I’m trying to solve for an orbital path, $C$ (preferably a parametric equation of $(X(t),Y(t))$ such that at $t=0$, it equal the starting pos), from a starting vector $V$ and position $P$. I also have a star at the origin with mass M that the object orbits. How can I find the orbital path if given V, P, and M?
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The orbit will be elliptical if $V$ does not exceed escape velocity (otherwise hyperbolic or, at the edge case, parabolic), in which case you might be able to use the ellipse's parametric setup (if it's symmetry axes are aligned with the coordinate axes):
$$\mathbf r(t)=\left(a\cos(wt+\theta) +c_1,b\sin(wt+/theta) +c_2\right).$$
Here $a,b$ are the two semi-axes (distances from centre to closest and farthest point, respectively), and $(c_1,c_2)$ is the ellipse centre. I have included constants $w, \theta$ in order to allow for adjustment of $t$ to start and behave as the intended time parameter. How you find the values of $a,b,w,\theta,c_1,c_2$ depends on what you know more precisely.
The star will be located at a focal point, which you mention to be at the coordinate system's origin, and the sum of the distance from both focii will be the same to any point. You should be able to find more of this kind of relations that link together the four values you seek.
Steeven
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