International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004 
  
and the former is only fitting one co-linear equation relative to 
x or y. The co-linear equation applied by feature point includes 
the parameters of feature line. For example, the coordinates of 
points in the line and circle can be expressed by their 
parameters, and substituted to the co-linear equation, and the 
parameters can be solved at one time. Besides physical and 
feature point, un-visual point----infinite point is also included in 
the category of generalized point. For example, vanish point---- 
the intersection of projects from a group of parallel line in the 
space is the project of the infinite point, and it is fitting the co- 
linear equation. Consequently, it is easy to reduce the point, line, 
circle, curve and infinite point in to one mathematical model: 
co-linear equation, and perform uniform adjustment. 
2. GENERALIZED POINT PHOTOGRAMMETRY 
In traditional photogrammetry, all physical points are fitting the 
co-linear equations: 
a(X-X)+b(Y -Ys) +e (2-25) 
a,(X - X,)* (Y - Y.) e(Z - Z,) 
a, (X— X,)+b,(Y -Y,)+c,(Z-Z,) (2) 
a,(X = X5) HD, = F5) + e(Z - Z5) 
  
(1) 
xexQ—f 
  
Nm yy f 
where x and y are the observations with corrections v,, WX, Y 
and Z are ground coordinates with corrections 4X, AY, AZ, Xj, 
Ys, Zs. 9, 0, K, f, Xo, yo are parameters to be solved, which can 
be expressed by approximations and corresponding corrections. 
If p;..,p, are additional parameters, the linearized error 
equations are: 
yug ea AY va AL og Aor ^o 
*Tag MC a Af daa duy, + DAX 
HhAY + DUAZ Ho pAp vu sepâp, —h (3) 
y du AX, +0, ,AY, +0 AZ +4, Ap +a,,A0 
TUS AKA 0 Af Fas AX, asy, T by AX 
TDSAY- DAZ ECQApU Y FC Ap, -1, (4) 
  
  
  
  
  
  
  
  
  
  
  
  
where 
Ox Ox ox 
= a,» = 
11 > 2 1 2 
OXs QYs 078 
ux Ox Ox 
Qu , se > o V NA 
op dw Ok 
oy Gy Qy 
an == a Ine , 
QXs QYs QZs 
Sv ^ ^ 
= Oy Op oy 
24 , SEE , 26 
op Qo OK 
b, au, Ds ay, by; == -a;, 
b, dys by =-—ay, by=-ay, 
Ox ox 
em = 
11 , 1 > 
Op, n Op, 
oy Oy 
Ca 7 Can 
OP, Op, 
Constants 7, — x-(x), /, — y-(y), where (x) and (y) are computed 
by the approximations of parameters substituted into equation 
(1) and (2). 
2.1 Vanish Point 
Let (Xx», Yx«), (Xyeyy«.) and (xz,, yz.) be the coordinates of 
intersections (vanish points) py, Pye and pz, of projects from 
three bundles of straight lines, which are parallel to X, Y and Z 
axes respectively. The numerator and denominator of equation 
(1) and (2) are divided by X, and let X tend towards to limitless, 
the co-linear equations of vanish point px, relative to the lines 
parallel to X axis can be acquired: 
  
a a 
= = 2 
Xx 7 Ka / yaoi. f 
a, a; 
The co-linear equations of vanish points py, and pz, relative to 
the lines parallel to ¥ and Z axis respectively can be acquired in 
the same way: 
  
b b 
; ni ni Oy i 2 
Tr m m frm pod ure 
73 24 
e c 
— — I er : 2 
X2 — Xo A ZL 2 Vie 3 Yo E y 
e C 
Above six equations show that the three interior parameters x, 
yo, f and three exterior direction parameters @, @ K can be 
computed using vanish points relative to three limitless points 
in X, Y and Z axes respectively 
2.2 Point in Feature Line 
Each point in the straight line and curve can be directly used to 
only one of two co-linear equations. 
2.2.1 Straight Line: Each point (x, v) in the line parallel to y 
axis on image is fitting equation (1) and (3), wether the vertical 
coordinate v is what value. In the same reason, each point (u, y) 
in the line parallel to x-axis on image is fitting equation (2) and 
(4), wether the horizontal coordinate u is what value. The 
equation of a line / with arbitrary other orientation 0 is 
ax+by+c=0, a#0, b#0, 0 = arctg(a/b), 0+ 0°, 0+ 90? 
When —45° < 8 < 45° or 135° < 8 < 225°, every point p(x, y) of 
line / is fitting equation (1) and (3), otherwise p is fitting 
equation (2) and (4). 
If the equation of spatial line L in plane Z ^ Z,, which project is 
/, is 
X z X, cos @ Y =Y,+1-sin0 (8) 
Substitute equation (8) into co-linear equations (1) and (2), then 
AU, T ICosU Y b ISMO +c (Z~Z,) 
GG, cos X. )4 5. (Y, rsinO—Y.)-c(Z-—-Z;) 
UN t cosD — X I t fsinO Y y+, (7-7) 
  
x=x 7 
  
Jy, e 
i a(x, +tcosd=X Yb. (Y. +isind-Y. Yo (Z—Z,) 
Because p;-X», p;-Y,, p;-0, so that among equation (3) and (4): 
bu D hb, 
Cj 7 4j. Ci27 045, C1 7 -Ed;, Sin 0, 
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2.2.2 Circle: If the equation of spatial circle with centre (Xe, Yo) 
and radius R in plane Z 7 Z, is: 
   
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