If you place a planet in orbit and try various initial velocities v (all in the same direction, 2i + j),
then you generate the family of orbits shown in green.
Yellow dot = sun (origin), orange = starting point (r), blue = apoapsis, cyan = periapsis, red = center of orbit (C).
Curiously, the center points (red) all lie on the same line.
The general formula for the center point C is
, the energy per unit mass of the planet (where M is the mass of the sun).
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