Trajektooride

Trajektooride is a hypothetical plane curve used in kinematic geometry to describe a trajectory whose local turning is tied to the rate of change of speed. The term blends trajeto, the root meaning trajectory, with -oride, a suffix used in some geometric names. In this framework, the curve is determined by a relationship between curvature and tangential acceleration.

Definition: Let γ(t) = (x(t), y(t)) be a twice-differentiable planar curve, with speed v(t) = |γ′(t)| and signed curvature κ(t) of γ at time t. The trajektooride is the trajectory that satisfies κ(t) = λ dv/dt, for a fixed constant λ > 0, where dv/dt is the time derivative of speed. The orientation determines the sign of κ.

Consequences: If dv/dt = 0, then κ = 0 and the path is a straight line. If dv/dt is constant, the curvature magnitude is constant, yielding a circular arc up to reparameterization. In general, the curve adapts its turning rate as acceleration changes, producing a family of curves with smoothly varying curvature tied to the kinematic profile.

Relation and examples: The trajektooride intersects standard plane-curve theory as a specialization where tangential acceleration governs curvature. It is used as a modeling device in robotic path planning and animation to study how speed changes influence turning. Construction can proceed by prescribing a speed function v(t) and integrating the differential constraint to obtain γ(t).

See also: curvature, differential geometry, plane curve, kinematics. References: The trajektooride is a hypothetical construct; background on curvature and plane curves can be found in standard texts on differential geometry and kinematics.