Kartenprojektionen: Maple Sheet Hauptstreckungen
| Maple Sheet Hauptverzerrungen |
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Parametergleichung links |
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X:=vector([R*cos(U)*cos(V),R*sin(U)*cos(V),R*sin(V)]); |
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Parametergleichung rechts |
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x:=vector([v*cos(u),v*sin(u)]); |
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Gaußsche Tangentenvektoren |
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links |
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XU:=vector([diff(X[1],U),diff(X[2],U),diff(X[3],U)]); |
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XV:=vector([diff(X[1],V),diff(X[2],V),diff(X[3],V)]); |
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rechts |
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xu:=vector([diff(x[1],u),diff(x[2],u)]); |
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xv:=vector([diff(x[1],v),diff(x[2],v)]); |
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erste Metriktensoren |
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links |
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Gl:=matrix(2,2): |
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Gl[1,1]:=simplify(dotprod(XU,XU,'orthogonal')): |
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Gl[1,2]:=simplify(dotprod(XU,XV,'orthogonal')): |
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Gl[2,1]:=Gl[1,2]: |
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Gl[2,2]:=simplify(dotprod(XV,XV,'orthogonal')): |
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print(Gl); |
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rechts |
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Gr:=matrix(2,2): |
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Gr[1,1]:=simplify(dotprod(xu,xu,'orthogonal')): |
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Gr[1,2]:=simplify(dotprod(xu,xv,'orthogonal')): |
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Gr[2,1]:=Gl[1,2]: |
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Gr[2,2]:=simplify(dotprod(xv,xv,'orthogonal')): |
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print(Gr); |
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Abbildungsgleichungen |
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fu:=U; |
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fv:=2*R*tan(Pi/4-V/2); |
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Linke Jacobimatrix |
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Jl:=matrix(2,2): |
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Jl[1,1]:=simplify(diff(fu,U)): |
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Jl[1,2]:=simplify(diff(fu,V)): |
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Jl[2,1]:=simplify(diff(fv,U)): |
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Jl[2,2]:=simplify(diff(fv,V)): |
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print(Jl); |
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Rechte Jacobimatrix incomplete! substitution muss angepasst werden. Besser isses, Jacobi so aufzustellen. |
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#Jr:=inverse(Jl); |
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#for i from 1 to 2 do |
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#for j from 1 to 2 do |
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#Gr[i,j]:=subs(v=2*R*tan(Pi/4-V/2),Gr[i,j]); |
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#od; |
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#od; |
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Cauchy Green Tensor |
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#for i from 1 to 2 do |
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#for j from 1 to 2 do |
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Gr[i,j]:=subs(v=2*R*tan(Pi/4-V/2),Gr[i,j]); |
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#od; |
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#od; |
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print(Gr); |
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Cl:=evalm(transpose(Jl)&*Gr&*Jl); |
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Hauptverzerrungen (Eigenwerte) bezüglich links |
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lambda[1]:=sqrt(1/2*(trace(evalm(Cl&*inverse(Gl))) + sqrt(trace(evalm(Cl&*inverse(Gl)))^2 - 4*det(evalm(Cl&*inverse(Gl)))))); |
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lambda[2]:=sqrt(1/2*(trace(evalm(Cl&*inverse(Gl))) - sqrt(trace(evalm(Cl&*inverse(Gl)))^2 - 4*det(evalm(Cl&*inverse(Gl)))))); |
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Eigenvektoren |
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EV[1]:=simplify(vector([(Cl[2,2]-lambda[1]^2*Gl[2,2])/sqrt( (Cl[2,2]-lambda[1]^2*Gl[2,2])^2*Gl[1,1] - 2*(Cl[1,2]-lambda[1]^2*Gl[1,2])*(Cl[2,2]-lambda[1]^2*Gl[2,2])*Gl[1,2] + (Cl[1,2]-lambda[1]^2*Gl[1,2])^2*Gl[2,2] ) , -(Cl[1,2]-lambda[1]^2*Gl[1,2])/sqrt( (Cl[2,2]-lambda[1]^2*Gl[2,2])^2*Gl[1,1] - 2*(Cl[1,2]-lambda[1]^2*Gl[1,2])*(Cl[2,2]-lambda[1]^2*Gl[2,2])*Gl[1,2] + (Cl[1,2]-lambda[1]^2*Gl[1,2])^2*Gl[2,2] )])); |
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EV[2]:=simplify(vector([-(Cl[1,2]-lambda[2]^2*Gl[1,2])/sqrt( (Cl[1,1]-lambda[2]^2*Gl[1,1])^2*Gl[2,2] - 2*(Cl[1,2]-lambda[2]^2*Gl[1,2])*(Cl[1,1]-lambda[2]^2*Gl[1,1])*Gl[1,2] + (Cl[1,2]-lambda[2]^2*Gl[1,2])^2*Gl[1,1] ), (Cl[1,1]-lambda[2]^2*Gl[1,1])/sqrt( (Cl[1,1]-lambda[2]^2*Gl[1,1])^2*Gl[2,2] - 2*(Cl[1,2]-lambda[2]^2*Gl[1,2])*(Cl[1,1]-lambda[2]^2*Gl[1,1])*Gl[1,2] + (Cl[1,2]-lambda[2]^2*Gl[1,2])^2*Gl[1,1] )])); |
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