Helmut Kager 
3.1 Metrification of Algebraic Error 
We assume the surface given by an implicit function ¢(z,y,2) = p(æ) = 0. Figure 1 shows this situation. 
   
Figure 1: Potential surfaces - Normal in footing point F. Figure 2: Potential surfaces - Normal in object point P. 
¢(F,a) = 0 resembles the true (resp. adjusted) surface with the footing point F for the adjusted point P. P itself is 
used as argument for the surface ¢ and yields a contradiction p(P,a) = w. This can be interpreted as if space is filled 
by a scalar field (like a potential field) and the desired surface represents potential zero; the final point P then will/might 
also lay on an iso-surface - but of potential w. 
Bearing in mind to measure the correction v along the surface’s normal in F 
Oo(F,a)V 
ne: n(F,a) = So = Vy(F,a) (13) 
oF 
we use the gradient of the surface q : 
P=F+v=F+pu-n therefore YU =p vVnne (14) 
with pe being a proportionality factor for scaling n= and qo: ut. 
We can describe w by Taylor-Series expansion of the zero-potential: 
Op(F, : 
w-qg(P,a)-q(F-cp:n,a) = @(F,a)+ Sen - UF + Ne + + - + higher order neglected 
= qq(F,a)- p: nzne — pe: nene (15) 
using (13) and since o(F,a) — 0 yielding 
w 
Y= T 
v Tie Tie 
For practical computation we have F not available at first. We could iterate that point F by some steepest-descent 
method as done in (Forkert, 1994) for ORIENT's 3D-splines, but here we want to circumvent that effort. 
  
  
He = er with (14) therefore (16) 
Assuming a "best corresponding" point Q on the final surface p(Q, a) = 0, we get analogously due to figure 2: 
ne := n(P,a) = CA = Vy(P,a) (17) 
we use the gradient of the surface ¢ : 
Q = P+w = P+pp-m therefore uw = p-vVnine (18) 
with ju being a proportionality factor for scaling ne and Vom uu. 
We can now describe the zero-potential by Taylor-Series expansion from potential w : 
0p(P,a) 
p(P,a)+ EP 
p(P, a) + pe : nene — pe : ng ne (19) 
0 = p(Q,a) = (P -* pe: ne, a) * [le * ie + - + - higher order neglected 
  
476 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.