Finding sphere of influence in multibody system I would like to draw a boundary illustrating the sphere of influence of each body in a 3 body system. What is the best way to find such a boundary?
My understanding is at the edge an SoI is the minimum of the sum of gravitational forces. My current method is, for each body, project a line from the body to some fixed point well beyond the size of the system, and find the minimum net gravitational force by binary intersection. Is there a faster way of doing this? 
 A: I think your definition of sphere of influence is incorrect. You could also be confusing the sphere of influence with the Hill sphere.
The sphere of influence has mainly an application in the patched conic approximation, and the word sphere is even another approximation. A related question asks about the derivation of the radius of this sphere. The core of the answer I gave is that the actual surface of the sphere of influence is defined by the ratio of primary and the perturbing gravity source. The perturbing gravity is not the same as its total gravity, since the primary source is also attracted to it. I am not certain if this method can be altered to be applied to three bodies, who exert a gravitational forces. You might be able to derive it for certain situations, such as for Trojan asteroids and just include the extra perturbation of Jupiter to the perturbation of the sun. However it will be harder for systems of more equal masses, such that there might be points in space where instead two, three spheres, or rather areas, of influences meet.
The Hill sphere is an approximated limit of the distance at which a satellite can still orbit a celestial body in a stable manner. And as David Hammen and the Wikipedia article stated, stable prograde orbits are limited to about 1/3 to 1/2 the Hill radius, while retrograde orbits are stable at larger distances.
A: find the center of mass of three bodies $M_3$, and three pairwise centers of masses. send the rays from $M_3$ to $M_{21},M_{13},M_{23}$, they'll divide the space to triangles of influence
