Even when a beam emittance is constant, its spatial extension can and usually does increase. This is easily seen by considering an ensemble of particles having initial positions between +d and —d and transversal velocities distributed uni­formly between +v and —v. As shown in figure III.5, at time t, particles with velocity +v will span positions between —d + vt and d + vt; those with velocity —v will span positions between —d — vt and d — vt. In phase space the initial rectangular volume ((—d, —v), (d, —v), (d, v), (—d, v)) transforms into a parallelepiped ((—d — vt, —v), (d — vt, —v), (d + vt, v), (—d + vt, v)). Thus, the beam spatial extension increases from 2d to 2(d + vt) while the phase space volume remains constant. Clearly, such an uncontrolled increase of the spatial extension of the beam is unacceptable in any accelerator, hence the need for spatial focusing.