Clear[ndf,pdf,cdf]
defaultBlue=ColorData[97][1];

ndf[θ_,α_]:=((α^2)/(π*(((((Cos[θ])^2)*((α^2)-1)+1))^2)))
(*cdf=Function[{θ,α},Evaluate[Integrate[2*π*ndf[θ,α]*Cos[θ]*Sin[θ],θ]]]*)
cdf[θ_,α_]:=((1-((Cos[θ])^2))/(1+((Cos[θ])^2)*((α^2)-1)))
pdf[θ_,α_]:=ndf[θ,α]*Cos[θ]
pdfCos[cosθ_,α_]:=((α^2)/(π*(((((cosθ)^2)*((α^2)-1)+1))^2)))*cosθ

Mip Expression

Clear[mu,mu2,mip2mu2,mu2mip2]
mipMax=8;

(*
mip2mu2[mip_]:=1-(1/3)(2^(2(mip-mipMax)))   (*Solid angles yields θ > (π/2)*)
mu2mip2[μ_]:= mipMax + (1/2)*Log2[3-3*μ]
*)
mip2mu[mip_]:=(1/(Sqrt[1+(2^(2(mip-mipMax)))]))
mu2mip[μ_]:= mipMax + (1/2)*Log2[(1/(μ^2))-1]

mip2mu[8]
2*ArcCos[mip2mu[8]]
N[mu2mip[(1/(Sqrt[2]))]]
N[mip2mu[mipMax]]
N[mu2mip[0.0001]]
N[mu2mip[mip2mu[3]]]
Plot[mu2mip[μ],{μ,(2/3),1},PlotRange->{{(1/(Sqrt[2])),1},{0,mipMax}}]

Compute the correlation between 2 neighbor NDFs separated by θ

Here we have 2 lobes aligned along 2 directions separated by an angle θ and we need to find the correlation least correlation between the 2 although there must be a little correlation anyway otherwise we could always choose roughness 0 for maximization.

It would look like we’re doing the same thing all over again but the normalization by pdf(0,α) makes much more sense than dividing by the cdf!

Showing angular correlation visually:

correl[θ_,α_,θ0_]:=((Max[0,Min[pdf[θ,α],pdf[θ0-θ,α]]])/(pdf[0,α]))
correl[θ_,α_,θ0_]:=((pdf[θ,α]*pdf[θ0-θ,α])/((pdf[0,α])^2))
Manipulate[ParametricPlot[ { 4*θ*{1,Tan[(π/2)-θ0]},{Sin[θ],Cos[θ]},4*θ*{Tan[((θ0)/2)],1},
((pdf[θ,α])/(pdf[0,α]))*{Sin[θ],Cos[θ]},
((pdf[θ0-θ,α])/(pdf[0,α]))*{Sin[θ],Cos[θ]},
correl[θ,α,θ0]*{Sin[θ],Cos[θ]},
(*((((1/(Sqrt[2]))))^2)*{Sin[θ],Cos[θ]}*)
},{θ,-(π/2),(π/2)},PlotRange->{{-1,1},{0,1}},PlotStyle->{LightGray,LightGray,Orange,defaultBlue,defaultBlue,Red,Green}],{{α,0.1},0,1},{{θ0,(π/2)},0,(π/2)}]

Plotting angular correlation as a function of angle:

Manipulate[Show[Plot[correl[θ,α,θ0],{θ,0,θ0},PlotRange->{0,Full}],ParametricPlot[{((θ0)/2),t},{t,-100,100},PlotStyle->Red],ParametricPlot[{t,((((1/(Sqrt[2]))))^2)},{t,0,θ0},PlotStyle->Red]],{{θ0,(π/10)},0,(π/2)},{{α,0.1},0,1}]

correl[(π/4),1,(π/2)]

Finding a Mapping

So okay, we can accept a certain amount of correlation that is given by the value at ((Subscript[θ, 0])/2)when α=1 and (Subscript[θ, 0])=(π/2)
Now we can find the roughness for a given (Subscript[θ, 0]) :

	c=(((((pdf(((Subscript[θ, 0])/2),α) )/(pdf(0,α)))))^2)

We can totally get rid of the squaring and rewrite the same expression for a new c.

