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GS = {A -> 3.41, B -> 0.54, C -> 2.27, D -> 1.07};GS = {A -> 14.47, C -> 9.57, B -> 2.28, D -> 4.51}; AngP[a_,b_]:=(a^2*(A*b^2*(B - C) + a^2*(B - C)^2 +     b^2*(B*(B + C) + 4*B*D + 2*D^2) +     (B + C)*Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +        4*a^2*b^2*(B + D)^2])^2)/ (2*(A^4*b^6 + a^6*(B - C)^2*(B^2 + C^2) +    b^4*B^3*(b^2*B + Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +        4*a^2*b^2*(B + D)^2]) +    A^3*b^4*(-2*b^2*B + 2*a^2*(B - C) +      Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +        4*a^2*b^2*(B + D)^2]) +    A^2*b^2*(a^4*B^2 + 3*a^2*b^2*B^2 + 2*b^4*B^2 -      2*a^4*B*C + 2*a^2*b^2*B*C + a^4*C^2 + a^2*b^2*C^2 +      8*a^2*b^2*B*D + 4*a^2*b^2*D^2 +      (-(b^2*B) + a^2*(B - C))*      Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2]) +    a^2*b^2*(b^2*(B^2*(B + C)*(3*B + C) + 4*B^2*(3*B + C)*        D + 2*B*(7*B + C)*D^2 + 8*B*D^3 + 2*D^4) +      (2*B + C)*(B*(B + C) + 4*B*D + 2*D^2)*      Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2]) +    a^4*((B - C)^2*(B + C)*Sqrt[(b^2*(A - B) + a^2*(B - C))^         2 + 4*a^2*b^2*(B + D)^2] +      b^2*(3*B^4 + 2*B^3*(C + 6*D) + 2*B*(C + 4*D)*        (C^2 + D^2) + 2*D^2*(2*C^2 + D^2) +        B^2*(3*C^2 + 4*C*D + 14*D^2))) +    A*b^2*(Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2]*(-(b^2*B^2) +        a^2*(3*B^2 - C^2 + 4*B*D + 2*D^2)) +      2*(-(b^4*B^3) + a^4*(B - C)*(2*B^2 + C^2 + 2*B*D +          D^2) + a^2*b^2*(B*(B^2 - 2*B*C - C^2) +          2*B*(B - C)*D + (B - C)*D^2)))));AngSV[a_,b_]:=-(a^2*(A*b^2*(B - C) + a^2*(B - C)^2 +      b^2*(B*(B + C) + 4*B*D + 2*D^2) -      (B + C)*Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2])^2)/ (2*(-(A^4*b^6) - a^6*(B - C)^2*(B^2 + C^2) +    b^4*B^3*(-(b^2*B) + Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +        4*a^2*b^2*(B + D)^2]) +    A^3*b^4*(2*b^2*B - 2*a^2*(B - C) +      Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +        4*a^2*b^2*(B + D)^2]) +    a^2*b^2*(-(b^2*(B^2*(B + C)*(3*B + C) +         4*B^2*(3*B + C)*D + 2*B*(7*B + C)*D^2 + 8*B*D^3 +         2*D^4)) + (2*B + C)*(B*(B + C) + 4*B*D + 2*D^2)*      Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2]) +    A^2*b^2*(-2*b^4*B^2 - a^4*(B - C)^2 -      (b^2*B + a^2*(-B + C))*      Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2] - a^2*b^2*(3*B^2 + C^2 +        4*D^2 + 2*B*(C + 4*D))) +    a^4*((B - C)^2*(B + C)*Sqrt[(b^2*(A - B) + a^2*(B - C))^         2 + 4*a^2*b^2*(B + D)^2] -      b^2*(3*B^4 + 2*B^3*(C + 6*D) + 2*B*(C + 4*D)*        (C^2 + D^2) + 2*D^2*(2*C^2 + D^2) +        B^2*(3*C^2 + 4*C*D + 14*D^2))) +    A*b^2*(Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2]*(-(b^2*B^2) +        a^2*(3*B^2 - C^2 + 4*B*D + 2*D^2)) +      2*(b^4*B^3 - a^4*(B - C)*(2*B^2 + C^2 + 2*B*D + D^2) +        a^2*b^2*(-B^3 + 2*B^2*(C - D) + C*D^2 +          B*(C^2 + 2*C*D - D^2))))));GruP[a_,b_]:=(b^2*(A^2*b^2 + a^2*A*B - 2*A*b^2*B + a^2*B^2 + b^2*B^2 -      a^2*A*C + a^2*B*C + 4*a^2*B*D + 2*a^2*D^2 +      (A + B)*Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2])^2 +   a^2*(A*b^2*(B - C) + a^2*(B - C)^2 +      b^2*(B*(B + C) + 4*B*D + 2*D^2) +      (B + C)*Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2])^2)/ (2*(A^2*b^4 + b^4*B^2 + a^4*(B - C)^2 -    2*A*b^2*(b^2*B + a^2*(-B + C)) +    2*a^2*b^2*(B*(B + C) + 4*B*D + 2*D^2))*  (b^2*(A + B) + a^2*(B + C) +    Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +      4*a^2*b^2*(B + D)^2]));GruSV[a_,b_]:=(b^2*(A^2*b^2 + a^2*A*B - 2*A*b^2*B + a^2*B^2 + b^2*B^2 -      a^2*A*C + a^2*B*C + 4*a^2*B*D + 2*a^2*D^2 -      (A + B)*Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2])^2 +   a^2*(A*b^2*(B - C) + a^2*(B - C)^2 +      b^2*(B*(B + C) + 4*B*D + 2*D^2) -      (B + C)*Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +         4*a^2*b^2*(B + D)^2])^2)/ (2*(A^2*b^4 + b^4*B^2 + a^4*(B - C)^2 -    2*A*b^2*(b^2*B + a^2*(-B + C)) +    2*a^2*b^2*(B*(B + C) + 4*B*D + 2*D^2))*  (b^2*(A + B) + a^2*(B + C) -    Sqrt[(b^2*(A - B) + a^2*(B - C))^2 +      4*a^2*b^2*(B + D)^2]));ParametricPlot[{Sqrt[GruSV[Cos[a], Sin[a]]] Sign[Sin[a]]Sqrt[          1 - AngSV[Cos[a], Sin[a]]],       Sqrt[GruSV[Cos[a], Sin[a]]]Sign[Cos[a]] Sqrt[AngSV[Cos[a], Sin[a]]]} /.     GS, {a, 0, 2 Pi}, AspectRatio -> Automatic,     PlotStyle->AbsoluteThickness[2]];ParametricPlot[{Sqrt[GruP[Cos[a], Sin[a]]] Sign[Sin[a]]Sqrt[          1 - AngP[Cos[a], Sin[a]]],       Sqrt[GruP[Cos[a], Sin[a]]]Sign[Cos[a]] Sqrt[AngP[Cos[a], Sin[a]]]} /.     GS, {a, 0, 2 Pi}, AspectRatio -> Automatic,     PlotStyle->AbsoluteThickness[2]];Show[%,%%,Frame->True,FrameLabel->{"Horizontal component(km/s)","Vertical component (km/s)",None,None},     PlotLabel->"Group Velocity Profiles (Wavefronts)"];Display["junk_ma.eps",%,"EPS"];

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