Solution of the shape equation -- Two cylinders on a membrane

Outer section

Shape in angle-arc length parametrization

Integrate[((-λ)/(Sqrt[2*(1 - Cos[ps])])), {ps, alpha0, psi}, 
      Assumptions -> {λ > 0, alpha0 > 0, alpha0 < 2  π, 
          psi < alpha0, psi > 0}]

Solve[% == s, psi]

psiout[s_, alpha0_, λ_] := 
      4 ArcTan[(E^(-(s/λ))) Tan[
              (alpha0/4)]]

FullSimplify[((λ^2)/2)*D[psiout[s, alpha0, λ], s]^2 + 
        1*Cos[psiout[s, alpha0, λ]]]

Curvature

K(s)

FullSimplify[-D[psiout[s, alpha0, λ], s], 
      Assumptions -> {λ > 0, alpha0 > 0, 
          alpha0 < 2  π}]

Kout[s_, 
        alpha0_, λ_] := ((4 (E^(s/λ)) Tan[(alpha0/4)])/(λ ((E^(((2 s)/λ))) + Tan[(alpha0/4)]^2)))

K(0)

FullSimplify[Kout[s, alpha0, λ] /. {s -> 0}]

Cartesian coordinates

Δx(s)

Integrate[
      1 - ((λ^2)/2) (((Kout[sp, alpha0, λ]))^2), {sp, 0, 
        s}, Assumptions -> {λ > 0}]

Δxout[s_, alpha0_, λ_] := 
      s + (((-1 + (E^(((2 s)/λ)))) λ (-1 + Cos[alpha0]))/(1 - Cos[(alpha0/2)] + (E^(((2 s)/λ))) (1 + Cos[(alpha0/2)])))

Δh(s)

FullSimplify[(λ^2)*(FullSimplify[
              Kout[s, alpha0, λ] /. {s -> 0}] - 
            Kout[s, alpha0, λ])]

Δhout[s_, alpha0_, λ_] := 
      2 λ (Sin[
              (alpha0/2)] - ((2 (E^(s/λ)) Tan[(alpha0/4)])/((E^(((2 s)/λ))) + Tan[(alpha0/4)]^2)))

Δh(s->∞)

FullSimplify[(λ^2)*
          Kout[s, alpha0, λ] /. {s -> 0}]

Plot of the shape

a0 = 80  °; lb = (Sqrt[2]);
ParametricPlot[{-Δxout[s, a0, 
              lb], Δhout[s, a0, lb]}, {s, -5, 10}, 
        AspectRatio -> Automatic, 
        AxesLabel -> {"-Δx", "Δh"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 300];

Inner section

a) 0 < η < 1

Shape in angle-arc length parametrization

psisym1[s_, η_, m_, λ_] := π + 
        2 JacobiAmplitude[(s/λ)*(Sqrt[(η/m)]) - EllipticK[m], 
            m]

(* check whether psisym1[s, η, 
          m, λ] solves the integral of the shape equation *)

FullSimplify[
        FullSimplify[
            TrigExpand[((λ^2)/2)*
                  D[psisym1[s, (m/(2 - m)), m, λ], s]^2 + 
                (m/(2 - m))*Cos[psisym1[s, (m/(2 - m)), m, λ]]], 
            Assumptions -> {λ > 0, m < 1, 
                m > 0}] /. {JacobiDN[((s )/(λ*(Sqrt[2 - m]))) - EllipticK[m], m]^2 -> 
              1 - m*JacobiSN[((s )/(λ*(Sqrt[2 - m]))) - EllipticK[m], m]^2}] /. {JacobiCN[(s/((Sqrt[2 - m]) λ)) - EllipticK[m], m]^2 + 
            JacobiSN[(s/((Sqrt[2 - m]) λ)) - EllipticK[m], m]^2 -> 1}

psisym1[0, η, m, λ]

Curvature

FullSimplify[D[psisym1[s, η, m, λ], s], 
      Assumptions -> {m > 0, m < 1}]

