Test plots of angle vs. radius

In[111]:=

  th01[r_,theta0_] = theta1[r,theta0] /.
{b -> 1, c -> 1} 

Out[111]=

                  1
  theta0 + ArcCos[-]
                  r

In[112]:=

  th02[r_,theta0_] = theta2[r,theta0]
/. {b -> 1, c -> 1} 

Out[112]=

                  1
  theta0 - ArcCos[-]
                  r

In order to get the entire path, before and after the point of closest approach, we need to use both functions for theta when plotting geodesics. Below shows why:

In[113]:=

  plot1 = ParametricPlot[{r Cos[th01[r,Pi/4]], r Sin[th01[r,Pi/4]]},
{r,1,5}, AspectRatio -> 1, PlotRange -> {{-3,4},{-3,4}}]

Out[114]=

  -Graphics-

Using the first function gives us the half of the path shown above, where the path does not cross theta=0 (the x-axis). (Remember the + sign in theta1.) Using the second function gives the below half of the path, which does pass through theta=0:

In[115]:=

  plot2 = ParametricPlot[{r Cos[th02[r,Pi/4]], r Sin[th02[r,Pi/4]]},
{r,1,5}, AspectRatio -> 1, PlotRange -> {{-3,4},{-3,4}}]

Out[116]=

  -Graphics-

If we plot the two halves together, we can see both halves of the trajectory, where theta increases from -Pi/4 to Pi/4 while r decreases from infinity to 1, followed by r increasing back to infinity again as theta keeps increasing from Pi/4 to 3 Pi/4.

In[117]:=

  Show[plot1,plot2, AspectRatio -> 1, PlotRange -> {{-3,4},{-3,4}}]

Out[118]=

  -Graphics-

Notice that if theta0<0, then theta1 is the function that passes through theta=0, as shown below:

In[119]:=

  plot3 = ParametricPlot[{r Cos[th01[r,-Pi/4]], r Sin[th01[r,-Pi/4]]},
{r,1,5}, AspectRatio -> 1,PlotRange -> {{-4,3},{-4,3}}]

Out[120]=

  -Graphics-

And for theta0<0, the theta2 branch is the half that doesn't pass through theta=0:

In[121]:=

  plot4 = ParametricPlot[{r Cos[th02[r,-Pi/4]], r Sin[th02[r,-Pi/4]]},
{r,1,5}, AspectRatio -> 1,PlotRange -> {{-4,3},{-4,3}}]

Out[122]=

  -Graphics-

Once again, the two halves give us the whole path:

In[123]:=

  Show[plot3,plot4,PlotRange -> {{-4,3},{-4,3}}]

Out[124]=

  -Graphics-

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