By fixing c = ((pdf((π/4),1) )/(pdf(0,1))) corresponding to the correlation of the worst mip that sees a (π/2) angle (a full face of the cube) and must correspond to the roughness value α=1.

We find:	c=(1/(Sqrt[2]))

((ndf[(π/4),1]*Cos[(π/4)])/(pdf[0,1]))
((pdf[(π/4),1])/(pdf[0,1]))

Clear[c,θ,α]
(*Solve[((pdf[θ,α])/(pdf[0,α]))==c,α]*)
Solve[(((α^4)*b)/((((b^2)((α^2)-1)+1))^2))==c,α]

Unfortunately, it’s impossible to invert the result...

(*Solve[(((α^4)*b)/((((b^2)((α^2)-1)+1))^2))==c,b]*)

sqAlpha:=Function[θ,
μ=Cos[θ];
c=(1/(Sqrt[2]));
(((μ^4)*c-(μ^2)*c-(Sqrt[μ*(((1-(μ^2)))^2)*c]))/((μ^4)*c-μ))
]
sqAlphaMu:=Function[μ,
c=(1/(Sqrt[2]));
(((μ^4)*c-(μ^2)*c-(Sqrt[μ*(((1-(μ^2)))^2)*c]))/((μ^4)*c-μ))
]

Plot[(Sqrt[sqAlpha[(θ/2)]]),{θ,0,(π/2)},PlotRange->{0,1},AxesLabel->{"θ","α"}]

Plot[{(Sqrt[sqAlpha[(θ/2)]]),sqAlpha[(θ/2)]},{θ,0,(π/2)},PlotRange->{0,1},AxesLabel->{"θ","α"}]
Plot[{(Sqrt[sqAlphaMu[μ]]),sqAlphaMu[μ]},{μ,(1/(Sqrt[2])),1},PlotRange->{{0.70,1},{0,1}},AxesLabel->{"cos(θ)","α"}]

sqRoughnessFromMip[mip_]:=sqAlphaMu[mip2mu[mip]]
(*sqRoughnessFromMip2[mip_]:=sqAlphaMu[mip2mu2[mip] ]*)
Table[N[(Sqrt[sqRoughnessFromMip[mip]])],{mip,0,mipMax}]
Plot[{sqAlpha[(θ/2)],(Sqrt[sqAlpha[(θ/2)]])},{θ,0,(π/2)},PlotRange->{0,1}]
(*Plot[{2*ArcCos[mip2mu[mip]],2*ArcCos[mip2mu2[mip]]},{mip,0,mipMax},PlotRange->{0,1.1*(π/2)}]*)
Plot[(Sqrt[sqRoughnessFromMip[mip]]),{mip,0,mipMax},PlotRange->{0,1},AxesLabel->{"mip","α"}]

(*Manipulate[ParametricPlot[ { 4*θ*{1,Tan[(π/4)-((θ0)/2)]},4*θ*{1,Tan[(π/4)+((θ0)/2)]},{Sin[θ],Cos[θ]},4*θ*{1,1},
((pdf[Abs[θ-((π/4)-((θ0)/2))],(Sqrt[sqAlpha[((θ0)/2)]])])/(pdf[0,(Sqrt[sqAlpha[((θ0)/2)]])]))*{Sin[θ],Cos[θ]},
((pdf[Abs[((π/4)+((θ0)/2))-θ],(Sqrt[sqAlpha[((θ0)/2)]])])/(pdf[0,(Sqrt[sqAlpha[((θ0)/2)]])]))*{Sin[θ],Cos[θ]},
(((pdf[Abs[θ-((π/4)-((θ0)/2))],(Sqrt[sqAlpha[((θ0)/2)]])]*pdf[Abs[((π/4)+((θ0)/2))-θ],(Sqrt[sqAlpha[((θ0)/2)]])]))/((pdf[0,(Sqrt[sqAlpha[((θ0)/2)]])])^2))*{Sin[θ],Cos[θ]},
((((1/(Sqrt[2]))))^2)*{Sin[θ],Cos[θ]}
},{θ,0,(π/2)},PlotRange->{{0,1},{0,1}},PlotStyle->{LightGray,LightGray,LightGray,Orange,defaultBlue,defaultBlue,Red,Green}],{{θ0,(π/2)},0.001,(π/2)}]*)