Ksym1[s_, η_, 
        m_, λ_] := ((2 (Sqrt[(η/m)]) JacobiDN[((s (Sqrt[(η/m)]))/λ) - EllipticK[m], m])/λ)

Cartesian coordinates

Δx(s)

(1/η)*
        Integrate[
          1 - ((λ^2)/2)*Ksym1[sp, η, m, λ]^2, {sp, 0, 
            s}] /. {JacobiDN[((s (Sqrt[(η/m)]))/λ) - 
              EllipticK[m], 
            m] -> (Sqrt[1 - m JacobiSN[((s (Sqrt[(η/m)]))/λ) - EllipticK[m], m]^2])}

(JacobiAmplitude[-EllipticK[m], 
            m] /. {m -> 0.6}) + (π/2)

EllipticE[((-π)/2), m]

Δxsym1[s_, η_, 
        m_, λ_] := ((s - 2 (Sqrt[(η/m)]) λ (EllipticE[JacobiAmplitude[((s (Sqrt[(η/m)]))/λ) - EllipticK[m], m], m] + EllipticE[m]))/η)

Δh(s)

FullSimplify[((λ^2)/η)*(Ksym1[s, η, m, λ] - 
            Ksym1[0, η, m, λ])]

Δhsym1[s_, η_, 
        m_, λ_] := ((2 (Sqrt[(η/m)]) λ (-(Sqrt[1 - m]) + JacobiDN[((s (Sqrt[(η/m)]))/λ) - EllipticK[m], m]))/η)

Plot of the shape

lb = (Sqrt[2]); ηplt = 0.98;
ParametricPlot[{-Δxsym1[
              s, ηplt, ((2*ηplt)/(ηplt + 1)), 
              lb], -Δhsym1[
              s, ηplt, ((2*ηplt)/(ηplt + 1)), 
              lb]}, {s, -9, 10}, AspectRatio -> Automatic, 
        AxesLabel -> {"-Δx", "-Δh"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 300];

b)  η = 0

Shape in angle-arc length parametrization

pssym2[s_, λ_] := ((Sqrt[2])/λ)*s

(* check whether psisym2[
          s, λ] solves the integral of the shape equation *)

((λ^2)/2)*D[pssym2[s, λ], s]^2 + 
      0*Cos[pssym2[s, λ]]

Curvature

Ksym2[s_, λ_] := ((Sqrt[2])/λ)

Cartesian coordinates

Δx(s)

Integrate[Cos[pssym2[sp, λ]], {sp, 0, s}]

Δxsym2[
        s_, λ_] := ((λ Sin[(((Sqrt[2]) s)/λ)])/(Sqrt[2]))

Δh(s)

Integrate[Sin[pssym2[sp, λ]], {sp, 0, s}, 
      Assumptions -> {λ > 0, s > 0}]

Δhsym2[
        s_, λ_] := (Sqrt[2]) λ Sin[(s/((Sqrt[2]) λ))]^2

Plot of the shape

lb = (Sqrt[2]);
ParametricPlot[{-Δxsym2[s, 
              lb], -Δhsym2[s, lb]}, {s, 0, 10}, 
        AspectRatio -> Automatic, 
        AxesLabel -> {"-Δx", "-Δh"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 300];

c) -1 < η < 0

Shape in angle-arc length parametrization

Integrate[(λ/(Sqrt[2*(1 - η*Cos[ps])])), {ps, 0, psi}, 
      Assumptions -> {λ > 0, 
          psi > 0, η < 0, η > -1}]

Solve[% == s, psi]

psisym3[s_, η_, m_, λ_] := 
      2 JacobiAmplitude[((s*(Sqrt[Abs])[η])/(λ*(Sqrt[m]))), 
          m]

(* check whether psisym3[s, η, 
          m, λ] solves the integral of the shape equation *)

