Geometry of Trajectories

The tangent vector $\boldsymbol{\gamma}'(t)$ encodes the local geometry of the trajectory. The tangent line to the curve at $\boldsymbol{\gamma}(t_0)$ is the line through $\boldsymbol{\gamma}(t_0)$ in direction $\boldsymbol{\gamma}'(t_0)$: $\{\boldsymbol{\gamma}(t_0) + s\boldsymbol{\gamma}'(t_0) : s \in \mathbb{R}\}$.

The arc length from $t=a$ to $t=b$ is $L = \int_a^b \|\boldsymbol{\gamma}'(t)\|\,dt$ — the integral of speed over time. Curvature $\kappa = \|\mathbf{T}'(t)\|/\|\boldsymbol{\gamma}'(t)\|$ (where $\mathbf{T} = \boldsymbol{\gamma}'/\|\boldsymbol{\gamma}'\|$ is the unit tangent) measures how quickly the curve turns.

Two key geometric facts about curves in the plane: (1) Constant speed ($\|\boldsymbol{\gamma}'\| = c$) means the speed vector has no tangential acceleration: $\boldsymbol{\gamma}'(t) \cdot \boldsymbol{\gamma}''(t) = 0$. (2) Circular trajectories have constant nonzero curvature.