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2 MODEL FORMATION OF THE SURFACE CONSTRAINT 
2.1 The First-Order Surface (Plane) Equation: 
Mathematically, three points, let’s say P(X,,Y,Z,), P,(X,,Y,,Z,) and P,(X,,Y,,Z,), define a plane. The plane 
equation with a, b and c being the function of coordinate components of three points can be seen as equation (1). 
aX +bY+cZ +d =0 (1) 
To calculate the distance along the surface normal between a point P(X pY,,Z,) and the plane aX +bY+cZ+d=0, 
one could use equation (2) shown as below, 
)- aX, *tbY, *cZ, t d 
Ja b vc 
2.2 The Surface Constraint: 
(2) 
Analytically, if a point is known to lie on the known plane, equation (1) would be automatically realized. A direct result 
of that is that the distance calculated by using equation (2) would end up with zero. 
Realistically, due to the measuring errors of the employed instrument, the physical truth of the plane could hardly be 
found by the collected data. Thus if the points are judged to lie on the known plane, one would compromise the 
observations by minimizing the distances along the normal, shown as equation (3), such that the imperfection of the 
measurements and the overall registration of the data set could be taken into account in an optimal way towards the 
solution. By that, the author proposes an algorithm in which the plane is hypothesized in the object space and confirmed 
by the surface points, thus forming the surface constraint into the adjustment for simultaneously determining the object 
points and the exterior orientation parameters via aerial triangulation procedures. 
Her em SER (3) 
da! b vc 
2.3 The Functional as well as the Stochastic Model of the Surface Constraint 
The symbols in equation (3) are classified into two groups, namely the observations and the unknowns. For this study, 
what is known are the surface points while those points P(X ,Y,,Z,) considered to lie on the planes are the unknowns 
through the photogrammetric measurements. One, therefore, is able to formulate functional model as well as stochastic 
model of the control surface constraint shown as equation (4) 
V 2pp^! 
Wil = Brx3q Y3gx1 = An3r7nx1 +B €3qx1> Be ~ (0, Y, =B XB F =s 00 BF, B 7) (4) 
where 
n : the number of object points from the photo measurements; 
m : the number of surface constraints ; 
q : the number of registered surface points; 
W: the discrepancy ((/” ) derived with the approximations of the unknowns and the observations; 
y : the observations of the known surface points; 
B : the coefficients of partial derivatives with respect to the surface points ; 
A : the coefficients of partial derivatives with respect to the unknowns of the object points ( may 
include 0 for those object points unable to find registered surface planes); 
mx X n—m ) 
X theunknowns of the object points ; €: the random errors of the surface points; 
E :the dispersion matrix of the surface points; P, : the weight matrix of the surface points; 
s , :the variance component of the surface points ; 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 445