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Trajectories

A trajectory (or parametric curve) is a vector-valued function $\boldsymbol{\gamma}: I \subseteq \mathbb{R} \to \mathbb{R}^m$ where the domain is an interval (1D). As the parameter $t$ varies over $I$ , the output $\boldsymbol{\gamma}(t)$ traces a curve through $\mathbb{R}^m$ .

Examples:

Straight line through $\mathbf{p}$ in direction $\mathbf{v}$ : $\boldsymbol{\gamma}(t) = \mathbf{p} + t\mathbf{v}$
Circle of radius $r$ in the plane: $\boldsymbol{\gamma}(t) = (r\cos t, r\sin t)$
Helix in $\mathbb{R}^3$ : $\boldsymbol{\gamma}(t) = (\cos t, \sin t, t)$

Trajectories are the case $n=1$ (one parameter) of vector-valued functions. They are fundamental in physics (position as function of time), differential equations (solution curves), and dynamical systems.

Formal View

Definition 13.5 — Trajectory (Parametric Curve)

A trajectory is a continuous function

\boldsymbol{\gamma}: I \to \mathbb{R}^m

where

I \subseteq \mathbb{R}

is an interval. The image of

\boldsymbol{\gamma}

is the set

\{\boldsymbol{\gamma}(t): t \in I\}

, the curve traced in

\mathbb{R}^m

The same curve can be traced by different parametrizations. The parametrization encodes not just the path but also the speed and direction of travel.

Interactive Visualization

Trajectory and Tangent Vector

Why This Matters

Trajectories are the mathematical language for describing motion, flow, and solution curves of differential equations.

Physics: position $\mathbf{r}(t)$ , velocity $\dot{\mathbf{r}}(t)$ , acceleration $\ddot{\mathbf{r}}(t)$ of a particle
Dynamical systems: solution trajectories of $\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x})$
Computer graphics and animation: smooth parametric paths for object motion

Learning Resources

Parametric Curves

Khan Academy

Introduction to parametric curves and trajectories.

10 min

Vector-Valued Functions and Curves

MIT OpenCourseWare

MIT lecture on parametric curves as vector-valued functions.

50 min

Quiz

Question 1

The trajectory $\boldsymbol{\gamma}(t) = (2\cos t, 2\sin t)$ for $t \in [0, 2\pi]$ traces:

Question 2

Two different parametrizations can trace the same set of points (same geometric curve) in $\mathbb{R}^m$ .

Common Mistakes

Confusing the trajectory (a function) with its image (a geometric curve).
Forgetting that trajectory derivatives give velocity, not just direction.
Assuming a parametrization is one-to-one — the same point can be visited multiple times.