trajektoriers

Trajectories, written as trajektoriers in Dutch contexts, refer to the paths traced by moving points or objects as time passes. They are a fundamental concept in physics, mathematics, engineering, and computer science, linking geometry with dynamics. In formal terms, a trajectory is a time-parametrized curve r(t) that records the position of a particle in space, such as R^2 or R^3, or on a more general manifold. The geometric locus formed by the path is the path, while the trajectory emphasizes the time dependence along that path.

In dynamical systems, trajectories are solutions to differential equations that describe motion. If a system is described by dx/dt = f(x,t) with an initial state x(t0) = x0, then x(t) traces a trajectory through the state space. Trajectories can be influenced by forces, constraints, or control inputs, which may produce smooth, piecewise, or abrupt changes in direction.

Representations of trajectories vary by application. Common forms include parametric curves, polylines, and interpolated splines such as Bezier or B-spline curves. In data analysis, trajectories may be reconstructed from discrete observations, requiring smoothing and interpolation. A key distinction is that a trajectory includes time parameterization, whereas the geometric path is simply the set of points visited.

Applications span projectile motion, orbital mechanics, and geodesic paths in astronomy and general relativity; in robotics and autonomous vehicles, trajectory planning and optimization address feasible, efficient, and safe motion under dynamic constraints. In computer graphics and animation, trajectory concepts drive particle motion and camera paths.

See also: path, curve, parametric equation, differential equation, trajectory planning, motion planning.