^3))+(4 (7-4 (Sqrt[3]))+(c^2))(1-(c^4))+20(2 (Sqrt[3])-3)+2(5-2 (Sqrt[3]))(c^3)+2(1+2 (Sqrt[3]))(c^5))//Simplify

(Subscript[s, 4])[1]-2((7-4(Sqrt[3]))(64+40(Sqrt[3])-7c)+c(((3-2(Sqrt[3])c))^2)+(c^3)+(c^3)(((3(Sqrt[3])-2-2c))^2))//Simplify

Lemma 7.2.

c=0

(Subscript[q, 4])[t,d]/.{c->0,s->1}//Simplify

(Subscript[q, 4])[τ,(2/(Sqrt[3]))](Subscript[q, 4])[τ,2]-(-(1/32) (6 (Sqrt[3])-8+3 (2-(Sqrt[3])) (1-(τ^2))))/.{c->0,s->1}//Simplify

Lemma 7.3

c=0

CoefficientList[(Subscript[q, 4])[t,d]/.{c->0,s->1},d][[-1]]//Simplify

(Subscript[q, 4])[τ,0]/.{c->0,s->1}//Simplify

lc[d_]=(3/64)(((2-(Sqrt[3])d))^2);

CoefficientList[(Subscript[q, 4])[t,d]/.{c->0,s->1},t][[-1]]-lc[d]//Simplify

lc[0]//Simplify

lc[2]-((6(2-(Sqrt[3])))/16)//Simplify

0<c<1

lc[x_]=(1/(64(c^2))) (9 (x^4)+12 (Sqrt[3]) c s (x^3)-6(3-5 (c^2)+2(c^4))(x^2)-12 (Sqrt[3])c (s^3)x+9(s^4));

CoefficientList[(Subscript[r, 4])[t,x],t][[-1]]-lc[x]/.{c->Cos[φ],s->Sin[φ]}//Simplify

lc[b4]/.{c->Cos[φ],s->Sin[φ]}//Simplify

lc[-((Sqrt[3])/3)s((Sqrt[3+(c^2)])+c)]/.{c->Cos[φ],s->Sin[φ]}//Simplify

D[lc[x],x]/.x->-((c s)/(Sqrt[3]))/.{c->Cos[φ],s->Sin[φ]}//Simplify

Numbers -((Sqrt[3])/3)s((Sqrt[3+(c^2)])+c),-((c s)/(Sqrt[3])),(Subscript[b, 4]) are different for all c<1:

Plot[{-((Sqrt[3])/3)s((Sqrt[3+(c^2)])+c)/.s->(Sqrt[1-(c^2)]),-((c s)/(Sqrt[3]))/.s->(Sqrt[1-(c^2)]),b4/.s->(Sqrt[1-(c^2)])},{c,0,1}]

lc[1]-lc[a4]-(1/64)(c(((4(Sqrt[3])-9+4(c^2)))^2)+12(Sqrt[3])c(Sqrt[1-(c^2)])+(24 (Sqrt[3])-36)(1-c)+3 (28 (Sqrt[3])-43) c(1-c)+(3+4 (Sqrt[3])) (c^2)(1-c)+(3+4 (Sqrt[3])) (c^3)(1-c)+5 (20 (Sqrt[3])-29) (c^4)(1-c)+(76 (Sqrt[3])-129) (c^5)(1-c)+(28 (Sqrt[3])-47) (c^6)(1-c)+(28 (Sqrt[3])-39) (c^7)(1-c)+(36 (Sqrt[3])-59) (c^8)(1-c)+(40 (Sqrt[3])-63) (c^9)(1-c)+8 (5 (Sqrt[3])-8) (c^10))/.s->(Sqrt[1-(c^2)])//Simplify

Lemma 7.4.

c=0

(Subscript[q, 4])[0,d]-(1/64)(52-12 (Sqrt[3]) d-9(d^2))/.{c->0,s->1}//Simplify

(Subscript[q, 4])[1,d]-(1-(1/8)(3 (Sqrt[3]) d))/.{c->0,s->1}//Simplify

c>0

CoefficientList[(Subscript[r, 4])[t,x],x][[-1]]//Simplify

(Subscript[f, 0])[x_]=D[(Subscript[r, 4])[0,x],x];