FullSimplify[
        FullSimplify[
            TrigExpand[((λ^2)/2)*
                  D[psisym3[s, (m/(m - 2)), m, λ], s]^2 + 
                (m/(m - 2))*Cos[psisym3[s, (m/(m - 2)), m, λ]]], 
            Assumptions -> {λ > 0, m < 1, 
                m > 0}] /. {JacobiDN[((s )/(λ*(Sqrt[2 - m]))), m]^2 -> 
              1 - m*JacobiSN[((s )/(λ*(Sqrt[2 - m]))), m]^2}, Assumptions -> {m < 1, λ > 0, 
            m > 0}] /. {JacobiCN[(s/((Sqrt[2 - m]) λ)), m]^2 + 
            JacobiSN[(s/((Sqrt[2 - m]) λ)), m]^2 -> 
          1}

Curvature

FullSimplify[D[psisym3[s, η, m, λ], s], 
      Assumptions -> {m < 1, m > 0}]

Ksym3[s_, η_, 
        m_, λ_] := ((2 (Sqrt[Abs])[η] JacobiDN[((s (Sqrt[Abs])[η])/((Sqrt[m]) λ)), m])/((Sqrt[m]) λ))

Cartesian coordinates

Δx(s)

(1/η)*
        Integrate[
          1 - ((λ^2)/2)*Ksym3[sp, η, m, λ]^2, {sp, 0, 
            s}] /. {JacobiDN[((s (Sqrt[Abs])[η])/((Sqrt[m]) λ)), 
            m] -> (Sqrt[1 - m JacobiSN[((s (Sqrt[Abs])[η])/((Sqrt[m]) λ)), m]^2])}

Δxsym3[s_, η_, 
        m_, λ_] := ((-s + ((2 λ (Sqrt[Abs])[η] EllipticE[JacobiAmplitude[((s (Sqrt[Abs])[η])/((Sqrt[m]) λ)), m], m])/(Sqrt[m])))/Abs)[η]

Δh(s)

FullSimplify[-((λ^2)/Abs)[η]*(Ksym3[s, η, 
              m, λ] - Ksym3[0, η, m, λ])]

Δhsym3[s_, η_, 
        m_, λ_] := -((2 λ (-1 + 
                  JacobiDN[((s (Sqrt[Abs])[η])/((Sqrt[m]) λ)), 
                    m]))/((Sqrt[m]) (Sqrt[Abs])[η]))

Plot of the shape

lb = (Sqrt[2]); ηplt = -0.98;
ParametricPlot[{-Δxsym3[
              s, ηplt, ((2  Abs[ηplt])/(1 + Abs[ηplt])), 
              lb], -Δhsym3[
              s, ηplt, ((2  Abs[ηplt])/(1 + Abs[ηplt])), 
              lb]}, {s, -5, 10}, AspectRatio -> Automatic, 
        AxesLabel -> {"-Δx", "-Δh"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 300];

d) η = -1

Shape in angle-arc length parametrization

-Integrate[((-λ)/(Sqrt[2*(1 + Cos[ps])])), {ps, 0, psi}, 
        Assumptions -> {λ > 0, psi > 0, 
            psi < π}]

FullSimplify[Solve[% == s, psi]]

psisym4[s_, λ_] := 
      4 ArcTan[Tanh[(s/(2 λ))]]

(* check whether psisym4[
          s, λ] solves the integral of the shape equation *)

FullSimplify[((λ^2)/2)*D[psisym4[s, λ], s]^2 - 
        1*Cos[psisym4[s, λ]]]

Curvature

FullSimplify[D[psisym4[s, λ], s], 
      Assumptions -> λ > 0]

Ksym4[
        s_, λ_] := ((2 Sech[(s/λ)])/λ)

Cartesian coordinates

Δx(s)

(1/(-1))*
      Integrate[1 - ((λ^2)/2)*Ksym4[sp, λ]^2, {sp, 0, s}, 
        Assumptions -> {λ > 0, Re[s] ≠ 0}]

Δxsym4[s_, λ_] := -s + 
        2 λ Tanh[(s/λ)]

Δh(s)

FullSimplify[((λ^2)/(-1))*(Ksym4[s, λ] - 
            Ksym4[0, λ])]

Δhsym4[
        s_, λ_] := -2 λ (-1 + 
            Sech[(s/λ)])