Manipulate[ParametricPlot[ { 4*θ*{1,Tan[(π/2)-θ0]},{Sin[θ],Cos[θ]},4*θ*{Tan[((θ0)/2)],1},
((pdf[θ,(Sqrt[sqAlpha[((θ0)/2)]])])/(pdf[0,(Sqrt[sqAlpha[((θ0)/2)]])]))*{Sin[θ],Cos[θ]},
((pdf[θ0-θ,(Sqrt[sqAlpha[((θ0)/2)]])])/(pdf[0,(Sqrt[sqAlpha[((θ0)/2)]])]))*{Sin[θ],Cos[θ]},
correl[θ,(Sqrt[sqAlpha[((θ0)/2)]]),θ0]*{Sin[θ],Cos[θ]},
((((1/(Sqrt[2]))))^2)*{Sin[θ],Cos[θ]}
},{θ,-(π/2),(π/2)},PlotRange->{{-1,1},{0,1}},PlotStyle->{LightGray,LightGray,Orange,defaultBlue,defaultBlue,Red,Green}],{{θ0,(π/2)},0.001,(π/2)}]

Fitting

The expression to get μ from roughness is a nightmare to invert so let’s do a curve fitting instead...

data=Table[{ N[sqAlphaMu[μ]],N[μ]},{μ,(1/(Sqrt[2])),1,((1-(1/(Sqrt[2])))/100)}]
ListLinePlot[data,PlotRange->Full]

Clear[a,b,c,d,e,f]
model=((a+b*x+c*(x^2))/(d+e*x+f*(x^2)))
model=((a+b*x+c*(x^2))/(a+e*x+f*(x^2)))

fSqRoughness2Cos2D=Function[x,Evaluate[model/.FindFit[data,model,{a,b,c,d,e,f},x]]]
Show[ListLinePlot[data,PlotRange->{(1/(Sqrt[2])),1},AxesLabel->{"(α^2)","cos(((Subscript[θ, 0])/2))"}],Plot[fSqRoughness2Cos2D[x],{x,0,1},PlotStyle->Red]]
(1/(Length[data]))Sum[(((fSqRoughness2Cos2D[data[[i]][[1]]] - data[[i]][[2]]))^2),{i,1,Length[data]}]
(Sqrt[2])fSqRoughness2Cos2D[1]

((1.2547008217731768` -1.870418495301146` x+0.72023546942613` (x^2))/(1.2547008217731768` -1.7522365852611441` x+0.6453293459673146` (x^2)))

Plot[mu2mip[fSqRoughness2Cos2D[(α^2)]],{α,0,1},PlotRange-> {0,mipMax},AxesLabel->{"α","mip"}]

3D Lobes Inter-Penetration Modeling

We’re starting over again but this time measuring the ratio of volume that get interpenetrated depending on the angle between lobes.
We know a volume element is given by dV=(r^2)sin(θ) dr dθ dϕ and the volume integral is:

	I=(∫_(-π)^π)(∫_0^(π/2))(∫_0^(r(θ,ϕ,α)))(r^2) sin(θ)dθdϕ=(∫_(-π)^π)(∫_0^(π/2)) (1/3)((r(θ,ϕ,α))^3) sin(θ)dθdϕ

In one case, we’re looking for the entire volume so r(θ,ϕ,α)=r(θ )=D(θ,α) cos(θ) the BRDF’s normal distribution function times the cosine of the angle.
In the second case we’re looking for the intersection of 2 volumes and r(θ,ϕ,α)=r(θ )=min( D(θ,α) cos(θ), D(θ',α) cos(θ') ).