Collect[(Subscript[f, 0])[x],x]

(Subscript[f, 1])[x_]=D[(Subscript[r, 4])[1,x],x];

Collect[(Subscript[f, 1])[x],x]

(Subscript[f, 0])[0]-(-(((Sqrt[3])(3+5 (c^2)) s)/(16 c)))//Simplify

(Subscript[f, 1])[0]-(-(((Sqrt[3])(3+(c^2)) s)/(8 c)))//Simplify

(Subscript[f, 0])[1]-(((9c+6 (c^3)+(Sqrt[3])s(6-5 (c^2)))/(16 c)))//Simplify

(Subscript[f, 1])[1]-(((12c+(Sqrt[3])s(6-(c^2)))/(8 c)))//Simplify

(Subscript[g, 01])[c_]=6(3+(Sqrt[3]))-9(-1+2(Sqrt[3]))c;
(Subscript[g, 02])[c_]=-5(Sqrt[3])(c^2)+(9+(Sqrt[3]))(c^5);
(Subscript[g, 03])[c_]=(c^3)(2(3+8(Sqrt[3]))-2(9+4(Sqrt[3]))c+(Sqrt[3])(c^2)(1-c));

(Subscript[f, 0])[a4]-((s(1-c))/(16c))((Subscript[g, 01])[c]+(Subscript[g, 02])[c]+(Subscript[g, 03])[c])/.{c->Cos[φ],s->Sin[φ]}//Simplify

(Subscript[g, 11])[c_]=6-4(3-(Sqrt[3]))c;
(Subscript[g, 12])[c_]=(c^2)(-13+6(Sqrt[3])+9c+(3-6(Sqrt[3]))(c^2)+3(c^3));

(Subscript[f, 1])[a4]-(((Sqrt[3])s(1-c))/(8c))((Subscript[g, 11])[c]+(Subscript[g, 12])[c])/.{c->Cos[φ],s->Sin[φ]}//Simplify

Algorithm for radial error

Solve[D[(Subscript[ψ, 4])[t,d],t]==0,t]//Simplify

φ0=(π/4); ϵ=(10^(-20));
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+((Sqrt[3])/2)(Sqrt[1- Cos[φ] d]){Sin[φ],Cos[φ]};
(Subscript[b, 2])={d,0};(Subscript[b, 3])={Cos[φ],Sin[φ]}+((Sqrt[3])/2)(Sqrt[1- Cos[φ] d]){Sin[φ],-Cos[φ]};(Subscript[b, 4])={Cos[φ],Sin[φ]};
τl=0;τr=1;
leftedge=((1-(b4^2))/c)/.{c->Cos[φ0],s->Sin[φ0]};
rightedge=((1-(a4^2))/c)/.{c->Cos[φ0],s->Sin[φ0]};
While[τr-τl>ϵ,τ0=(1/2) (τr+τl);
d0=Select[d/.NSolve[(Subscript[ψ, 4])[τ0,d]==0/.{φ->φ0},d,Reals],#>leftedge&&#<rightedge&][[1]];
If[N[(Subscript[ϕ, 4])[0,d0]+(Subscript[ϕ, 4])[(√(21-6 (d0^2)-10 d0 Cos[φ0]-3 Cos[2 φ0]-2 d0 Cos[3 φ0]-4 d0 (Sqrt[3-3 d0 Cos[φ0]]) Sin[φ0]-6 (Sqrt[3-3 d0 Cos[φ0]]) Sin[2 φ0]))/(√(-7-6 (d0^2)+14 d0 Cos[φ0]+Cos[2 φ0]-2 d0 Cos[3 φ0]+8 d0 (Sqrt[3-3 d0 Cos[φ0]]) Sin[φ0]-4 (Sqrt[3-3 d0 Cos[φ0]]) Sin[2 φ0])),d0]/.{φ->φ0},20]>0,τr=τ0,τl=τ0]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b42[t,φ0,d0],{t,-1,1},PlotStyle->Red],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])}]}],AspectRatio->Automatic,PlotRange->All],Plot[(Subscript[ϕ, 4])[t,d0]/.{φ->φ0},{t,-1,1},GridLines->{{},{-(Subscript[ϕ, 4])[0,d0],(Subscript[ϕ, 4])[0,d0]}/.{φ->φ0}}]}]/.{φ->φ0,d->d0}
b42[t,φ0,d0]//N//Simplify

