This paper studies the trajectory instability and convergence of the curvilinear motion of a hard projectile in deep penetration, based on rigid body dynamics and the differential area force law. It is observed that both sudden (instable) and gradual (curvilinear) changes of the projectile trajectory may happen under certain conditions depending on projectile geometry, target property, impact velocity and other striking parameters. A criterion is introduced to determine the occurrence of trajectory instability. Two characteristic velocities are identified, i.e. (a) the critical impact velocity for the occurrence of trajectory instability, and (b) the characteristic convergent velocity, at which the projectile trajectory turns to a straight line (linear motion) from the curvilinear motion, which is supported by experimental evidence. It is shown that both the critical impact velocity and the characteristic convergent velocity depend linearly on the relative location of projectile's centre of mass. However, the characteristic convergent velocity is independent of the non-axisymmetrical disturbance and impact velocity.