Here, cos(θ')= sin( θ ) cos( ϕ ) sin( (Subscript[θ, 0])) + cos( θ ) cos( (Subscript[θ, 0]) ) and (Subscript[θ, 0]) is the cone’s aperture angle.

ParametricPlot[Cos[θ]*{Sin[θ],Cos[θ]},{θ,-(π/2),(π/2)}]

cosAngleChange[θ_,ϕ_,γ_]:=Sin[θ]*Cos[ϕ]*Sin[γ]+Cos[θ]*Cos[γ]
penelobe[θ_,ϕ_,γ_,α_]:=π*Max[0,Min[pdf[θ,α],pdfCos[cosAngleChange[θ,ϕ,γ],α]]]
Manipulate[((2π)/3)*NIntegrate[((Cos[θ])^3)*Sin[θ],{θ,0,(π/2)}],{{α,1},0.001,1}]
Manipulate[((2π)/3)*NIntegrate[(((π*pdf[θ,α]))^3)*Sin[θ],{θ,0,(π/2)}],{{α,1},0.001,1}]
Manipulate[(1/3)NIntegrate[((penelobe[θ,ϕ,θ0,α])^3)*Sin[θ],{θ,0,(π/2)},{ϕ,-π,π}],{{α,1},0.001,1},{{θ0,(π/2)},0,(π/2)}]

Manipulate[ParametricPlot3D[{
((pdf[θ,α])/(pdf[0,α]))*{Sin[θ]*Cos[ϕ], Sin[θ]*Sin[ϕ], Cos[θ]},
((pdfCos[cosAngleChange[θ,ϕ,θ0],α])/(pdf[0,α]))*{Sin[θ]*Cos[ϕ], Sin[θ]*Sin[ϕ], Cos[θ]},
((Min[pdf[θ,α],pdfCos[cosAngleChange[θ,ϕ,θ0],α]])/(pdf[0,α]))*{Sin[θ]*Cos[ϕ], Sin[θ]*Sin[ϕ], Cos[θ]}
},{θ,0,(π/2)},{ϕ,-π,π},PlotRange->{{-1,1},{-1,1},{0,1}},AxesLabel->{"x","y","z"},PlotStyle->{Opacity[0.125],Opacity[0.125],Directive[Red,Opacity[1]]}],{{α,1},0.0001,1},{{θ0,(π/2)},0.0001,(π/2)}]

SetDirectory["D:\\Workspaces\\Patapom.com\\Web\\blog\\patapom\\docs\\BRDF"]; 
countX=128;
countY=64;
fileName="VolumeInterPenetrationProper_"<>ToString[countX]<>"x"<>ToString[countY]<>".csv"

(*
ta=ParallelTable[{N[θ0],N[α],
(1/3)NIntegrate[((penelobe[θ,ϕ,θ0,α])^3)*Sin[θ],{θ,0,(π/2)},{ϕ,-π,π}, PrecisionGoal->2,MaxRecursion->4,WorkingPrecision->$MachinePrecision]
}
,{θ0,0,(π/2),((π/2)/countX)}
,{α,0.001,1,((1-0.001)/countY)}]
Export[fileName,ta,"CSV"];
*)

ta2=Import[fileName,"CSV"];
ta2=ToExpression[ta2]

Let’s create a normalized table by dividing by the total lobe volume

ta2[[129]][[65]]

lobeVolume[α_]:=((2π)/3)*NIntegrate[(((π*pdf[θ,α]))^3)*Sin[θ],{θ,0,(π/2)}]
taLobeVolumes=Table[lobeVolume[0.001+((1-0.001)/countY)*y],{y,0,countY}]

taNorm=Table[ta2[[x]][[y]]*{1,1,(1/(taLobeVolumes[[y]]))},{x,1,countX+1},{y,1,countY+1}]
ref=taNorm[[countX+1]][[countY+1]][[3]]

