SL(2,R)

Gömbök

π/4<a<π/2

Vr[a_,s_]:=ArcSinh[Cos[a]/Sqrt[-Cos[2*a]]*Sinh[s*Sqrt[-Cos[2*a]]]]
Vtheta[a_,s_]:=-ArcTan[Sin[a]/Sqrt[-Cos[2*a]]*Tan[s*Sqrt[-Cos[2*a]]]]
Vphi[a_,s_]:=s*2*Sin[a]-ArcTan[Sin[a]/Sqrt[-Cos[2*a]]*Tan[s*Sqrt[-Cos[2*a]]]]
Vx[a_,s_]:=Tan[Vphi[a,s]]
Vy[a_,s_]:=Tanh[Vr[a,s]]*Cos[Vtheta[a,s]-Vphi[a,s]]/Cos[Vphi[a,s]]
Vz[a_,s_]:=Tanh[Vr[a,s]]*Sin[Vtheta[a,s]-Vphi[a,s]]/Cos[Vphi[a,s]]

0<a<π/4

Wr[a_,s_]:=ArcSinh[Cos[a]/Sqrt[Cos[2*a]]*Sinh[s*Sqrt[Cos[2*a]]]]
Wtheta[a_,s_]:=-ArcTan[Sin[a]/Sqrt[Cos[2*a]]*Tanh[s*Sqrt[Cos[2*a]]]]
Wphi[a_,s_]:=s*2*Sin[a]-ArcTan[Sin[a]/Sqrt[Cos [2*a]]*Tanh[s*Sqrt[Cos[2*a]]]]
Wx[a_,s_]:=Tan[Wphi[a,s]]
Wy[a_,s_]:=Tanh[Wr[a,s]]*Cos[Wtheta[a,s]-Wphi[a,s]]/Cos[Wphi[a,s]]
Wz[a_,s_]:=Tanh[Wr[a,s]]*Sin[Wtheta[a,s]-Wphi[a,s]]/Cos[Wphi[a,s]]

a=π/4

Zr[s_]:=ArcSinh[s/Sqrt[2]]
Ztheta[s_]:=-ArcTan[s/Sqrt[2]]
Zphi[s_]:=s*Sqrt[2]-ArcTan[s/Sqrt[2]]
Zx[s_]:=Tan[Zphi[s]]
Zy[s_]:=Tanh[Zr[s]]*Cos[Ztheta[s]-Zphi[s]]/Cos[Zphi[s]]
Zz[s_]:=Tanh[Zr[s]]*Sin[Ztheta[s]-Zphi[s]]/Cos[Zphi[s]]

s:=1.5;
t1:=ParametricPlot3D[{{Sqrt[Vy[a, s]^2+Vz[a, s]^2]*Cos[omega],Sqrt[Vy[a, s]^2+Vz[a, s]^2]*Sin[omega],Vx[a,s]}, {Sqrt[Vy[a, s]^2+Vz[a, s]^2]*Cos[omega],Sqrt[Vy[a, s]^2+Vz[a, s]^2]*Sin[omega],-Vx[a,s]}},{a, π/4, π/2}, {omega, -π, π }, PlotStyle->{{LightBlue,Specularity[White,40], Opacity[0.7]},{ LightBlue,Specularity[White,40], Opacity[0.7]}}, Mesh->None]
t2:=ParametricPlot3D[{{Sqrt[Wy[a, s]^2+Wz[a, s]^2]*Cos[omega],Sqrt[Wy[a, s]^2+Wz[a, s]^2]*Sin[omega],Wx[a,s]}, {Sqrt[Wy[a, s]^2+Wz[a, s]^2]*Cos[omega],Sqrt[Wy[a, s]^2+Wz[a, s]^2]*Sin[omega],-Wx[a,s]}},{a, 0, π/4}, {omega, -π, π },
PlotStyle->{{Orange,Specularity[White,10], Opacity[0.7]},{ Orange,Specularity[White,10], Opacity[0.7]}}, Mesh->None]
t3:=ParametricPlot3D[{{Sqrt[Zy[ s]^2+Zz[s]^2]*Cos[omega],Sqrt[Zy[ s]^2+Zz[ s]^2]*Sin[omega],Zx[s]}, {Sqrt[Zy[s]^2+Zz[ s]^2]*Cos[omega],Sqrt[Zy[ s]^2+Zz[ s]^2]*Sin[omega],-Zx[s]}}, {s, 0,π/2}, {omega, -π, π }, PlotStyle->{{Yellow,Specularity[White,10], Opacity[0.6]},{Yellow,Specularity[White,10], Opacity[0.6]}}, Mesh->None]

t1
t2
t3

Show[t1,t2,t3]

