(16) 
0 planes are 
n alternative 
two tangent 
| lies on the 
(17) 
T 
endicular to 
» 1, it can be 
(18) 
on by angle, 
;ark, 1993; 
atrix, R, we 
(19) 
igonometric 
; for the two 
(20) 
These vectors may then be rotated into the camera co- 
ordinate system, and the equations for the planes in camera 
space will be given by, 
(x-x,)e(Rt)=0 (21) 
with notation as in $2.1. 
The intersection of these two planes with the focal plane, 
give us the equations of the two lines forming the occluding 
edges of the cylinder in the image, in the form of equation 
(7). 
32  Tangent Observations to a Cylinder 
By direct analogy with the equations derived in $2 for a 3D 
line, we obtain the following equation for the observations, 
in an image, of points along the occluding edge of a 
cylinder, 
[[R(x-x,)J«[19 (a-x.)]] = {rx- x, |} (22) 
3.3 Tangent Planes to a Cylinder: 
Again by direct analogy, we can derive the equations for 
planes to be tangent to a cylinder, 
nel=0 (23) 
{nea+d} =r’ (24) 
The relationship between equations (24) and (22) is not so 
obvious in this case, but vector, n, is still given by equation 
(11), and we note that, 
neX, =—d (25) 
3.4 Projected Cylinder End-Caps 
To determine the equations of the ellipses forming the 
projection of a cylinder’s end-caps into an image, we follow 
a similar procedure to that in the previous sections. 
However, in this instance, we determine the equation of the 
cone whose base is the end-cap of the cylinder, and whose 
apex is the optical centre of the camera. 
Let us retain the definition of a cylinder given in §3.1, but 
further define the point, P, to be the centre of one of the 
cylinder’s end-caps. Thus, 
P=(P. P, BY (26) 
We can therefore define the circular edge of the end-cap to be 
the intersection of the following two surfaces, a plane and a 
sphere, 
(X-P)el=0 (27) 
(X-P)e(X-P)-r? -0 (28) 
The cone we are seeking to define is that surface generated by 
the straight line passing through the point, X,, which 
intersects the curve defined by equations (27) and (28). Let us 
define this straight line as follows, 
X = X, +at (29) 
where, 
t-(t u v) 
Substitute for, X, from equation (29) into both equations 
(27) and (28), and then eliminate, œ, between them. Upon 
gathering terms we reach the following equation, 
(tot)(X,~P)el] 
-2[t«(X, - P)|(X, - P)eijte1) G0) 
«(X, -P)«(X, -P)-r te)? 20 
Now, equation (30) is a homogeneous equation which the 
direction-cosines, t, must satisfy for the line to pass through 
the optical centre of the camera and the edge of the circular 
end-cap. From (Bell, 1950) we can therefore state that the 
equation of the cone we are seeking is given by the same 
homogeneous equation as below, 
-J(x -X,)»(x, -P)[ X, -P)e1[(X -X,)e1] (31) 
4X, - P)«(X, -P)-z](x-x,)eif «0 
To derive the equation of the ellipse forming the projected 
view of the cylinder end-cap, is now numerically a 
straightforward two stage process. To start we transform this 
cone into our camera space, and then we determine the 
intersection of this transformed cone with the focal plane of 
the camera. The algebra of this process is not detailed in this 
paper, but results in an equation of the form, 
Ax! - By? 4 2Hxy -* 28x - 2Fy C 20 (32) 
4. APPLICATION EXAMPLES 
The equations derived in the previous sections have all been 
developed for inclusion in software incorporated into a 
digital photogrammetric measurement system (HAZMAP), 
see (Chapman et al., 1992), with the aim of increasing the 
automation of the CAD modelling process. The work to date 
has concentrated upon the modelling of pipework and 
cylindrical vessels, using the equations derived. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996