Time 0Burned 0 haHead ROS 0 m/minFlame 0 mreadyFPS 0

Drag to orbit, wheel or pinch to zoom, tap or click the hillside to start another fire. Orange: burning, dark: burnt. The white ring is the ideal ellipse for the first ignition (uniform fuel, local wind). The wide arrow shows where the wind blows to. Changing a slider re-plans the whole fire from its ignition times.

Wildfire Elliptical Spread: Length-to-Breadth, Huygens Wavelets, Backing Fire and Firebreaks

A fire from a point grows into an ellipse: fast at the head, slow at the back. Wind and slope stretch it; a lake, a firebreak and wet fuel bend it.

Weather and terrain

Fuel

Firebreak (road, x = 130 m)

View

Live readouts (first ignition)

Where does it spread fastest? Spread rate by direction

Polar plot of the distance a point fire covers per minute in each direction (ellipse with the ignition at the rear focus). The dot marks the wind or upslope direction.

Burned area against time

Experiment: how does wind stretch the fire?

Flat, uniform fuel, no lake or break. Each row ignites one point and stops the clock when the head has travelled 100 m. The numeric area is counted from the arrival-time solution; the ideal ellipse area is π a b. Above length-to-breadth of about 3 the 5 m grid cannot resolve the narrow ellipse and the numeric area is too small.

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom. Tap or click the hill to add an ignition (up to 6).
Wind direction is the direction the wind blows towards. Slope adds an upslope wind-equivalent; fire runs uphill and creeps downhill.
A firebreak holds only if it is wider than the spotting distance.
Speed is simulated seconds per real second.

The science

Ellipse. In uniform fuel, wind and slope, a fire from a point grows into an ellipse. With the ignition at the rear focus, the distance covered per unit time in a direction ψ from the head is r(ψ) = R_h(1 − e)/(1 − e cos ψ). Head R_h, flank R_h(1 − e) and back R_h(1 − e)/(1 + e). The eccentricity follows from the length-to-breadth ratio, e = √(1 − 1/LB²).

Length-to-breadth. Anderson (1983): LB = 0.936 e^{0.2566 U} + 0.461 e^{−0.1548 U} − 0.397 with U the effective wind in mph (capped at 8). It equals 1 (a circle) in calm air. Here U is the magnitude of the vector sum of the midflame wind and a slope-equivalent wind 15 mph × tan(slope) × slope-influence, pointing upslope; the slope coefficient is illustrative.

Head rate. R_h = R₀(1 − M/M_x)(1 + k U^1.5): base rate R₀ (grass 0.06, shrub 0.025, litter 0.008 m/s), moisture M with extinction M_x = 30 %, wind factor with k = 0.25 to 0.35. These constants are illustrative, in the spirit of Rothermel-type models, and not calibrated to a particular fuel model.

Huygens principle. Every burning point sends out its own elliptical wavelet; the fire front is the envelope. We solve for the earliest arrival time of fire at every node of a 129 × 129 grid (5 m cells) with Dijkstra's algorithm over 48 directions, using the local ellipse at every node, so heterogeneous wind (slope), a lake and a firebreak just change the local ellipse. The ellipse is convex and contains its focus, so its gauge function is subadditive and the straight line is the fastest path: the discrete solution converges to the ellipse.

Intensity. Byram: I = H w R_h with H = 18 600 kJ/kg and fuel load w (grass 0.3, shrub 1.0, litter 1.2 kg/m²), and flame length L = 0.0775 I^0.46. The containment guide is a common rule of thumb (hand crews below about 1.2 m flame length, machinery below about 2.4 m, direct attack not safe above about 3.5 m).

Simplifications. Steady spread rates (no acceleration after ignition), no crown fire, spotting reduced to one rule (a break is crossed when it is not wider than the spotting distance, then at 35 % of the rate), constant wind over the domain, fixed residence time, no fire-atmosphere coupling, and a coarse grid that under-resolves very elongated fires. Educational model, not for operational use. Models built with Blender, scene by Three.js.