Eltolás

x:={1,0.35,0.35,0}
T={{x[[1]],x[[2]],x[[3]],x[[4]]},{-x[[2]],x[[1]],x[[4]],-x[[3]]},{x[[3]],x[[4]],x[[1]],x[[2]]},{x[[4]],-x[[3]],-x[[2]],x[[1]]}}

MatrixForm[T]

v1:={1,Vx[a,s],Sqrt[Vy[a,s]^2+Vz[a,s]^2]*Cos[omega],Sqrt[Vy[a,s]^2+Vz[a,s ]^2]*Sin[omega]}
v2:={1,-Vx[a,s],Sqrt[Vy[a,s]^2+Vz[a,s]^2]*Cos[omega],Sqrt[Vy[a,s]^2+Vz[a,s ]^2]*Sin[omega]}
Vu1:=v1.T
Vu2:=v2.T
w1:={1,Wx[a,s],Sqrt[Wy[a,s]^2+Wz[a,s]^2]*Cos[omega],Sqrt[Wy[a,s]^2+Wz[a,s ]^2]*Sin[omega]}
w2:={1,-Wx[a,s],Sqrt[Wy[a,s]^2+Wz[a,s]^2]*Cos[omega],Sqrt[Wy[a,s]^2+Wz[a,s ]^2]*Sin[omega]}
Wu1:=w1.T
Wu2:=w2.T
z1:={1,Zx[s],Sqrt[Zy[s]^2+Zz[s]^2]*Cos[omega],Sqrt[Zy[s]^2+Zz[s ]^2]*Sin[omega]}
z2:={1,-Zx[s],Sqrt[Zy[s]^2+Zz[s]^2]*Cos[omega],Sqrt[Zy[s]^2+Zz[s ]^2]*Sin[omega]}
Zu1:=z1.T
Zu2:=z2.T

kortefa:=ParametricPlot3D[{{Vu1[[3]]/Vu1[[1]],Vu1[[4]]/Vu1[[1]],Vu1[[2]]/Vu1[[1]]},{Vu2[[3]]/Vu2[[1]],Vu2[[4]]/Vu2[[1]],Vu2[[2]]/Vu2[[1]]}},{a, π/4, π/2}, {omega, -π, π }, PlotStyle->{{LightBlue,Specularity[White,40], Opacity[0.7]},{ LightBlue,Specularity[White,40], Opacity[0.7]}}, Mesh->None]
almafa:=ParametricPlot3D[{{Wu1[[3]]/Wu1[[1]],Wu1[[4]]/Wu1[[1]],Wu1[[2]]/Wu1[[1]]},{Wu2[[3]]/Wu2[[1]],Wu2[[4]]/Wu2[[1]],Wu2[[2]]/Wu2[[1]]}},{a, 0, π/4}, {omega, -π, π }, PlotStyle->{{Orange,Specularity[White,10], Opacity[0.7]},{ Orange,Specularity[White,10], Opacity[0.7]}}, Mesh->None]
cseresznyefa:=ParametricPlot3D[{{Zu1[[3]]/Zu1[[1]],Zu1[[4]]/Zu1[[1]],Zu1[[2]]/Zu1[[1]]},{Zu2[[3]]/Zu2[[1]],Zu2[[4]]/Zu2[[1]],Zu2[[2]]/Zu2[[1]]}}, {s, 0,π/2},{omega, -π, π }, PlotStyle->{{Yellow,Specularity[White,10], Opacity[0.6]},{Yellow,Specularity[White,10], Opacity[0.6]}}, Mesh->None]

kortefa

almafa

cseresznyefa

Show[kortefa, almafa]

Zu1[[4]]/Zu1[[1]]

Wu1[[3]]/Wu1[[2]]

