X, sinP, sinf, -cosp sinp,cos. 
Y, 7 À,| cosP sin, +sinB,sinB,cosp, 
Z, cosp.cosp., (2.2) 
cos p,cos p., 
*q|-sin p,cos B, 
sin p, 
(7x «(9-93 -r? (2.3) 
V-V=R? (2.4) 
V-5=0 (2.5) 
P-[X,-X) @-Y) G-2)] (2.6) 
P=[cosa cosa, sina cosa, sina,]" (2.7) 
2.2 Geometric Constraints 
One of the rich sources of information when dealing 
with linear features is the existence of various types of 
geometric constraints in the object space among such 
features. These are of two types: one providing relative 
information, such as parallel, perpendicular, coplanar, 
etc, and the other partial absolute information with 
respect to the reference coordinate system, such as 
horizontal, vertical, etc., features. Constraints among 
straight lines include: relative 2 parallel lines (2 
Equations); 2 perpendicular lines (1 Eq.) 2 coplanar 
lines (1 Eq.); partial absolute: line parallel to X-, Y-, or 
Z-axis (vertical) each provides 2 Eq., horizontal line (1 
Eq.) Constraints among circular features include: 
relative. 2 parallel circles (2 Eq.); 2 coplanar circles (3 
Eq.); 2 circles in perpendicular planes (1 Eq.); partial 
absolute. circle in XY (horizontal), YZ, or ZX planes, 
each provides 3 constraint equations. Constraints 
between straight lines and circles include: 1 line 
coplanar with 1 circle (2 Eq.); 1 line perpendicular to 
circle plane (2 Eq.), 1 line passing through circle center 
(2 Eq.), all of these provide relative information. 
23 Photogrammetric Conditions 
Classical photogrammetric condition equations were all 
derived on the basis of point features and therefore 
need to be re-developed for linear features. Each type 
of linear features requires a suitable form. For a 
straight line feature, an equivalent pair of collinearity 
equations relating the line image parameters, p,« to its 
object descriptors, q, p,, p.,, Bs: 
536 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
(Dm, * fcosam,, *fsinam,)(myq -X,) 
*(Dm,, * fcosam,, +fsinam,2)(ng214 - Y,) (2.8) 
*(Dm,, * fcosam,, * finum, )(m,,,q -Z;)- 0 
(Dm, * fcosam,, * fsinaum,) 
+(Dm,, +fcosam,, + fsinam,,) (2.9) 
*(Dm,, * fcosam,, - fsinam,;) =0 
D=p-x,cos« -y,sinæ (2.10) 
in which m, are elements of the image orientation 
matrix, mp; are elements of the line rotation matrix. 
For a circular feature in the object space, the 
collinearity condition reduces to a single equation for 
each image 
points, ij, on its image. The image vector is 
  
Px X; X 
P; i 
and the condition equation is 
X, X, i 
Ü- y, x, |. 505732 *0-19 222) | 2.3) 
P,P,*D,p, *P.p, 
Z, -Z, P; 
in which x, y,.f, X p YpZ,, M represent the interior (IO) 
and exterior (EO) image parameters, 
X, Y, Z Rp. D,.D, (elements of p, see Eq. (2.7)) the 
circle descriptors in the object space. 
2.4 Line-Based Photogrammetric Operations 
Line features, like point features, may be used as pass 
and control features. Therefore, all photogrammetric 
operations executed with point features can similarly be 
performed on the basis of linear features. Here are 
examples: 
Resection: 3 control straight lines or 2 control circles 
are the minimum required to estimate the six exterior 
orientation elements of a single photograph. If the 
interior orientation elements are to be also recovered, 
two additional control straight lines would be required 
for a minimum solution. Combination of features and 
more than the minimum control may be used. 
Relative Orientation (RO). Pass straight lines do not 
contribute to RO of a pair. For a triplet, however, a 
pass line in 3 images contributes 2 equations to RO. A 
     
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