Quintic case

(Subscript[B, 0])[u_]=(((1-u))^5);(Subscript[B, 1])[u_]=5(((1-u))^4)u;(Subscript[B, 2])[u_]=10(((1-u))^3)(u^2);(Subscript[B, 3])[u_]=10(((1-u))^2)(u^3);(Subscript[B, 4])[u_]=5(1-u)(u^4);(Subscript[B, 5])[u_]=(u^5);
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+d{Sin[φ],Cos[φ]};
(Subscript[b, 2])={((5 d (4-5 (d^2)) Cos[φ]+4 (2+5 (d^2)) Sin[φ])/(4 (5 d+2Sin[φ]Cos[φ]))),-((5 d ((4-5 (d^2)) Sin[φ]-6 d Cos[φ]))/(4 (5 d+2Sin[φ]Cos[φ])))};
(Subscript[b, 3])={((5 d (4-5 (d^2)) Cos[φ]+4 (2+5 (d^2)) Sin[φ])/(4 (5 d+2Sin[φ]Cos[φ]))),((5 d ((4-5 (d^2)) Sin[φ]-6 d Cos[φ]))/(4 (5 d+2Sin[φ]Cos[φ])))};(Subscript[b, 5])={Cos[φ],Sin[φ]};(Subscript[b, 4])={Cos[φ],Sin[φ]}+d{Sin[φ],-Cos[φ]};
b53[t_,φ_,d_]=Sum[(Subscript[b, j])(Subscript[B, j])[(t+1)/2],{j,0,5}];//Simplify;
(Subscript[ψ, 5])[t_,d_]=b53[t,φ,d][[1]]^2+b53[t,φ,d][[2]]^2-1//Simplify;
(Subscript[ϕ, 5])[t_,d_]=(Sqrt[(Subscript[ψ, 5])[t,d]+1])-1;

(Subscript[q, 5])[t_,d_]=(1/(1024 (((5 d+2c s))^2)))(-16 (c^6) (36-16 (t^2)+25 (d^2) (-1+(t^2)))+(c^2) (-15625 (d^6) (-1+(t^2))+64 (-59+4 (t^2))+400 (d^2) (69+5 (t^2))+2500 (d^4) (-27+8 (t^2)))+25 (64+625 (d^6) (t^2)-300 (d^4) (-3+2 (t^2))+16 (d^2) (-34+9 (t^2)))+8 (c^4) (344-64 (t^2)+625 (d^4) (1+(t^2))-50 (d^2) (36+13 (t^2)))+20 c d (-200 (d^2) (-11+2 (t^2))+16 (c^4) (-3+2 (t^2))+625 (d^4) (-3+2 (t^2))-16 (19+6 (t^2))-8 (c^2) (-44-8 (t^2)+25 (d^2) (3+2 (t^2)))) s);

(Subscript[ψ, 5])[t,d]-((((t^2)-1))^4) (Subscript[q, 5])[t,d]/.{c->Cos[φ],s->Sin[φ]}//Simplify

α5=((2s)/5);a5=((16s)/(25+15c));b5=((2s)/(3+2c));β5=((4s)/(5(1+c)));

Plot[{α5/.{s->(Sqrt[1-(c^2)])},a5/.{s->(Sqrt[1-(c^2)])},b5/.{s->(Sqrt[1-(c^2)])},β5/.{s->(Sqrt[1-(c^2)])}},{c,0,1}]

lc[d_]=(1/(1024 (((5 d+2 c s))^2)))((c^6) (256-400 (d^2))+25 (d^2) (((12-25 (d^2)))^2)+8 (c^4) (-64-650 (d^2)+625 (d^4))+(c^2) (256+2000 (d^2)+20000 (d^4)-15625 (d^6))+640 (c^5) d s-320 (c^3) d (-4+25 (d^2)) s+40 c d (-48-200 (d^2)+625 (d^4)) s);

