1366 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975 
Ellipse 
  
  
  
Fıc. 7. Error ellipse. 
Let the three orthogonal axes of the ellipsoid be denoted asx’, y’, and z’. Furthermore, let the 
orientation of the ellipsoid be defined by the rotations w, ¢, and x about the x’—, y’—, and z’ — 
axes respectively, and let R be a rotational matrix such that 
020 0 Puy lay 73, 92 Ory Og Tii “Tie - fis 
0 02, 0 FT: as Na Or, Oy, Oy Tai T3 Tas (22) 
0 0 a2 713 125 733 On 0,02 Far Tan T7353 
1.0. 6 = RoR” (23) 
in which the elements r;/'s are functions of the three rotation parameters w, ©, and k. 
Equation 22 consists of the following six independent equations: 
Ty d3iT i1. dao 12 1313) T 18217117 022712 T d 23113) 
ris giri. 32012 T 033713) 702, (24) 
T2(d 31721 G12022 T 13123) +T20( 21121 + A 29729 + A237 23) 
ras 31721 +A327 229 + A 33193) =0Ÿ (25) 
T31(@11731 FA 19732 HA 13133) HT3( A211731 FA 99732 HA 28733) 
+735 A31731 + Agar 35+ A337 33) 702: (26) 
T11(@11721 +127 20 +A 13725) +T1( 21791 (22122 (23123) 
tris 31721 +A 39 2+ A33T 93) =0 (27) 
Ty dy1 31 12132 O13 33) +71 121731 (22133 - 0.23133) 
ras da1T31 T 132132 0.33733) =0 (28) 
T2(031731 12733 T G 13133) ^ T25( 021731 7-023132 - 0.23133) 
Tros 31131 (35132 T 1.33133) =0. (29) 
These six equations involve six unknowns: a, Oy, 0.1, €, Q, and x. The three rotation 
parameters can be obtained by solving equations 27, 28, and 29. Since these equations are 
non-linear, they must first be linearized and then solved by an iterative procedure. The 
computed values ofo, ¢, and» can then be substituted into equations 24, 25, and 26 to solve for 
0, 0,» and g,-. 
There is only a 20 per cent probability that the true position of point j would fall within the 
ellipsoid defined by C = 1 in the following expression: 
x'2 y"? z'2 
e anoo nie Cid (30) 
  
  
There is 90 per cent and 99 per cent probability that the true position would fall within the 
ellipsoid defined by C = 6.25 and C = 11.34 respectively. 
Most photogrammetric mapping problems involve the determination of positions in three- 
dimensional space. However, because ofthe involved computation, it is neither practical nor