Computer Aided Engineering Design

GEOMETRIC MODELING USING POINT CLOUDS 305

be performed using the least square approach. In case of prismatic parts, algebraic surfaces (planar,
cylindrical) are fit. Note that the edges determined during segmentation may be error prone and that
they may not be considered as the bounding edges of the surface patches, The new bounding edges
may be taken as the intersection of these surface patches, illustrated in Figure 10.12. The segmented
cloud is shown in Figure 10.12(a). The unbounded algebraic surfaces fitted to these clouds by least
squares method are shown in Figure 10.12(b). The edges obtained by intersecting these surfaces are
used to trim the surfaces and the resulting bounded surface network is shown in Figure 10.12(c).

Figure 10.12 (a) Segmented cloud (offset), (b) algebraic surfaces fitted to segmented cloud and

(c) solid model after intersection and trimming.

(a) (b) (c)

10.6.2 Segmentation and Surface Fitting for Freeform Shapes

Both edge-based and face-based methods cannot be used for representing a complex free form
surface, as encountered in sculpted objects, since this will result in many small pieces of say planar
or quadratic surfaces, which is not the desired result. However, edge-based methods can be used for
segmentation of the point cloud. An alternative approach may be that a user segments the cloud
interactively. A general methodology to obtain free form geometry from a point cloud is as follows:

(a) Segment the cloud into regions, each of which are representable by parametric surfaces such

as Bézier or B-spline patches.

(b) Parameterize the points.

(c) Determine B-spline patches using least square fit maintaining appropriate continuity between

the adjacent patches.

For surface fitting, a large crudely approximated four-sided patch is chosen by the user. Boundaries
are chosen such that the points of interest in the cloud lie within the boundary of the surface. The
points are then projected on to the surface to find corresponding points on the approximating surface.
The distance between the points in the cloud and the corresponding points on the surface is minimized
using the least-square method. If the error is too large after least square fitting, iterative parameterization
and refit are performed with better approximating surfaces. Figure 10.13 schematically illustrates an
iterative step of the approach.
B-splines are used predominantly in free form curve and surface fitting. Two approaches are usually
employed for surface fitting of point cloud. The first is to fit directly a B-spline patch to the cloud