CoefficientList[(Subscript[q, 5])[t,d],t][[-1]]-lc[d]//Simplify

Manipulate[Plot[{(Subscript[q, 5])[t,d]/.{c->Cos[φ],s->Sin[φ]},5lc[d]/.{c->Cos[φ],s->Sin[φ]}},{d,0.3,0.9},GridLines->{{α5,a5,b5,β5}/.{c->Cos[φ],s->Sin[φ]},{}},PlotStyle->{Thick,Dashed}],{t,0,1},{{φ,(π/2)},0,(π/2)}]

Lemma 8.1.

r[d_]=((125s (d^3)+100c (d^2)-20s(3+(c^2))d+16c (s^2))/(32(5d+2s c)));

lc[d]-((r[d])^2)/.{c->Cos[φ],s->Sin[φ]}//Simplify

r'[d]-((5 (125 s (d^3)+25 c (2+3 (s^2))(d^2) +40 (c^2) d s-4 c (s^2)(5+ (c^2))))/(16 (((5 d+2 c s))^2)))//Simplify

r'[α5]-((5 (1-c))/(8 (((1+c))^2)))(1+c+2(c^2))/.{c->Cos[φ],s->Sin[φ]}//Simplify

r[b5]-(-(((((1-c))^2)(5-2c+2(c^2))s)/(4 (((3+2 c))^2) (5+3 c+2 (c^2)))))/.{c->Cos[φ],s->Sin[φ]}//Simplify

r[β5]-(((((1-c))^2)s)/(4 (1+c) (2+c+(c^2))))/.{c->Cos[φ],s->Sin[φ]}//Simplify

Lemma 8.2.

g[x_]=(Subscript[q, 5])[t,((4s)/(5(1+c)))x];

(Subscript[a, 0])[t_]=(((1+c))^5) (75+15 (c^2)+23 (c^4)-(c^6)+(45 (c^2)+22 (c^4)+(c^6)) (t^2));
(Subscript[a, 1])[t_]=(((1+c))^4) (60 c+136 (c^3)-68 (c^5)+75 (1+c)+15 (c^2) (1+c)+23 (c^4) (1+c)-(c^6) (1+c)+(-60 c-112 (c^3)-52 (c^5)+45 (c^2) (1+c)+22 (c^4) (1+c)+(c^6) (1+c)) (t^2))//Simplify;
(Subscript[a, 2])[t_]=(((1+c))^3) (75+210 c+366 (c^2)+166 (c^3)-474 (c^4)-22 (c^5)+2 (c^6)-2 (c^7)-(c^8)+(-60 c-159 (c^2)-22 (c^3)+147 (c^4)-8 (c^5)+19 (c^6)+2 (c^7)+(c^8)) (t^2))//Simplify;
(Subscript[a, 3])[t_]=(((1+c))^2) (75+573 c+576 (c^2)-812 (c^3)-308 (c^4)+48 (c^5)-20 (c^6)-3 (c^8)-(c^9)+(-252 c-219 (c^2)+395 (c^3)+125 (c^4)-117 (c^5)+11 (c^6)+21 (c^7)+3 (c^8)+(c^9)) (t^2));
(Subscript[a, 4])[t_]=(-1+(c^2))  (-219-867 c-624 (c^2)-388 (c^3)-500 (c^4)-240 (c^5)-28 (c^6)-8 (c^7)-5 (c^8)-(c^9)+(96+348 c-189 (c^2)-365 (c^3)+203 (c^4)+195 (c^5)+61 (c^6)+29 (c^7)+5 (c^8)+(c^9)) (t^2));
(Subscript[a, 5])[t_]=(-1+(c^2)) (-219-291 c-624 (c^2)-964 (c^3)-500 (c^4)-240 (c^5)-28 (c^6)-8 (c^7)-5 (c^8)-(c^9)+(96-612 c-189 (c^2)+211 (c^3)+203 (c^4)+195 (c^5)+61 (c^6)+29 (c^7)+5 (c^8)+(c^9)) (t^2));
(Subscript[a, 6])[t_]=(-1+c) (-219-510 c-1299 (c^2)-1588 (c^3)-1464 (c^4)-740 (c^5)-268 (c^6)-36 (c^7)-13 (c^8)-6 (c^9)-(c^10)+(-288-516 c-417 (c^2)+22 (c^3)+414 (c^4)+398 (c^5)+256 (c^6)+90 (c^7)+34 (c^8)+6 (c^9)+(c^10)) (t^2));

