Boehm, Jan 
  
  
  
  
  
  
  
  
ID Type CH K» 
Frature - dd 
tam 31 PRO SRF PLANE 0.000000 0.000000. 
2191 PRO_SRF_CYL -0.006250 -0.000000 Sassi: > Basia: 
2424  PRO_SRF_TORUS 0.243827 0.051440 0.235 0.045653 ... 
2642 USR.SRF SPHERE -9.250000 
2705 . USR.SRF.SPHERE -0.100000 2 s RN 
? ; 0.114790 .005 0.114783 0.005108 .., 
le — PROCSRE-FIL 0.114784 -0.005108 0.114780. ~0.005110 |. 
2786 PRO_SRF_FIL -0.114790 -0.005105 -0.114783 -0.005108 ... 
2791 PRO_SRF_FIL -0.114784 -0.005108 -0.114780 -0.005110 |." 
2917  USR_SRF_SPHERE 0.333333 0.111111 
3202  PRO_SRF_CYL 0.166667 0.000000 
707 SRF_PLANE 0.000000 0.0 0 e 
7082 PRO SRF-TABCYL 0.033435 0.000000 0.000000 ... 
7086  PRO_SRF_TABCYL -0.019724 -0. 000000 0.019724 =0.000000 . 
7088  PRO_SRF_TABCYL -0.001299 -0.000000 -0.001299 -0.000000... 
7382  PRO_SRF_CYL 0.166667 0.000000 
  
(b) Example output 
(a) screenshot of the CAD system 
Figure 2: The algorithm is fully integrated into the CAD system. 
  
  
  
  
  
0.015 rx : 1 T 
x plane’ © | 
001 7%, ü AT e 
Sn LE 'sphere' E | 
g 0.005 x M 
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3 0 EUR I na + e 
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2 zi = x : - 
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S X 
$ -0.015 | E = 
3 5 i 
S 002} x I + 
-0.025 + x : - 
-0.03 ; i 
-0.15 -0.1 -0.05 0 0.05 
   
Mean Curvature (H) 
(a) plot in HK space (b) cylinder (c) sphere (d) torus 
Figure 3: Some typical surfaces and their signature in HK space. 
sphere. For other surface types we retrieve the parametric representation F' : x(u,v) of the surface. We evaluate the 
surface at a discrete set of points {(u1, v1 ), …, (Ui, v;)}. The coordinates [x, y, z] of the point, the surface normal N = 
[Nz, N,, N. z] and the derivatives Xy; Xy, Xyy; Xyy, Xyy are computed at each sampled point. From these the mean and 
Gaussian curvature are computed using the well known formulas 1 and 2 shown in figure 4, see for example (do Carmo, 
1976). The list of HK values form the footprint of the surface in HK space. After all surfaces have been visited, we have 
obtained a list of surface features for our CAD model. Figure 2(b) shows a typical output of our algorithm. 
However this list has to be post-processed. Features which are too similar in HK space can not be distinguished and have 
to be merged. This is particularly true for features which are exactly the same. For example, if the CAD model contains 
several planar surfaces or several cylindrical surfaces of the same radius, each will be reported separately. Merging of 
features introduces ambiguities in the subsequent matching process. More formal, if the model contains features F; and F; 
which can not be distinguished they are merged temporarily to feature T,. After classification. when a feature f; from the 
scene falls into the category of T; it has a possible match to both F; and F; or Q(f;) = {F;, F;). While pattern matching 
processes are designed to handle a certain amount of ambiguity, the complexity increases exponentially with the number 
of possible pairings. So clearly if the object is a polyhedron thus containing only planar surfaces (all with H = K = 0) 
our approach will perform poorly. We depend on the surfaces having distinguishable curvature characteristics. 
FE = XuXy BF == XuXy G= XyXw 1 
L= Nu Mz Nx N= Nx 
— EN+GL-2FM — DN-—M? 
H = 2(EG-— F2) K = EG-F? (2) 
Figure 4: Formulas for derivation of curvature. 
  
  
78 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
  
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