Plot of the shape

lb = (Sqrt[2]);
ParametricPlot[{-Δxsym4[s, 
              lb], -Δhsym4[s, lb]}, {s, -5, 10}, 
        AspectRatio -> Automatic, 
        AxesLabel -> {"-Δx", "-Δh"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 300];

e)  η < -1

Shape in angle-arc length parametrization

psisym5[s_, η_, m_, λ_] := 
      Re[ArcCos[
          1 - (2/m)*JacobiSN[(s/λ)*(Sqrt[Abs])[η], (1/m)]^2]]

(* check whether psisym5[s, η, 
          m, λ] solves the integral of the shape equation *)

psisym5test[s_, η_, m_, λ_] := 
      ArcCos[1 - 
          (2/m)*JacobiSN[(s/λ)*(Sqrt[Abs])[η], (1/m)]^2]

FullSimplify[
        FullSimplify[
            TrigExpand[((λ^2)/2)*
                  D[psisym5test[s, (m/(m - 2)), m, λ], s]^2 + 
                (m/(m - 2))*
                  Cos[psisym5test[s, (m/(m - 2)), m, λ]]], 
            Assumptions -> {λ > 0, m < 2, 
                m > 1}] /. {JacobiDN[ (((((m^2)/(((-2 + m))^2))))^(1/4)) (s/λ), (1/m)]^2 -> 
              1 - (1/m)*
                  JacobiSN[(((((m^2)/(((-2 + m))^2))))^(1/4)) (s/λ), (1/m)]^2}, 
        Assumptions -> {λ > 0, m < 2, 
            m > 1}] /. {-2*
              JacobiCN[(((((((m^2)/(((-2 + m))^2))))^(1/4)) s)/λ), (1/m)]^2 - 
            2*JacobiSN[(((((((m^2)/(((-2 + m))^2))))^(1/4)) s)/λ), (1/m)]^2 -> -2}

Curvature

FullSimplify[FullSimplify[(λ^(-1))*(Sqrt[2*(1 + Abs[η]*Cos[psisym5test[s, η, m, λ]])]), 
            Assumptions -> {λ > 0, m < 2, 
                m > 1}] /. {JacobiSN[((s (Sqrt[Abs])[η])/λ), (1/m)]^2 -> 1 - JacobiCN[((s (Sqrt[Abs])[η])/λ), (1/m)]^2} /. 
        Abs[η] -> (m/(2 - m)), 
      Assumptions -> {λ > 0, m < 2, m > 1}]

Ksym5[s_, η_, 
        m_, λ_] := ((2*(Sqrt[Abs])[η] JacobiCN[((s (Sqrt[Abs])[η])/λ), (1/m)])/((Sqrt[m ]) λ ))

Cartesian coordinates

Δx(s)

Integrate[(1 - 
            (2/m)*JacobiSN[(sp/λ)*(Sqrt[Abs])[η], (1/m)]^2), {sp, 0, 
          s}] /. {JacobiDN[((s (Sqrt[Abs])[η])/λ), 
            (1/m)] -> (Sqrt[1 - (1/m)*JacobiSN[((s (Sqrt[Abs])[η])/λ), (1/m)]^2])}

Δxsym5[s_, η_, 
        m_, λ_] := -s + ((2 λ EllipticE[JacobiAmplitude[((s (Sqrt[Abs])[η])/λ), (1/m)], (1/m)] )/(Sqrt[Abs]))[η]

Δh(s)

FullSimplify[-((λ^2)/Abs)[η] (Ksym5[s, η, 
              m, λ] - Ksym5[0, η, m, λ])]

Δhsym5[s_, η_, 
        m_, λ_] := -((2 λ (-1 + 
                  JacobiCN[((s (Sqrt[Abs])[η])/λ), 
                    (1/m)]))/(Sqrt[m Abs[η]]))