Show[ListPlot3D[Flatten[taNorm,1],PlotRange->{0,1},AxesLabel->{"θ","α","Lobe Penetration Ratio"}],Plot3D[ref,{x,0,(π/2)},{y,0,1},PlotStyle->Directive[Green,Opacity[0.5]]]]

ListInterpolation[Table[taNorm[[1]][[y]][[3]],{y,1,countY+1}],{0,1}]

Clear[α]
taRoughnessFunctions=Table[ListInterpolation[Table[taNorm[[θ0]][[y]][[3]],{y,1,countY+1}],{0,1}],{θ0,1,countX+1}];
Plot[taRoughnessFunctions[[64]][α],{α,0,1}]
ref
NSolve[taRoughnessFunctions[[64]][α]==ref,α]

Clear[α]
roughnessMap=Table[{N[((θ0-1)/(countX+1))*(π/4)],α/.NSolve[taRoughnessFunctions[[θ0]][α]==ref,α][[1]]},{θ0,2,countX+1}];
roughnessMapCos=Table[{N[((Cos[((θ0-1)/(countX+1))*(π/4)])^2)],(α^2)/.NSolve[taRoughnessFunctions[[θ0]][α]==ref,α][[1]]},{θ0,2,countX+1}];
ListLinePlot[roughnessMap]
ListLinePlot[roughnessMapCos]

roughnessMap=Delete[roughnessMap,100];
roughnessMapCos=Delete[roughnessMapCos,{{123},{100}}];
ListLinePlot[roughnessMap]
ListLinePlot[roughnessMapCos]
roughnessMapCos[[1]]
roughnessMapCos[[ Length[roughnessMapCos] ]]

Let’s fit this...

Clear[a,b,c,d,e,f,x,model,SqCos2SqRoughness3D]

roughnessMapCos;
roughnessMapCosTweaked=Map[Function[x,{2*(1-x[[1]]),x[[2]]}],roughnessMapCos];

minSqRoughness=0.0001;
model=((a+b*x+c*(x^2))/(d+e*x+f*(x^2)));
model=minSqRoughness+0*a+b*x+c*(x^2)+d*(x^3);
model
fTemp=Function[x,Evaluate[model/.FindFit[roughnessMapCosTweaked,model,{a,b,c,d,e,f},x]]]
(*SqCos2SqRoughness3D=Function[{x},fTemp[2*(1-x)]];*)
SqCos2SqRoughness3D=Function[y,x=2*(1-y); 0.0001 +1.5691217003175857*x-1.2279356901888097*(x^2)+0.658713989871224*(x^3)]; (* Tweaked so roughness = 0.01 at x=0.5 *)

(1/(Length[roughnessMapCosTweaked] ))Sum[(fTemp[roughnessMapCosTweaked[[i]][[1]]]-roughnessMapCosTweaked[[i]][[2]])^2,{i,1,Length[roughnessMapCosTweaked]}]
(1/(Length[roughnessMapCos])) Sum[(SqCos2SqRoughness3D[roughnessMapCos[[i]][[1]]]-roughnessMapCos[[i]][[2]])^2,{i,1,Length[roughnessMapCos]}]
{SqCos2SqRoughness3D[1/2],SqCos2SqRoughness3D[1]}

Show[ListLinePlot[roughnessMapCosTweaked,AxesLabel->{"2*(1-cos(((Subscript[θ, 0])/2))^2)","(α^2)"}],Plot[fTemp[x],{x,0,1},PlotRange->{0,1},PlotStyle->Red]]
Show[ListLinePlot[roughnessMapCos,AxesLabel->{"cos(((Subscript[θ, 0])/2))^2","(α^2)"}],Plot[SqCos2SqRoughness3D[x],{x,(1/2),1},PlotRange->{0,1},PlotStyle->Red,PlotRange->Full]]

