Linear Interpolation Along a Trajectory

A trajectory is a piecewise-linear path through a sequence of waypoints. To track an aircraft's continuous position along this path, we need to interpolate between consecutive waypoints.

Linear interpolation between two waypoints $\mathbf{A}$ and $\mathbf{B}$ at parameter $t \in [0,1]$ is defined as

$$\mathbf{P}(t) = (1-t)\,\mathbf{A} + t\,\mathbf{B}.$$

The endpoints correspond to $t=0$ (giving $\mathbf{A}$) and $t=1$ (giving $\mathbf{B}$). The midpoint is at $t = \tfrac{1}{2}$. Each coordinate is interpolated independently.

For example, with $\mathbf{A} = (4, 10)$, $\mathbf{B} = (12, 30)$, and $t = 0.25{:}$

$$\begin{align*} \mathbf{P}(0.25) &= 0.75 \cdot (4, 10) + 0.25 \cdot (12, 30) \\[3pt] &= (3, 7.5) + (3, 7.5) \\[3pt] &= (6, 15). \end{align*}$$