Intersections of Loran Lines of Position 

 (Plane Approximation) 



P. D. Thomas, Mathematician 



Consider the hyperbolic system as shown in Figure 24. The hyperbolic locus with foci F, F' 

 has for equation 



(c^ - a^) x^ - ^Y = a^ (c^ - a^), (e = - > 1) (1) 



a 



As a varies (a < c) all the hyperbolas with the fixed foci F, F' (which are 2c apart) are 

 generated. 



The hyperbolic locus with the fixed foci F, F"when referred to the same coordinate system 

 as (1), has for equation 



Ax' + Bxy + Cy' + Dx + Ey + F = 0, (e = d/b > 1). (2) 



where one may first compute r = b' - d% /z = d cos a, v = d sin a, S = r - c pt , and then 

 A = fi^" - b% B = l^v, C = I/' - b\ D = 2(rfi - c A), E = 2Si/, F = S' - b'c\ 



As b varies (b < d) all the hyperbolas with the fixed foci F, F" (which are 2d apart) are 

 generated. 



The respective pairs of constants c, a; d, b for each hyperbola are related to the fundamental 

 constants of a Loran line by 



c = VV>J1, a = kV,/2; d = kBj/2, b - kVj/2 (2.1) 



where v- = t-, t- is the time difference, vj is the difference of light microseconds, Bj is the 



length (measured in light microseconds) of the direct line (baseline) between two Loran stations, 

 k is the length of a light microsecond in the linear units in which x and y are expressed. 



Since five distinct points determine a conic uniquely, two conies can have at most four 

 points in common. For the hyperbolas (1) and (2) there will always be four real points of 

 intersection except when F ', F, F"are coUinear (a = 0) and then there will be two. 



ALGEBRAIC SOLUTIONS 

 I. If equations (1) and (2) are solved simultaneously for x one obtains the quartic equation 



X* + Hx' + Jx' + Mx + N = (3) 



where one may first compute G = c' - a', jSo = CG + Aa', co = F - CG, § = BEG, 

 y = a'B' - E% L = /8o' - G B'a% and then H = 2a' (D^^o - S)/L, J = aMa'D'+2/3o w + Gy)/L, 



'Loran; Pierce, McKenzie, Woodward," McGraw Hill, 1948, pages 52, 53, 174. 



101 



