Hyperbolic orbits correspond to eccentricities e > 1,

(1)

where a is the semimajor axis and v is the true anomaly. But r > 0, so

(2)

(3)

Define

(4)

and the ellipse equation becomes

(5)

Define

(6)

then

(7)

(8)

Let

(9)

(10)

then

(11)

Circular Orbit, Elliptical Orbit, Orbit, Parabolic Orbit, Two-Body Problem