g''[x]-(((((1-c))^1)/((2^3) (((1+c))^2)(((2x+c+(c^2)))^4))))((Subscript[a, 0])[t](1-x)+(Subscript[a, 1])[t]x(1-x)+(Subscript[a, 2])[t](x^2)(1-x)+(Subscript[a, 3])[t](x^3)(1-x)+(Subscript[a, 4])[t](x^4)(1-x)+(Subscript[a, 5])[t](x^5)(1-x)+(Subscript[a, 6])[t](x^6))/.s->Sqrt[1-c^2]//Simplify

All coefficients (Subscript[a, j])[t]are even polynomials of degree 2 and all (Subscript[a, j])[0] and (Subscript[a, j])[1] are polynomials in c:

Grid[Table[{(Subscript[a, j])[0],(Subscript[a, j])[1]}//Simplify,{j,0,6}],Frame->All]

(*) Polynomial p(t)=(Subscript[c, 0])+(Subscript[c, 1])t+(Subscript[c, 2])(t^2)+...+(Subscript[c, n])(t^n) is nonnegative on [0,1] if (Subscript[c, 0])+(Subscript[c, 1])+...+(Subscript[c, k])≥0 for all k=0,...,n.

By (*) for all j=0,...,6 we have  (Subscript[a, j])[0]≥0:

Grid[Table[Table[Total[CoefficientList[(Subscript[a, j])[0],c][[1;;k]]],{k,1,12}],{j,0,6}],Frame->All]

By (*) for all j=0,...,3 we have  (Subscript[a, j])[1]≥0:

Grid[Table[Table[Total[CoefficientList[(Subscript[a, j])[1],c][[1;;k]]],{k,1,10}],{j,0,6}],Frame->All]

For j=4,5,6, a polynomial  (Subscript[a, j])[1] is of the form  (Subscript[a, j])[1]=(1-c)(Subscript[aa, j]) and by (*) for all j=4,5,6 we have  (Subscript[aa, j])≥0:

Grid[Table[Table[Total[CoefficientList[PolynomialQuotient[(Subscript[a, j])[1],1-c,c],c][[1;;k]]],{k,1,9}],{j,4,6}],Frame->All]

Lemma 8.3.

(Subscript[ψ, 5])[t,α5]-(((((1-c))^5)(((1-(t^2)))^4)(t^2))/(16(1+c)))/.{c->Cos[φ],s->Sin[φ]}//Simplify

(Subscript[ψ, 5])[t,a5]-(-(((1+c)(((1-c))^4)(((1-(t^2)))^4))/(1024(((5+3c))^4)(((8+5c+3(c^2)))^2)))(48(15925+4065c+1338(c^2)-3846(c^3)-1071(c^4)-27(c^5))-64(1-c)(((44+3c+9(c^2)))^2)(t^2)))/.{c->Cos[φ],s->Sin[φ]}//Simplify

(Subscript[ψ, 5])[t,b5]-(((1+c)(((1-c))^4)(((1-(t^2)))^4))/(64(((3+2c))^4)(((5+3c+2(c^2)))^2)))(16c(180+169c+123(c^2)+27(c^3)+(c^4))+4(1-c)(((5-2c+2(c^2)))^2)(t^2))/.{c->Cos[φ],s->Sin[φ]}//Simplify

Lemma 8.4.

f[d_]=((8(5+3(c^2))s+20c(9-(c^2))d+150s (d^2)-125c (d^3))/(32(5d+2s c)));