Plot of the shape

lb = (Sqrt[2]); ηplt = -1.04;
ParametricPlot[{-Δxsym5[
              s, ηplt, ((2  Abs[ηplt])/(1 + Abs[ηplt])), 
              lb], -Δhsym5[
              s, ηplt, ((2  Abs[ηplt])/(1 + Abs[ηplt])), 
              lb]}, {s, -13, 15}, AspectRatio -> Automatic, 
        AxesLabel -> {"-Δx", "-Δh"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 300];

a) η > 1

Shape in angle-arc length parametrization

psianti1[s_, η_, m_, λ_] := 
      Re[ArcCos[
          (2/m)*JacobiSN[(s/λ)*(Sqrt[η]) - EllipticK[(1/m)], (1/m)]^2 - 
            1]]

(* check whether psianti1[s, η, 
          m, λ] solves the integral of the shape equation *)

psianti1test[s_, η_, m_, λ_] := 
      ArcCos[(2/m)*
            JacobiSN[(s/λ)*(Sqrt[η]) - EllipticK[(1/m)], (1/m)]^2 - 
          1]

FullSimplify[
        FullSimplify[
            TrigExpand[((λ^2)/2)*
                  D[psianti1test[s, (m/(2 - m)), m, λ], s]^2 + 
                (m/(2 - m))*
                  Cos[psianti1test[s, (m/(2 - m)), 
                      m, λ]]]] /. {JacobiDN[(Sqrt[-(m/(-2 + m))]) (s/λ) - EllipticK[(1/m)], (1/m)]^2 -> 
              1 - (1/m)*
                  JacobiSN[(Sqrt[-(m/(-2 + m))]) (s/λ) - EllipticK[(1/m)], (1/m)]^2}] /. {-2*
              JacobiCN[(((Sqrt[-(m/(-2 + m))]) s)/λ) - EllipticK[(1/m)], (1/m)]^2 - 
            2*JacobiSN[(((Sqrt[-(m/(-2 + m))]) s)/λ) - EllipticK[(1/m)], (1/m)]^2 -> -2}

Curvature

FullSimplify[FullSimplify[(λ^(-1))*(Sqrt[2*(1 - η*Cos[psianti1test[s, η, m, λ]])]), 
            Assumptions -> {λ > 0, m > 1, 
                m < 2}] /. {JacobiSN[((s (Sqrt[η]))/λ) - EllipticK[(1/m)], (1/m)]^2 -> 
              1 - JacobiCN[((s (Sqrt[η]))/λ) - EllipticK[(1/m)], (1/m)]^2} /. {η -> (m/(2 - m))}, 
      Assumptions -> {λ > 0, m < 2, m > 1}]

Kanti1[s_, η_, 
        m_, λ_] := ((2*(Sqrt[(η/m)]) JacobiCN[(((Sqrt[η]) s)/λ) - EllipticK[(1/m)], (1/m)] )/(λ ))

Cartesian coordinates

Δx(s)

FullSimplify[
      Integrate[((2/m)*
                JacobiSN[(sp/λ)*(Sqrt[η]) - EllipticK[(1/m)], (1/m)]^2 - 
              1), {sp, 0, 
            s}] /. {JacobiDN[((s (Sqrt[η]))/λ) - EllipticK[(1/m)], 
              (1/m)] -> (Sqrt[1 - (1/m)*JacobiSN[((s (Sqrt[η]))/λ) - EllipticK[(1/m)], (1/m)]^2])}]

(JacobiAmplitude[-EllipticK[(1/m)], 
            (1/m)] + (π/2)) /. {m -> 1.4}

EllipticE[((-π)/2), (1/m)]

Δxanti1[s_, η_, m_, λ_] := 
      s - ((2 λ EllipticE[JacobiAmplitude[((s (Sqrt[η]))/λ) - EllipticK[(1/m)], (1/m)], (1/m)] + 2 λ EllipticE[(1/m)])/(Sqrt[η]))

Δh(s)

((λ^2)/η)*(Kanti1[s, η, m, λ] - 
          Kanti1[0, η, m, λ])

Δhanti1[s_, η_, 
        m_, λ_] := ((2 λ JacobiCN[((s (Sqrt[η]))/λ) - EllipticK[(1/m)], (1/m)])/(Sqrt[m η]))

