Equation 9 can be simplified for ease of computation with a 

 less than 1 percent loss in accuracy by eliminating the terms 

 after M^,^: 



For graphic presentation of the fire shape, the perimeter is 

 plotted by using the intercept of the major and minor axes, 0, 

 as the origin. This is possible because the minor axis is com- 

 mon to both semiellipses and both semimajor axes can be 

 defined in terms of d, the forward distance traveled along the 

 major axis, and U, the windspeed. Any point on the perimeter 

 is defined by: 



If Cos >:0, a positive value: 

 x = (a2CosO)d 

 y = (bSin0)d 



If Cos < 0, a negative value: 

 X = (a,Cos9)d 

 y = (bSin0)d 



where = angular degrees from the forward direction with 

 as the origin. The origin of the fire is defined as c - aj, the 

 focus of the semiellipse containing the backing fire. 



With these equations and conditional statements, it is possi- 

 ble to predict the area burned by a fire and the distance around 

 its perimeter. Windspeed and the forward rate of spread are the 

 only inputs needed. 



ANALYSIS RELATED TO FIRE 

 SIZE AND SHAPE 



Just how well these mathematical models match Fons' 

 graphic data (fig. 2) was analyzed by comparing measured fire 

 data with calculated values. The fire shapes of figure 2 were 

 scaled at 1 inch = 1,000 ft (8.3 cm = 1 km). As size and 

 shape dimensions were calculated, a plot of the fire shape was 

 generated so computations could be compared to results ob- 

 tained from figure 2 and the plotted shapes. The plotted shapes 

 were prepared for windspeeds of 2, 6, and 12 mi/h (3.2, 9.6, 

 and 19.3 km/h). The values computed and those measured 

 were found to be within ± 2 percent of each other. Fons' fire 

 shapes of figure 2 and the model generated plots were within 

 ± 9 percent of each other for area measurements and within 

 ± 3 percent for perimeter measurements. The area and 

 perimeter measurements were made with a compensating polar 

 planimeter for area and a map measurer for the perimeter. The 

 measured and computed values are presented in table 5 and 

 shown in figures 3 and 4. 



Perimeter may be underestimated because of the natural vari- 

 ability that exists in the field. A few of the variables contribut- 

 ing to an irregular and longer fire edge are windspeed and 

 direction, slope and topography, and changes in fuel 

 distribution. 



The greatest benefit of using these equations is that only two 

 input variables — wind at midflame height and rate of spread 

 (distance for a given time) — are needed to compute area, 

 perimeter, backing fire distance, flanking fire distance, their 

 ratios to the heading fire, and the maximum length to width 

 ratio. These estimates have proven valuable to various elements 

 of fire management, but it must be remembered that the orig- 

 inal data were taken on fires burning through pine needle beds 

 without variation in wind direction. Outputs such as the length 

 to width ratio may show that fuel size (surface area to volume 

 ratio) and fuel bed packing ratio (fuel volume per unit volume) 

 have an influence. 



10 r 



2 4 6 8 10 12 



WINDSPEED (Ml/ H) 



Figure 3.— Deviations of mathematical model 

 versions from Pons' diagrams for fire size and 

 shape, area. 



2 4 6 8 10 12 



WINDSPEED mi H) 

 Figure 4.— Deviations of mathematical model 

 versions from Fons' diagrams for fire shape, 

 perimeter. 



5 