0.0001` +1.5691217003175857` x-1.221818447413178` (x^2)+0.658713989871224` (x^3)
0.0001 +1.5691217003175857-1.221818447413178+0.658713989871224
0.0001 +1.5691217003175857-(1.221818447413178+0.0061172427756317)+0.658713989871224
1.221818447413178+0.0061172427756317
1.2279356901888097`
0.0001 +1.5691217003175857-1.2279356901888097+0.658713989871224

sqRoughnessFromMip3D[mip_]:=SqCos2SqRoughness3D[((mip2mu[mip])^2)]
Plot[{((sqAlpha[(θ/2)])^(1/2)),((SqCos2SqRoughness3D[((Cos[(θ/2)])^2)])^(1/2))},{θ,0,(π/2)},PlotRange->{0,1},AxesLabel->{"θ","α"}]
Plot[{(Sqrt[sqRoughnessFromMip[mip]]),(Sqrt[sqRoughnessFromMip3D[mip]])},{mip,0,mipMax},PlotRange->{0,1},AxesLabel->{"mip","α"}]
Plot[(Sqrt[sqRoughnessFromMip3D[mip]]),{mip,0,mipMax},PlotRange->{0,1},AxesLabel->{"mip","α"}]

Let’s map the inverse function now:

data=Table[{ N[SqCos2SqRoughness3D[(μ^2)]],N[(μ^2)]},{μ,(1/(Sqrt[2])),1,((1-(1/(Sqrt[2])))/100)}];
(* Let's remap results in [0,1] ascending *)
dataTweaked=Table[{ N[SqCos2SqRoughness3D[(μ^2)]],N[2*(1-(μ^2))]},{μ,(1/(Sqrt[2])),1,((1-(1/(Sqrt[2])))/100)}];
ListLinePlot[data,PlotRange->Full]
ListLinePlot[dataTweaked,PlotRange->Full,AxesLabel->{"(α^2)","2*(1-(μ^2))"}]
dataTweaked[[1]]
dataTweaked[[Length[dataTweaked]]]

Clear[a,b,c,d,e,f,x,y]
model=0+b*x+c*(x^2)+d*(x^3);
model=((a+b*x+c*(x^2))/(d+e*x+f*(x^2)));
model=((0+b*x+c*(x^2))/(a+e*x+f*(x^2)));
model


(*fSqRoughness2SqCos3D=Function[x,Evaluate[model/.FindFit[data,model,{a,b,c,d,e,f},x]]] (* Directly fit data (error^2)= 2.993373400293064`*^-7 *) *)

fTemp2=Function[x,Evaluate[model/.FindFit[dataTweaked,model,{a,b,c,d,e,f},x]]] (* Fit tweaked data with simpler model (error^2)= 4.006623686440905`*^-7 *)
fTemp2=Function[x,((1.3675780565697848` x-0.46349896024802306` (x^2))/(2.259534233469395` -2.4219546454815024` x+1.0664995083338693` (x^2)))] (* Tweaked so it's exactly 1 when x=1 *)
fSqRoughness2SqCos3D=Function[x,1-((fTemp2[x])/2)]

(1/(Length[data]))Sum[(((fSqRoughness2SqCos3D[data[[i]][[1]]] - data[[i]][[2]]))^2),{i,1,Length[data]}]
fSqRoughness2SqCos3D[1]
fSqRoughness2SqCos3D[0]

Show[ListLinePlot[dataTweaked,PlotRange->{0,1},AxesLabel->{"(α^2)","2*(1-cos(((Subscript[θ, 0])/2))^2)"}],Plot[fTemp2[x],{x,0,1},PlotStyle->Red]]
Show[ListLinePlot[data,PlotRange->{0.98*(1/2),1},AxesLabel->{"(α^2)","cos(((Subscript[θ, 0])/2))^2"}],Plot[fSqRoughness2SqCos3D[x],{x,0,1},PlotStyle->Red]]

Let’s tweak the fitting so that it’s exactly 1 when x=1

((1.3675780565697848` x-0.46648359430374686` (x^2))/(2.259534233469395` -2.4219546454815024` x+1.0664995083338693` (x^2)))