Plot of the shape

lb = (Sqrt[2]); ηplt = 1.04;
ParametricPlot[{-Δxanti1[
              s, ηplt, ((2*ηplt)/(1 + ηplt)), 
              lb], -Δhanti1[
              s, ηplt, ((2*ηplt)/(1 + ηplt)), 
              lb]}, {s, -8, 10}, AspectRatio -> Automatic, 
        AxesLabel -> {"-Δx", "-Δh"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 300];

b) η < -1

Shape in angle-arc length parametrization

psianti2[s_, η_, m_, λ_] := 
      Re[ArcCos[
          1 - (2/m)*
              JacobiSN[(s/λ)*(Sqrt[Abs])[η] - EllipticK[(1/m)], (1/m)]^2]]

(* check whether psianti2[s, η, 
          m, λ] solves the integral of the shape equation *)

psianti2test[s_, η_, m_, λ_] := 
      ArcCos[1 - 
          (2/m)*JacobiSN[(s/λ)*(Sqrt[Abs])[η] - EllipticK[(1/m)], (1/m)]^2]

FullSimplify[
        FullSimplify[
            TrigExpand[((λ^2)/2)*
                  D[psianti2test[s, -(m/(2 - m)), m, λ], s]^2 - (m/(2 - m))*
                  Cos[psianti2test[s, -(m/(2 - m)), m, λ]]], 
            Assumptions -> {λ > 0, m < 2, 
                m > 1}] /. {JacobiDN[(Sqrt[-(m/(-2 + m))]) (s/λ) - EllipticK[(1/m)], (1/m)]^2 -> 
              1 - (1/m)*
                  JacobiSN[(Sqrt[-(m/(-2 + m))]) (s/λ) - EllipticK[(1/m)], (1/m)]^2}, 
        Assumptions -> {λ > 0, m < 2, 
            m > 1}] /. {-2*
              JacobiCN[(((Sqrt[-(m/(-2 + m))]) s)/λ) - EllipticK[(1/m)], (1/m)]^2 - 
            2*JacobiSN[(((Sqrt[-(m/(-2 + m))]) s)/λ) - EllipticK[(1/m)], (1/m)]^2 -> -2}

Curvature

FullSimplify[
        FullSimplify[-(λ^(-1))*(Sqrt[2*(1 - η*Cos[psianti2test[s, η, m, λ]])]), 
            Assumptions -> {λ > 0, m > 1, m < 2}] /. {η -> 
              (m/(m - 2))}] /. {JacobiSN[((s (Sqrt[Abs])[(m/(-2 + m))])/λ) - EllipticK[(1/m)], (1/m)]^2 -> 
          1 - JacobiCN[((s (Sqrt[Abs])[(m/(-2 + m))])/λ) - EllipticK[(1/m)], (1/m)]^2}

Kanti2[s_, η_, 
        m_, λ_] := -((2*(Sqrt[Abs[η]/m]) JacobiCN[(((Sqrt[Abs])[η] s)/λ) - EllipticK[(1/m)], 
                (1/m)] )/(λ ))

Cartesian coordinates

Δx(s)

Δxanti2[s_, η_, 
        m_, λ_] := -s + ((2 λ EllipticE[JacobiAmplitude[((s (Sqrt[Abs])[η])/λ) - EllipticK[(1/m)], (1/m)], (1/m)] + 2 λ EllipticE[(1/m)])/(Sqrt[Abs]))[η]

Δh(s)

Δhanti2[s_, η_, 
        m_, λ_] := ((2 λ JacobiCN[((s (Sqrt[Abs])[η])/λ) - EllipticK[(1/m)], (1/m)])/(Sqrt[m*Abs[η]]))

Plot of the shape

lb = (Sqrt[2]); ηplt = -1.04;
ParametricPlot[{Δxanti2[
            s, ηplt, ((2  Abs[ηplt])/(1 + Abs[ηplt])), 
            lb], Δhanti2[
            s, ηplt, ((2  Abs[ηplt])/(1 + Abs[ηplt])), 
            lb]}, {s, -8, 10}, AspectRatio -> Automatic, 
        AxesLabel -> {"Δx", "Δh"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 300];