(Subscript[q, 5])[0,d]-(((f[d])^2)-1)/.{c->Cos[φ],s->Sin[φ]}//Simplify

g[d_]=-4(5-6(c^2)+(c^4))s+60(s^2)c d+75(s^3)(d^2)-125c (d^3);

f'[d]-(5/(16(((5d+2 s c))^2)))g[d]/.{c->Cos[φ],s->Sin[φ]}//Simplify

a5-(((2(1-c)s)/(75+95c+30(c^2)))x+((16s)/(25+15c)))/.x->0//Simplify

b5-(((2(1-c)s)/(75+95c+30(c^2)))x+((16s)/(25+15c)))/.x->1//Simplify

g[((2(1-c)s)/(75+95c+30(c^2)))x+((16s)/(25+15c))]-((4(s^3)(((1-c))^2))/((((3+2 c))^3)(((5+3 c))^3)))((3+2c)^3(335+297c+189c^2+27c^3)+30(3+2c)^2(8+5c+3c^2)x+3(5+2c+c^2)(3+2c)(1-3c)x^2-2(1-c)c x^3)/.{c->Cos[φ],s->Sin[φ]}//Simplify

h[d_]=(Subscript[q, 5])[1,d];

h'[a5]-5((4 (((1-c))^2) s )/(256(((5+3 c))^3)(((8+5 c+3 (c^2)))^3))) (440863+1380116 c+1950998 (c^2)+1731140 (c^3)+984672 (c^4)+368604 (c^5)+80946 (c^6)+8748 (c^7)+729 (c^8))/.{c->Cos[φ],s->Sin[φ]}//Simplify

h''[b5]-((375(1-c))/(32(((3+2 c))^2) (((5+3 c+2 (c^2)))^4))) (2755+9325 c+15649 (c^2)+16453 (c^3)+11136 (c^4)+4638 (c^5)+688 (c^6)-356 (c^7)-240 (c^8)-48 (c^9))/.{c->Cos[φ],s->Sin[φ]}//Simplify

h'''[b5]-(-(375/(16(3+2 c)(((5+3 c+2 (c^2)))^5)s)) (650+8850 c+24645 (c^2)+41580 (c^3)+53039 (c^4)+52746 (c^5)+40515 (c^6)+22080 (c^7)+7255 (c^8)+216 (c^9)-1016 (c^10)-480 (c^11)-80 (c^12)))/.{c->Cos[φ],s->Sin[φ]}//Simplify

h''''[d]-(1875/128) (1+((320 (((5+2 (c^2)+(c^4)))^2) (s^4))/(((5 d+2 c s))^6)))/.{c->Cos[φ],s->Sin[φ]}//Simplify

r[β5]-r[a5]-(((((1-c))^2) s(288+791 c+915 (c^2)+624 (c^3)+260 (c^4)+57 (c^5)+9 (c^6)))/(4 (((1+c))^2) (((5+3 c))^2) (2+c+(c^2)) (8+5 c+3 (c^2))))/.{c->Cos[φ],s->Sin[φ]}//Simplify

