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I am trying to get my head around these equations. I am not sure they are correct. My logic is an orbit exists at a starting point (x0,y0,z0) with a starting velocity at time zero (vx0,vy0,vz0) changing with time (dx/dt, dy/dt, dz/dt). The gravity is (d^2x/dt^2, d^2y/dt^2, d^2z/dt^2). How do you turn a sinusoid into a elliptic orbit?

x = (-1 / 2 * G * M / r ^ 2 * Cos(h) * Cos(p)) * t ^ 2 + vx0 * t + x0

y = (-1 / 2 * G * M / r ^ 2 * Sin(h) * Cos(p)) * t ^ 2 + vy0 * t + y0

z = (-1 / 2 * G * M / r ^ 2 * Sin(p)) * t ^ 2 + vz0 * t + z0

where

r=(x^2+y^2+z^2)^0.5

h=atan(y/x)

p=acos(z/r)

x = (-1 / 2 * G * M / r ^ 2 * Cos(h) * Cos(p)) * t ^ 2 + vx0 * t + x0

y = (-1 / 2 * G * M / r ^ 2 * Sin(h) * Cos(p)) * t ^ 2 + vy0 * t + y0

z = (-1 / 2 * G * M / r ^ 2 * Sin(p)) * t ^ 2 + vz0 * t + z0

where

r=(x^2+y^2+z^2)^0.5

h=atan(y/x)

p=acos(z/r)

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