If two dimensionless particles collide, then the vectors of their momenta before the collision must intersect. These two vectors thus define a plane, and the sum of their momenta before the collision lies in this plane.
The usual assumption for a collision is that, during the short duration of the collision, the impulse provided by external forces is negligible in comparison with the total momentum, so momentum is conserved:
p1f and p2f also intersect and the sum of these vectors must, by conservation of momentum, also lie in this plane too. So two-particle collisions can be considered as collisions in two dimensions.
Two further simplifications are possible: one can choose a frame of reference in which one particle is stationary and the other travels initially in the x direction. These simplifications are made in the case shown at right, but using billiard balls, not particles, so an offset displacement between their centres is possible.