Computer Aided Engineering Design

GEOMETRIC MODELING USING POINT CLOUDS 307

u 1 ,.. ., uj corresponding to each data point and the knot vectors, the matrix C can be obtained. A user
may want to specify the degree of a B-spline curve as cubic. The number of control vertices Mis
chosen depending on the complexity of the point cloud shape. The values u 1 , u 2 , ..., uj may be
determined using some parameterization technique. Also, the knot vector may be determined consistent
with the above parameterization to minimize the computation for convergence. The three kinds of
parameterization mostly used are uniform, centripetal and chord length methods as discussed in
Section 5.10. A generalization of all the above parameterization models is

uuui i i i ij e

e

s

j

s s e

1–1

–1

=1

–1

+1

= 0; = +

| – |

| – |

with 2 , 0 1

PP

Σ PP

≤≤ ≤ ≤ (10.6)

wherej is the total number of data points specified on the curve. For e = 0, 1, and 0.5, the above
equation yields uniform, chord length and centripetal parameterization, respectively.
Least square minimization may commence with the parameter values in Eq. (10.6). Using these,
a B-spline curve may be determined using Eqs. (10.2) and (10.5). Thereafter, the error between the
fitted B-spline curve and the corresponding points in the cloud is computed. If the error is large,
parameter values ui are optimized by iteratively improving them using a first order Taylor correction
for the error expression. An example B-spline curve fitting is illustrated in Figure 10.14(a). Once four
such boundary curves for a segmented cloud are determined, a linear or cubic (Hermite) blended
Coon’s patch (section 7.2.1) can be developed as illustrated in Figure 10.14(b). Note that the cross
boundary tangents and twist vectors can be determined in a manner such that the Coon’s patch
represents the best fit for the segmented point cloud in the least square sense.

After surface fitting is performed for all points in the cloud, the final step is to fine tune the patches
to obtain the required continuity across the patch boundaries and to address other engineering constraints
like symmetry. The procedure for enforcing tangent plane continuity between adjacent patches is
discussed in Section 7.3. Other important geometric properties such as symmetry, parallelism,
orthogonality and concentricity, which are essential attributes, have to be enforced to convert the B-
rep model to a valid solid model for further downstream applications. This stage requires user
interaction, or, artificial intelligence techniques may also help.

Figure 10.14 (a) Boundary B-spline curves with the segmented cloud (b) Coon’s

patch fit using the boundary curves

(a) (b)