SL(2,R) JAV

Gömbök

π/4<a<π/2

Vr[a_,s_]:=ArcSinh[((Cos[a])/(Sqrt[-Cos[2a]]))Sinh[s(Sqrt[-Cos[2*a]])]]
Vtheta[a_,s_]:=-ArcTan[Sin[a]/Sqrt[-Cos[2*a]]*Tan[s*Sqrt[-Cos[2*a]]]]
Vφ[a_,s_]:=s*2*Sin[a]-ArcTan[Sin[a]/Sqrt[-Cos[2*a]]*Tan[s*Sqrt[-Cos[2*a]]]]
Vx[a_,s_]:=Tan[Vφ[a,s]]
Vy[a_,s_]:=Tanh[Vr[a,s]]((Cos[Vϑ[a,s]-Vφ[a,s]])/(Cos[Vφ[a,s]]))
Vz[a_,s_]:=Tanh[Vr[a,s]]((Sin[Vϑ[a,s]-Vφ[a,s]])/(Cos[Vφ[a,s]]))

0<a<π/4

Wr[a_,s_]:=ArcSinh[Cos[a]/Sqrt[Cos[2*a]]*Sinh[s*Sqrt[Cos[2*a]]]]
Wtheta[a_,s_]:=-ArcTan[Sin[a]/Sqrt[Cos[2*a]]*Tanh[s*Sqrt[Cos[2*a]]]]
Wphi[a_,s_]:=s*2*Sin[a]-ArcTan[Sin[a]/Sqrt[Cos [2*a]]*Tanh[s*Sqrt[Cos[2*a]]]]
Wx[a_,s_]:=Tan[Wphi[a,s]]
Wy[a_,s_]:=Tanh[Wr[a,s]]*Cos[Wtheta[a,s]-Wphi[a,s]]/Cos[Wphi[a,s]]
Wz[a_,s_]:=Tanh[Wr[a,s]]*Sin[Wtheta[a,s]-Wphi[a,s]]/Cos[Wphi[a,s]]

a=π/4

Vr[s_]:=ArcSinh[(s/(Sqrt[2]))]
Vϑ[s_]:=-ArcTan[(s/(Sqrt[2]))]
Vφ[s_]:=s (Sqrt[2])-ArcTan[(s/(Sqrt[2]))]
Vx[s_]:=Tan[Vφ[s]]
Vy[s_]:=Tanh[Vr[s]]((Cos[Vϑ[s]-Vφ[s]])/(Cos[Vφ[s]]))
Vz[s_]:=Tanh[Vr[s]]((Sin[Vϑ[s]-Vφ[s]])/(Cos[Vφ[s]]))

s:=1.5;
t1:=ParametricPlot3D[{{Sqrt[Vy[a, s]^2+Vz[a, s]^2]*Cos[omega],Sqrt[Vy[a, s]^2+Vz[a, s]^2]*Sin[omega],Vx[a,s]}, {Sqrt[Vy[a, s]^2+Vz[a, s]^2]*Cos[omega],Sqrt[Vy[a, s]^2+Vz[a, s]^2]*Sin[omega],-Vx[a,s]}},{a, π/4, π/2}, {omega, -π, π }, PlotStyle->{{LightBlue,Specularity[White,40], Opacity[0.7]},{ LightBlue,Specularity[White,40], Opacity[0.7]}}, Mesh->None]
t2:=ParametricPlot3D[{{Sqrt[Wy[a, s]^2+Wz[a, s]^2]*Cos[omega],Sqrt[Wy[a, s]^2+Wz[a, s]^2]*Sin[omega],Wx[a,s]}, {Sqrt[Wy[a, s]^2+Wz[a, s]^2]*Cos[omega],Sqrt[Wy[a, s]^2+Wz[a, s]^2]*Sin[omega],-Wx[a,s]}},{a, 0, π/4}, {omega, -π, π },
PlotStyle->{{Orange,Specularity[White,10], Opacity[0.7]},{ Orange,Specularity[White,10], Opacity[0.7]}}, Mesh->None]
t3:=ParametricPlot3D[{{Sqrt[Zy[ s]^2+Zz[s]^2]*Cos[omega],Sqrt[Zy[ s]^2+Zz[ s]^2]*Sin[omega],Zx[s]}, {Sqrt[Zy[s]^2+Zz[ s]^2]*Cos[omega],Sqrt[Zy[ s]^2+Zz[ s]^2]*Sin[omega],-Zx[s]}}, {s, 0,π/2}, {omega, -π, π }, PlotStyle->{{Yellow,Specularity[White,10], Opacity[0.6]},{Yellow,Specularity[White,10], Opacity[0.6]}}, Mesh->None]

t1
t2
t3

Show[t1,t2,t3]

