2-2

Geodesics on Paraboloid & Hexenhut

Nicholas Wheeler
13 February 2016

Unit Paraboloid

paraboloid10=ParametricPlot3D[{(Sqrt[u])Cos[v],(Sqrt[u])Sin[v],u},{u,0,10},{v,0,2π},Axes->False,PlotStyle->{Opacity[0.3]},Boxed->False,PlotLabel->"Geodesic on Unit Paraboloid",LabelStyle->Directive[Bold]]

vplus=(Subscript[v, 0])+(ArcSin[(Sqrt[((u-(c^2))/(u(1+4(c^2))))])]+2c Log[((2(Sqrt[u-(c^2)])+(Sqrt[4u+1]))/(Sqrt[1+4(c^2)]))]);
vminus=(Subscript[v, 0])-(ArcSin[(Sqrt[((u-(c^2))/(u(1+4(c^2))))])]+2c Log[((2(Sqrt[u-(c^2)])+(Sqrt[4u+1]))/(Sqrt[1+4(c^2)]))]);

(Subscript[v, 0])=-π/2;
c=1;

plusgeodesic10=ParametricPlot3D[{(Sqrt[u])Cos[vplus],(Sqrt[u])Sin[vplus],u},{u,0,10},Axes->False,PlotStyle->{Thick},Boxed->False];
minusgeodesic10=ParametricPlot3D[{(Sqrt[u])Cos[vminus],(Sqrt[u])Sin[vminus],u},{u,0,10},Axes->False,PlotStyle->{Thick},Boxed->False];

Show[paraboloid10, plusgeodesic10, minusgeodesic10]

paraboloid120=ParametricPlot3D[{(Sqrt[u])Cos[v],(Sqrt[u])Sin[v],u},{u,0,120},{v,0,2π},Axes->False,PlotStyle->{Opacity[0.3]},Boxed->False,PlotLabel->"Geodesic Crossings on Unit Paraboloid",LabelStyle->Directive[Bold]]

plusgeodesic120=ParametricPlot3D[{(Sqrt[u])Cos[vplus],(Sqrt[u])Sin[vplus],u},{u,0,120},Axes->False,PlotStyle->{Thick},Boxed->False];
minusgeodesic120=ParametricPlot3D[{(Sqrt[u])Cos[vminus],(Sqrt[u])Sin[vminus],u},{u,0,120},Axes->False,PlotStyle->{Thick},Boxed->False];

Show[paraboloid120, plusgeodesic120, minusgeodesic120]

(Subscript[v, 0])=.
Clear[c]

Unit Hexenhut

hexenhut10=ParametricPlot3D[{(1/(Sqrt[u]))Cos[v],(1/(Sqrt[u]))Sin[v],u},{u,0,10},{v,0,2π},Axes->False,PlotStyle->{Opacity[0.3]},Boxed->False,PlotLabel->"Geodesic Near Base of Hexenhut",LabelStyle->Directive[Bold]]

hexvplus=(Subscript[v, 0])+((2(2+(c^2)u)(Sqrt[1-(c^2)u])-4)/(3(c^3)));
hexvminus=(Subscript[v, 0])-((2(2+(c^2)u)(Sqrt[1-(c^2)u])-4)/(3(c^3)));

c=0.1;
(Subscript[v, 0])=-π/2;

hexminusgeodesic10=ParametricPlot3D[{(1/(Sqrt[u]))Cos[hexvminus],(1/(Sqrt[u]))Sin[hexvminus],u},{u,0,10},Axes->False,PlotStyle->{Thick},Boxed->False];

Show[hexenhut10,hexminusgeodesic10]

highhexenhut=ParametricPlot3D[{20(1/(Sqrt[u]))Cos[v],20(1/(Sqrt[u]))Sin[v],u},{u,90,110},{v,0,2π},Axes->False,PlotStyle->{Opacity[0.3]},Boxed->False,PlotLabel->"Hexenhut Geodesic at Turning Point",LabelStyle->Directive[Bold]]

turningpoint=ParametricPlot3D[{20(1/(Sqrt[u]))Cos[hexvminus],20(1/(Sqrt[u]))Sin[hexvminus],u},{u,90,110},Axes->False,PlotPoints->{1000},Boxed->False]

Show[highhexenhut, turningpoint]

Number of meridian crossings:

(8/3)((c^(-3))/(2π))