The movement of fluid particles around a running cylinder or a sphere is considered. Particle trajectories viewed from a fixed object are contours of the stream function and well known in many cases. Here, we are concerned with trajectories viewed from the absolute coordinates where the object is moving. In 1870, Maxwell considered the problem in irrotational flow of inviscid fluid, and found that the trajectory of a particle is a curve of elastica having a self-intersection point. We consider here a similar problem in three-dimensional (3D) irrotational flow, 3D Stokes flow around a sphere and Brinkman's porous-media flow. In the 3D Stokes case, we found that the trajectories are unbounded and have no self-intersection. In the Brinkman case, we treated both flow around a cylinder and flow around a sphere: our numerical examinations revealed both self-intersecting and non-self-intersecting trajectories.