Again the relationship between equations (8) and (10) is 
obvious. If one considers the identities of the triple scalar 
product, we see that, 
n 2 [R'(x- x,)]81 (11) 
Note that equation (10) relates the distance from our point on 
the line to a plane, and equation (9) constrains us to a 
particular plane. 
3. CYLINDER PROJECTION 
Cylinders in the form of pipes and vessels are abundant in 
the typical industrial plant, and dominate the CAD models of 
that plant. The equations of the two tangent planes to the 
cylinder constrained to pass through the optical centre of the 
camera will be derived. As with the straight line the 
intersection of these two planes with the focal plane of the 
camera yields the 2D line equations of the two occluding 
edges of the observed cylinder. 
As before, a similar approach yields the equation linking the 
planes tangent to a cylinder, to the parameters of that 
cylinder. 
The picture may be completed by considering the equation of 
the cone whose base is the edge of the circular end-cap of a 
cylinder in object space, and whose apex is the optical centre 
of the camera. Once again the intersection of this cone with 
the focal plane of the camera yields the equation of the 
ellipse representing the observed image of the circular end- 
cap. 
3.1 Projected Occluding Edges of a Cylinder 
In $2 we used the triple scalar product to establish the co- 
planarity of our 3D line and our image line. Since the triple 
scalar product may also used to determine the distance 
between two lines, it can be used in an identical manner to 
that in $2 to give an equation for the tangent planes to a 
cylinder. 
Let us define our cylinder as follows, 
a= (a, a, 2, ) - point on the cylinder axis (12) 
1=(1 m n) - cylinder axis vector (13) 
r -cylinder radius (14) 
The distance between a line tangent to the cylinder, and the 
cylinder axis will be, r. Therefore we can write, 
(er [Re Re T 
The term on the right hand side is scaled since, x-x,, is not a 
unit vector. 
We can remove the ambiguity of the sign on the right hand 
side of equation (15), by squaring both sides. Therefore, 
f(x.) «[R[re (a - x )]] = {rx-x,|} (16) 
Unfortunately the individual equations of the two planes are 
not readily extracted from equation (16). As an alternative 
we can derive the unit normal vectors to the two tangent 
planes as follows. 
Let point, P, be that point closest to X, which lies on the 
line defined by the axis of the cylinder. 
Then we have, 
P=a+[(X,-a)ei}l (17) 
  
  
Figure 1 Normal Vectors to a Cylinder 
Now by definition, the vector, (X, - P), is perpendicular to 
the surface of the cylinder. If we consider Figure 1, it can be 
seen that 
t- R n) 
= E (18) 
where, R,, is a rotation matrix providing a rotation by angle, 
t0, about axis, l, see (Bowyer & Woodwark, 1993; 
Thompson, 1969). 
Upon substitution for the terms of the rotation matrix, R,, we 
find, 
NEP) incor | Qi P) 
t- [m ® | sin(0) + Ez cos(0) (19) 
and therefore upon substitution for the trigonometric 
functions (refer to Figure 1) we get the equations for the two 
tangent normal vectors, 
ts (X, - P) &[(X, -a)ei]/Ix, -[ -:? 
20 
x, -p (20) 
286 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
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