For a hyperbolic trajectory the total energy must be positive. E > 0 means epsilon > 1 (hyperbola). For E = 0 you get eps = 1 (parabola) and for E < 1 you get eps < 1 (ellips or circle when eps = 0). Read this too: en.wikipedia.org/wiki/Kepler_orbit#Properties_of_trajectory_equation
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You just arrived my friend the equation for the equation of the path of any orbiting body in the influence of gravity which is a conic section
Any videos related to eq of orbit in polar coordinates
Why do you Cross (B sin phi)?
Why can r not be smaller than r(hyperbola)?
For a hyperbolic trajectory the total energy must be positive. E > 0 means epsilon > 1 (hyperbola).
For E = 0 you get eps = 1 (parabola) and for E < 1 you get eps < 1 (ellips or circle when eps = 0).
Read this too: en.wikipedia.org/wiki/Kepler_orbit#Properties_of_trajectory_equation
A much better derivation, easiest and simplest, is given at Natural theory of relativity_inertia@gravitation_gravitivity