The algorithm for the radial error

Solve[D[(Subscript[ψ, 5])[t,d],t]==0,t]//Simplify

φ0=(π/4); ϵ=(10^(-20));
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+d{Sin[φ],Cos[φ]};
(Subscript[b, 2])={((5 d (4-5 (d^2)) Cos[φ]+4 (2+5 (d^2)) Sin[φ])/(4 (5 d+2Sin[φ]Cos[φ]))),-((5 d ((4-5 (d^2)) Sin[φ]-6 d Cos[φ]))/(4 (5 d+2Sin[φ]Cos[φ])))};
(Subscript[b, 3])={((5 d (4-5 (d^2)) Cos[φ]+4 (2+5 (d^2)) Sin[φ])/(4 (5 d+2Sin[φ]Cos[φ]))),((5 d ((4-5 (d^2)) Sin[φ]-6 d Cos[φ]))/(4 (5 d+2Sin[φ]Cos[φ])))};(Subscript[b, 5])={Cos[φ],Sin[φ]};(Subscript[b, 4])={Cos[φ],Sin[φ]}+d{Sin[φ],-Cos[φ]};
τl=0;τr=1;
leftedge=a5/.{c->Cos[φ0],s->Sin[φ0]};
rightedge=b5/.{c->Cos[φ0],s->Sin[φ0]};
While[τr-τl>ϵ,τ0=(1/2) (τr+τl);
d0=Select[d/.NSolve[(Subscript[ψ, 5])[τ0,d]==0/.{φ->φ0},d,Reals],#>leftedge&&#<rightedge&][[1]];
If[N[(Subscript[ϕ, 5])[0,d0]+(Subscript[ϕ, 5])[(√(896-9130 (d0^2)-13750 (d0^4)+9375 (d0^6)+(-1248+11575 (d0^2)-55000 (d0^4)+15625 (d0^6)) Cos[2 φ0]+(384-2470 (d0^2)+750 (d0^4)) Cos[4 φ0]-32 Cos[6 φ0]+25 (d0^2) Cos[6 φ0]-2072 d0 Sin[2 φ0]+32800 (d0^3) Sin[2 φ0]-35000 (d0^5) Sin[2 φ0]+1120 d0 Sin[4 φ0]-2000 (d0^3) Sin[4 φ0]-56 d0 Sin[6 φ0]))/((Sqrt[2]) (Sqrt[-(((-4 (1+25 (d0^2)) Cos[φ0]+4 Cos[3 φ0]+5 d0 (14-25 (d0^2)+2 Cos[2 φ0]) Sin[φ0]))^2)])),d0]/.{φ->φ0},20]>0,τl=τ0,τr=τ0]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b53[t,φ0,d0],{t,-1,1},PlotStyle->Red],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4]),(Subscript[b, 5])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4]),(Subscript[b, 5])}]}],AspectRatio->Automatic,PlotRange->All],Plot[(Subscript[ϕ, 5])[t,d0]/.{φ->φ0},{t,-1,1},GridLines->{{},{-(Subscript[ϕ, 5])[0,d0],(Subscript[ϕ, 5])[0,d0]}/.{φ->φ0}}]}]/.{φ->φ0,d->d0}
b53[t,φ0,d0]//N//Simplify

The algorithm for the radial error for nonoptimal solutions

φ0=(π/4); ϵ=(10^(-20));
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+d{Sin[φ],Cos[φ]};
(Subscript[b, 2])={((5 d (4-5 (d^2)) Cos[φ]+4 (2+5 (d^2)) Sin[φ])/(4 (5 d+2Sin[φ]Cos[φ]))),-((5 d ((4-5 (d^2)) Sin[φ]-6 d Cos[φ]))/(4 (5 d+2Sin[φ]Cos[φ])))};
(Subscript[b, 3])={((5 d (4-5 (d^2)) Cos[φ]+4 (2+5 (d^2)) Sin[φ])/(4 (5 d+2Sin[φ]Cos[φ]))),((5 d ((4-5 (d^2)) Sin[φ]-6 d Cos[φ]))/(4 (5 d+2Sin[φ]Cos[φ])))};(Subscript[b, 5])={Cos[φ],Sin[φ]};(Subscript[b, 4])={Cos[φ],Sin[φ]}+d{Sin[φ],-Cos[φ]};
τl=0;τr=1;
leftedge=a5/.{c->Cos[φ0],s->Sin[φ0]};
rightedge=b5/.{c->Cos[φ0],s->Sin[φ0]};
While[τr-τl>ϵ,τ0=(1/2) (τr+τl);
d0=Select[d/.NSolve[(Subscript[ψ, 5])[τ0,d]==0/.{φ->φ0},d,Reals],#>rightedge&][[1]];
If[N[(Subscript[ϕ, 5])[0,d0]+(Subscript[ϕ, 5])[(√(896-9130 (d0^2)-13750 (d0^4)+9375 (d0^6)+(-1248+11575 (d0^2)-55000 (d0^4)+15625 (d0^6)) Cos[2 φ0]+(384-2470 (d0^2)+750 (d0^4)) Cos[4 φ0]-32 Cos[6 φ0]+25 (d0^2) Cos[6 φ0]-2072 d0 Sin[2 φ0]+32800 (d0^3) Sin[2 φ0]-35000 (d0^5) Sin[2 φ0]+1120 d0 Sin[4 φ0]-2000 (d0^3) Sin[4 φ0]-56 d0 Sin[6 φ0]))/((Sqrt[2]) (Sqrt[-(((-4 (1+25 (d0^2)) Cos[φ0]+4 Cos[3 φ0]+5 d0 (14-25 (d0^2)+2 Cos[2 φ0]) Sin[φ0]))^2)])),d0]/.{φ->φ0},20]>0,τl=τ0,τr=τ0]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b53[t,φ0,d0],{t,-1,1},PlotStyle->Red],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4]),(Subscript[b, 5])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4]),(Subscript[b, 5])}]}],AspectRatio->Automatic,PlotRange->All],Plot[(Subscript[ϕ, 5])[t,d0]/.{φ->φ0},{t,-1,1},GridLines->{{},{-(Subscript[ϕ, 5])[0,d0],(Subscript[ϕ, 5])[0,d0]}/.{φ->φ0}}]}]/.{φ->φ0,d->d0}
b53[t,φ0,d0]//N//Simplify

