Marching-Squares Contour Extraction

In live air-routing, hazard fields such as convective intensity, icing severity, or turbulence index arrive as scalar values sampled on a regular grid. To plan around hazards, we need the iso-contour of the field at a chosen threshold $\tau$ — the curve along which the field equals $\tau$. Marching squares is the standard per-cell algorithm: it walks every $2 \times 2$ cell of the grid and emits the contour segments inside that cell.

For each cell, label the four corners $0$–$3$ counterclockwise starting from the bottom-left:

$$\begin{array}{cc} \text{corner }3\text{ (TL)} & \text{corner }2\text{ (TR)} \\ \text{corner }0\text{ (BL)} & \text{corner }1\text{ (BR)} \end{array}$$

Compare each corner value to $\tau$. A corner is inside if its value is $\geq \tau$ and outside otherwise. Packing the four inside/outside bits into a single integer gives the case index:

$$c = b_0 \cdot 1 + b_1 \cdot 2 + b_2 \cdot 4 + b_3 \cdot 8,$$

where $b_i = 1$ if corner $i$ is inside and $b_i = 0$ otherwise. The case index takes integer values from $0$ to $15$ and tells us which contour-segment pattern the cell emits.

For example, a cell with corner values $1.2,\,4.7,\,5.0,\,0.8$ (in BL, BR, TR, TL order) and threshold $\tau = 3.0$ has $b_0=0$, $b_1=1$, $b_2=1$, $b_3=0$, giving

$$c = 0 + 2 + 4 + 0 = 6.$$