Space plots

In[147]:=

  ParametricPlot[{{x[t,1,Pi/4],y[t,1,Pi/4]},{x[t,.7,Pi/4],y[t,.7,Pi/4]},
{x[t,.5,Pi/4],y[t,.5,Pi/4]}},{t,-5,5}, AspectRatio -> 1]

Out[148]=

  -Graphics-

In[149]:=

  ParametricPlot[{{x[t,1,-Pi/4],y[t,1,-Pi/4]},
{x[t,.7,-Pi/4],y[t,.7,-Pi/4]},{x[t,.5,-Pi/4],y[t,.5,-Pi/4]}},
{t,-5,5}, AspectRatio -> 1]

Out[150]=

  -Graphics-

Now let's start examining the patterns when we keep the deficit angle fixed and vary the turning point at theta0:

In[151]:=

  ParametricPlot[{{x[t,.7,Pi/4],y[t,.7,Pi/4]},
{x[t,.7,-Pi/4],y[t,.7,-Pi/4]},{x[t,.7,Pi/8],y[t,.7,Pi/8]},
{x[t,.7,-Pi/8],y[t,.7,-Pi/8]}},
{t,-5,5}, AspectRatio -> 1]

Out[152]=

  -Graphics-

Now let's look at the asymptotic values of theta:

In[153]:=

  thout = PowerExpand[Limit[theta[t,c,theta0],t -> Infinity]]

Out[153]=

  Pi
  --- + theta0
  2 c

In[154]:=

  thin = PowerExpand[Limit[theta[t,c,theta0],t -> -Infinity]]

Out[154]=

  -Pi
  --- + theta0
  2 c

In[155]:=

  thout - thin

Out[155]=

  Pi
  --
  c

If we plot geodesics with values of theta0 near Pi/2c, so that the incoming paths are close to the x-axis, we can see that the geodesics cross each other in pairs on the other side of our point mass located at x=0, y=0:

In[156]:=

  ParametricPlot[{{x[t,.7,Pi/1.45],y[t,.7,Pi/1.45]},
{x[t,.7,-Pi/1.45],y[t,.7,-Pi/1.45]},{x[t,.7,Pi/1.7],y[t,.7,Pi/1.7]},
{x[t,.7,-Pi/1.7],y[t,.7,-Pi/1.7]},{x[t,.7,Pi/2],y[t,.7,Pi/2]},
{x[t,.7,-Pi/2],y[t,.7,-Pi/2]}},{t,-5,5}, AspectRatio -> 1, PlotRange -> {-4,4}]

Out[157]=

  -Graphics-

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