1.3675780565697848-0.46648359430374686
2.259534233469395 -2.4219546454815024+1.0664995083338693
0.9040790963217618-0.901094462266038
0.0029846340557238005
1.3675780565697848-(0.46648359430374686-0.0029846340557238005)
0.46648359430374686-0.0029846340557238005

((1.3675780565697848` x-0.46349896024802306` (x^2))/(2.259534233469395` -2.4219546454815024` x+1.0664995083338693` (x^2)))

Let’s plot and compare with the 2D model:

Plot[{fSqRoughness2Cos2D[β],((fSqRoughness2SqCos3D[β])^(1/2))},{β,0,1},AxesLabel->{"(α^2)","cos(((Subscript[θ, 0])/2))"}]

Show a plot of mip level as a function of square roughness

Plot[mu2mip[((fSqRoughness2SqCos3D[(α^2)])^(1/2))],{α,0,1},PlotRange-> {0,mipMax},AxesLabel->{"α","mip"}]

Plot[{mu2mip[fSqRoughness2Cos2D[(α^2)]],mu2mip[((fSqRoughness2SqCos3D[(α^2)])^(1/2))]},{α,0,1},PlotRange-> {0,mipMax},AxesLabel->{"α","mip"}]

Analytical Inverse

It just came to me that we could totally reverse the fSqRoughness2SqCos expression instead of fitting another approximate curve to something that’s already approximate!

Clear[a,b,c,d]
Solve[((a+b*(x^2))/(a+c*x+d*(x^2)))==y,x]
Solve[((a+b*(x^2))/(a+c*x+d*(x^2)))==y,x]/.{a->5.238881411332154,b ->  -1.5797129260127443`,c->1.5265349072841383`,d->0.6923074066106989`}

Clear[a,b,c,d,x,y]
Solve[a+b*x+c*(x^2)+d*(x^3)==y,x]
Solve[a+b*x+c*(x^2)+d*(x^3)==y,x]/.{a->0.01,b-> 1.4827516714288516, c-> -1.0381908729571114, d-> 0.5454431449123816}

Clear[x,y,fSqRoughness2SqCos2]
fSqCos2SqRoughness2=Function[{y},
a=5.238881411332154;
b =  -1.5797129260127443`;
c=1.5265349072841383`;
d=0.6923074066106989`;
((c y-(Sqrt[(c^2) (y^2)-4 (a-a y) (b-d y)]))/(2 (b-d y)))
];
Plot[{fSqCos2SqRoughness[μ2],fSqCos2SqRoughness2[μ2]},{μ2,(1/2),1}]

Clear[fSqRoughness2SqCos,fSqRoughness2SqCos2]
fSqRoughness2SqCos[x_]:= ((5.238881411332154` -1.5797129260127443` (x^2))/(5.238881411332154` +1.5265349072841383` x+0.6923074066106989` (x^2)))
fSqRoughness2SqCos2=Function[x,
y=0.634463238830562` -((1.038247677392446`)/(((-5.399114064617517`+8.03272205696155` y+(Sqrt[9.807202175962217` +(((-5.399114064617517`+8.03272205696155` y))^2)])))^(1/3)))+0.4850493483371199` (((-5.399114064617517`+8.03272205696155` y+(Sqrt[9.807202175962217` +(((-5.399114064617517`+8.03272205696155` y))^2)])))^(1/3));
1-(y/2)
];
Plot[{fSqRoughness2SqCos[β],fSqRoughness2SqCos2[β]},{β,0,1}]

D[(((R^2)-(r^2))/((R^2)-2r*(R^2)*Cos[θ-ϕ]+(r^2))),{r,2}]
D[(((R^2)-(r^2))/((R^2)-2r*(R^2)*Cos[θ-ϕ]+(r^2))),{θ,2}]

Integrate[((Cos[θ]*Sin[θ])/π),{ϕ,0,2π},{θ,0,(π/2)}]
Integrate[((Cos[θ]*Sin[θ])/π),θ]