RS 
netric 
ulated 
n and 
)bject 
[S are 
ligital 
ks are 
points 
role, 
> von 
erden 
st. In 
1. Die 
enige 
chem 
zwar 
fgabe 
it der 
3.41. 
from 
ation. 
vided 
Legenstein, Dietmar 
  
(b) the tangent plane lies on an edge of the object forming the 
boundary of the image of the object (e.g. cube). 
Case (b) has already been solved, so in the present work points 
of contours satisfying condition (a) are the only concern. From 
this property the condition of contour is derived: the vector s 
from the camera position X, to the point on the contour X is 
perpendicular to the normal vector n of the surface ® (Figure 
1). 
In general, a point on a given surface can be described by its 
image: the three unknown space co-ordinates can be 
determined using the two co-ordinates in the image together 
with the condition, that the point is on the surface. Therefore 
the three unknowns can be calculated from three equations. In 
the case of a point on the contour, the additional information of 
the condition of contour leads to an over-determined system of 
equations. In the case of a full control point three additional 
equations are given whereas in the present case only one 
additional condition leads to the overdetermination. 
  
$ 
X 
Figure 1: Condition of contour between surface, 
projection ray and point on contour. 
0 
2 MATHEMATICAL PRELIMINARIES 
The following conventions are used for the notation of points and vectors: In general points and vectors are printed in 
bold letters. For algebraic treatment in calculations tensor-notation is used. If indices are used for scalar components of 
the elements they are printed in non-bold letters. Einstein summation convention is used for all formulas. 
2. The Condition of Contour 
A point X on a surface ® is on a contour from the projection centre X, if and only if the normal vector n of the surface 
® is perpendicular to vector s from the camera position to the point on the contour. This condition can be expressed 
with the inner product of the two vectors: 
E, ns; z0 (2.1-1) 
As mentioned above this leads to an overdetermined system of equations that can be solved using least squares 
approximation. Due to overdetermination discrepancies, which have to be minimised, from (2.1-1) i.e. from O will 
occur. The inner product n's; = 0 does not allow a vivid geometric interpretation; Therefore further conditions can be 
used: 
n's. 
E, E Q.1-2) 
nn, 
n's. 
E; = —= =0 (2.1-3) 
win Jes, 
In case of (2.1-2) the normal distance from the projection ray to the surface will be minimised, in case of (2.1-3) the 
discrepancy of the angle between n and s and a right angle will be minimised. By using one of these alternatives to (2.1- 
1) the user can select a suitable criterion for his application. : i 
In photogrammetry several co-ordinate systems are used. Surfaces are often described in local co-ordinates because a 
suitable choice of the system can lead to a simple description. Therefore n is given in the model co-ordinate system 
(index M); on the other hand the vector s is determined in the image co-ordinate system (index B). For the conditions 
(2.1-1, 2.1-2, 2.1-3) both co-ordinate systems are transformed in a global reference system (without index). To 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 473