Collision Detection between Accelerating Spheres

Question

I am writing a physics engine/simulator which incorporates 3D space flight, planetary/stellar gravitation, ship thrust and relativistic effects. So far, it is going very well,

Answer 1

On the webpage AShelley referred to, the Closest Point of Approach method is developed for the case of two objects moving at constant velocity. However, I believe the same vector-calculus method can be used to derive a result in the case of two objects both moving with constant non-zero acceleration (quadratic time dependence).

In this case, the time derivative of the distance-squared function is 3rd order (cubic) instead of 1st order. Therefore there will be 3 solutions to the Time of Closest Approach, which is not surprising since the path of both objects is curved so multiple intersections are possible. For this application, you would probably want to use the earliest value of t which is within the interval defined by the current simulation step (if such a time exists).

I worked out the derivative equation which should give the times of closest approach:

0 = |D_ACC|^2 * t^3 + 3 * dot(D_ACC, D_VEL) * t^2 + 2 * [ |D_VEL|^2 + dot(D_POS, D_ACC) ] * t + 2 * dot(D_POS, D_VEL)

where:

D_ACC = ob1.ACC-obj2.ACC

D_VEL = ob1.VEL-obj2.VEL (before update)

D_POS = ob1.POS-obj2.POS (also before update)

and dot(A, B) = A.x*B.x + A.y*B.y + A.z*B.z

(Note that the square of the magnitude |A|^2 can be computed using dot(A, A))

To solve this for t, you'll probably need to use formulas like the ones found on Wikipedia.

Of course, this will only give you the moment of closest approach. You will need to test the distance at this moment (using something like Eq. 2). If it is greater than (obj1.Radius + obj2.Radius), it can be disregarded (i.e. no collision). However, if the distance is less, that means the spheres collide before this moment. You could then use an iterative search to test the distance at earlier times. It might also be possible to come up with another (even more complicated) derivation which takes the size into account, or possible to find some other analytic solution, without resorting to iterative solving.

Edit: because of the higher order, some of the solutions to the equation are actually moments of farthest separation. I believe in all cases either 1 of the 3 solutions or 2 of the 3 solutions will be a time of farthest separation. You can test analytically whether you're at a min or a max by evaluating the second derivative with respect to time (at the values of t which you found by setting the first derivative to zero):

D''(t) = 3 * |D_ACC|^2 * t^2 + 6 * dot(D_ACC, D_VEL) * t + 2 * [ |D_VEL|^2 + dot(D_POS, D_ACC) ]

If the second derivative evaluates to a positive number, then you know the distance is at a minimum, not a maximum, for the given time t.