Ermes, Pierre 
  
  
x CV X 
Figure 3. The cylinder and the torus with their local 
coordinate system. 
The pose of the cylinder in object space is defined by a 3D translation T,,, and a 3x3 rotation matrix R ,,, according to: 
X - . t 
T! my QR, x A t (1) 
= . . tz 
The translation is parameterized by its three elements x, y, and z, while the rotation is parameterized by the elements £, , 
f,and f, of a 3D unit vector. Since a cylinder is symmetric around its Z-axis, the degrees of freedom concerning its 
rotation are reduced from 3 to 2. The rotation parameterization in (1) requires an extra constraint on the length of the 
vector (it must be equal to 1) but provides a representation that is free of singularities. We will refer to this rotation 
parameterization as a ‘trinion’. The dotted elements in the rotation matrix are not relevant for this parameterization 
because of the rotation symmetry in the model. When necessary they can be computed to make R a valid rotation 
matrix. 
A torus, as shown in Figure 3, has three shape parameters; a minor radius, a major radius R, and an angle c. The base 
is located in its local coordinate system at (0,0,0) and the curve bends towards the Y-axis. The pose of the cylinder in 
object space is defined by a 3D translation T jorus and a 3x3 rotation matrix R, according to: 
4 -4;-4;*4;  2X44,*4,4,) 2(q,4, — 9294) 
Tous =) 9 bRaors =) 2(94;-44 -0+02-01+4} — 2(9,4, 4,4.) (2) 
z 2(q,9, + 9.94) 2(444,—4,4,) | —4,-4; *4; *dq, 
The translation is parameterized by its three elements x, y, and z, while the rotation is parameterized by four quaternion 
elements. A 3D rotation has three degrees of freedom and, therefore, the rotation parameterization in (2) requires an 
extra constraint on the length of the quaternion (must equal 1) but it does provide a representation free of singularities. 
When piping elements are connected to each other, continuity requirements impose constraints on the radii of the 
elements, the positions of both connecting ends, and the orientation of the elements. For the radii r and r, of two 
elements the constraint equation is: 
y-5-£ (3) 
Where £ is the residual, or difference between the two radii. For the positions P, and P, of two connecting ends the 
three constraint equations are (for x, y, and z): 
P-P=¢ (4) 
Where &, is a vector containing the residuals for the position (in x, y and z). For the directions D, and D, of two 
connecting ends, the three constraint equations are (for x, y, and z): 
D, -D,-&, (5) 
Where €, is a vector containing the residuals for the direction (in x, y, and z). Constraining the direction of a 
connection restricts two degrees of freedom and still leaves one degree of freedom for a rotation. The use of three 
  
218 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
  
  
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