Eltolás

x:={1,0.35,0.35,0}
T={{x[[1]],x[[2]],x[[3]],x[[4]]},{-x[[2]],x[[1]],x[[4]],-x[[3]]},{x[[3]],x[[4]],x[[1]],x[[2]]},{x[[4]],-x[[3]],-x[[2]],x[[1]]}}

MatrixForm[T]

v1:={1,Vx[a,s],Sqrt[Vy[a,s]^2+Vz[a,s]^2]*Cos[omega],Sqrt[Vy[a,s]^2+Vz[a,s ]^2]*Sin[omega]}
v2:={1,-Vx[a,s],Sqrt[Vy[a,s]^2+Vz[a,s]^2]*Cos[omega],Sqrt[Vy[a,s]^2+Vz[a,s ]^2]*Sin[omega]}
Vu1:=v1.T
Vu2:=v2.T
w1:={1,Wx[a,s],Sqrt[Wy[a,s]^2+Wz[a,s]^2]*Cos[omega],Sqrt[Wy[a,s]^2+Wz[a,s ]^2]*Sin[omega]}
w2:={1,-Wx[a,s],Sqrt[Wy[a,s]^2+Wz[a,s]^2]*Cos[omega],Sqrt[Wy[a,s]^2+Wz[a,s ]^2]*Sin[omega]}
Wu1:=w1.T
Wu2:=w2.T
z1:={1,Zx[s],Sqrt[Zy[s]^2+Zz[s]^2]*Cos[omega],Sqrt[Zy[s]^2+Zz[s ]^2]*Sin[omega]}
z2:={1,-Zx[s],Sqrt[Zy[s]^2+Zz[s]^2]*Cos[omega],Sqrt[Zy[s]^2+Zz[s ]^2]*Sin[omega]}
Zu1:=z1.T
Zu2:=z2.T

kortefa:=ParametricPlot3D[{{Vu1[[3]]/Vu1[[1]],Vu1[[4]]/Vu1[[1]],Vu1[[2]]/Vu1[[1]]},{Vu2[[3]]/Vu2[[1]],Vu2[[4]]/Vu2[[1]],Vu2[[2]]/Vu2[[1]]}},{a, π/4, π/2}, {omega, -π, π }, PlotStyle->{{LightBlue,Specularity[White,40], Opacity[0.7]},{ LightBlue,Specularity[White,40], Opacity[0.7]}}, Mesh->None]
almafa:=ParametricPlot3D[{{Wu1[[3]]/Wu1[[1]],Wu1[[4]]/Wu1[[1]],Wu1[[2]]/Wu1[[1]]},{Wu2[[3]]/Wu2[[1]],Wu2[[4]]/Wu2[[1]],Wu2[[2]]/Wu2[[1]]}},{a, 0, π/4}, {omega, -π, π }, PlotStyle->{{Orange,Specularity[White,10], Opacity[0.7]},{ Orange,Specularity[White,10], Opacity[0.7]}}, Mesh->None]
cseresznyefa:=ParametricPlot3D[{{Zu1[[3]]/Zu1[[1]],Zu1[[4]]/Zu1[[1]],Zu1[[2]]/Zu1[[1]]},{Zu2[[3]]/Zu2[[1]],Zu2[[4]]/Zu2[[1]],Zu2[[2]]/Zu2[[1]]}}, {s, 0,π/2},{omega, -π, π }, PlotStyle->{{Yellow,Specularity[White,10], Opacity[0.6]},{Yellow,Specularity[White,10], Opacity[0.6]}}, Mesh->None]

kortefa

almafa

cseresznyefa

Show[kortefa, almafa]

Zu1[[4]]/Zu1[[1]]

Wu1[[3]]/Wu1[[2]]