Employing polar coordinates, let r(e, t) be the distance from the point of ignition 

 to the edge of the fire at time t and in the direction given by the angle 6 (measured 

 from an arbitrary fixed direction). Here we insist that r(9, t) be single valued, 

 possess a positive time derivative, and be dif ferentiable with respect to 9 . If the 

 rate of progress of a fire-suppression crew, working at the edge of the fire, can be 

 expressed as A (6, t) , then we can write the equations for the generation of the final 

 boundary, R(9) . 



FIRE BOUNDARY 



CONTROL LINE SEGMENT 



OF LENGTH Adt CONSTRUCTED 



IN TIME dt 



a / 



r (0+de, t+dt 



r (e,t) 



Figicpe 1. — Generation of fi-nal 

 boundary of burned area by crew 

 working at the edge of a fire. 



Referring to figure 1, assume a differential element dt of time to elapse. During 

 this time the crew will construct an element Adt of the final boundary. In order to 

 remain in contact with the advancing edge of the fire, this element of the boundary 

 curve must be constructed at an angle a (measured counterclockwise) with respect to the 

 direction 9. We can write the formulae for the components of the arc Adt in the tan- 

 gential and radial directions from inspection: 



de = (Adt) sin a/r(e,t) (1) 



dR 



(Adt) cos a = r(9 + de,t + dt) - r(9,t) = |^ d9 + |j dt (2) 



do at 



Substituting d6 in equation 2 from equation 1, and employing the short notation 



3r ^ - l£ = ■ 



39 " ^ ' 3t ~ ^ 



we find 



A cos a = (r'/r) A sin a + r (3) 

 Squaring both sides of equation 3 and solving for sin a we obtain 



(r/A) (rVr) +Vl + (rVr)^ - (r/A)^ (4) 



sm a = — — — — 7-, — r-7 



1+ (r /r)^ 



(r/A) + (rVrjVl + (rVrj- - (r/A)^ (5) 



cos a - ^ r ' I ^ 



1+ (r /r) 



2 



