continuousmotion

Continuousmotion is the property of motion that proceeds without abrupt jumps in position as time advances. In mathematics and physics, a trajectory is continuous if the position function r(t) changes smoothly with time; no sudden discontinuities in r(t) occur. If r is differentiable, velocity v(t) = dr/dt is defined and describes the instantaneous rate of change of position; further differentiability yields acceleration.

In physical systems, continuity of motion corresponds to the absence of instantaneous spatial jumps. Real collisions involve impulses that can change velocity instantaneously while maintaining a continuous position over the collision event, assuming rigid bodies and finite contact duration.

In practice, continuousmotion is important for modeling and simulation. In animation and computer graphics, continuous motion is achieved by interpolating between keyframes using curves such as splines to avoid visible jumps, producing smooth transitions. In robotics and control theory, trajectory planning imposes continuity constraints (C0, C1, C2) on position, velocity, and acceleration to ensure feasible and smooth movement, reduce wear, and improve control.

From a theoretical perspective, some models treat time as continuous, forming the basis of classical mechanics and differential equations, while discrete-time models approximate continuous motion and are common in digital simulations.

Examples include celestial motion along smooth orbital paths, a robotic arm tracing a curved trajectory, or a vehicle smoothly following a highway lane. Continuousmotion thus underpins both foundational physics and practical engineering applications.