φ0=(π/4); ϵ=(10^(-20));
(Subscript[b, 0])={Cos[φ],-Sin[φ]};(Subscript[b, 1])={Cos[φ],-Sin[φ]}+d{Sin[φ],Cos[φ]};
(Subscript[b, 2])={((5 d (4-5 (d^2)) Cos[φ]+4 (2+5 (d^2)) Sin[φ])/(4 (5 d+2Sin[φ]Cos[φ]))),-((5 d ((4-5 (d^2)) Sin[φ]-6 d Cos[φ]))/(4 (5 d+2Sin[φ]Cos[φ])))};
(Subscript[b, 3])={((5 d (4-5 (d^2)) Cos[φ]+4 (2+5 (d^2)) Sin[φ])/(4 (5 d+2Sin[φ]Cos[φ]))),((5 d ((4-5 (d^2)) Sin[φ]-6 d Cos[φ]))/(4 (5 d+2Sin[φ]Cos[φ])))};(Subscript[b, 5])={Cos[φ],Sin[φ]};(Subscript[b, 4])={Cos[φ],Sin[φ]}+d{Sin[φ],-Cos[φ]};
τl=0;τr=1;
leftedge=a5/.{c->Cos[φ0],s->Sin[φ0]};
rightedge=b5/.{c->Cos[φ0],s->Sin[φ0]};
While[τr-τl>ϵ,τ0=(1/2) (τr+τl);
d0=Select[d/.NSolve[(Subscript[ψ, 5])[τ0,d]==0/.{φ->φ0},d,Reals],#>0&&#<leftedge&][[1]];
If[N[(Subscript[ϕ, 5])[0,d0]+(Subscript[ϕ, 5])[(√(896-9130 (d0^2)-13750 (d0^4)+9375 (d0^6)+(-1248+11575 (d0^2)-55000 (d0^4)+15625 (d0^6)) Cos[2 φ0]+(384-2470 (d0^2)+750 (d0^4)) Cos[4 φ0]-32 Cos[6 φ0]+25 (d0^2) Cos[6 φ0]-2072 d0 Sin[2 φ0]+32800 (d0^3) Sin[2 φ0]-35000 (d0^5) Sin[2 φ0]+1120 d0 Sin[4 φ0]-2000 (d0^3) Sin[4 φ0]-56 d0 Sin[6 φ0]))/((Sqrt[2]) (Sqrt[-(((-4 (1+25 (d0^2)) Cos[φ0]+4 Cos[3 φ0]+5 d0 (14-25 (d0^2)+2 Cos[2 φ0]) Sin[φ0]))^2)])),d0]/.{φ->φ0},20]>0,τl=τ0,τr=τ0]];
GraphicsRow[{Show[ParametricPlot[{Cos[φ],Sin[φ]},{φ,-φ0,φ0},PlotStyle->{Blue,Dashed},Ticks->{{0,0.5,1,1.5},{-1,-0.5,0,0.5,1}}],ParametricPlot[b53[t,φ0,d0],{t,-1,1},PlotStyle->Red],ListPlot[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4]),(Subscript[b, 5])},PlotStyle->{PointSize[0.02],Black}],Graphics[{Black,Line[{(Subscript[b, 0]),(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4]),(Subscript[b, 5])}]}],AspectRatio->Automatic,PlotRange->All],Plot[(Subscript[ϕ, 5])[t,d0]/.{φ->φ0},{t,-1,1},GridLines->{{},{-(Subscript[ϕ, 5])[0,d0],(Subscript[ϕ, 5])[0,d0]}/.{φ->φ0}}]}]/.{φ->φ0,d->d0}
b53[t,φ0,d0]//N//Simplify