Computer Aided Engineering Design

GEOMETRIC MODELING USING POINT CLOUDS 301

a parameter value on one contour corresponds to the same parameter value on all the other contours,
as illustrated in Figure 10.7, in which a line segment connects corresponding points on two adjacent
contours. However, as shown in Figure 10.8(a), this may lead to twisting or shearing of the tiling
surface. A better approach, since the contours are closed, may be to use the angular parameter for all
contours, using cylindrical coordinates by placing the origin at the geometric center of one of the
contours, say Ci. Parameter correspondence between two adjacent contours Ci and Ci+1 may then be
established by taking equal angular spacing. This angular correspondence between points on adjacent
contours would avoid twisting of the spline surface fitted through them. The approach is illustrated
in Figure 10.8(b). Surface reconstruction from planar contours gets complicated if the object branches
wherein more than one contour may be present on any slice. We must then connect either a single
contour on one slice with several contours on an adjacent slice (one-to-many tiling or skinning), or
many contours on one slice to many contours on an adjacent slice (many-to-many tiling or skinning).

Figure 10.7 Linear parameter correspondence in tiling surface fitting

Curve 1

Tilling surface

Curve 2

Corresponding points

Figure 10.6 Edge tracking with edge points: (a) edge tracking illustrated at a point of branching and
(b) illustration of closed B-spline contour fitting through the tracked edge points

Linking direction

Leading edge

Disjointed edge

Following edge

Endpoint of a boundary curve

(a)

r

q 2

r

q 1

r

p

θ 1

θ 2

(b)

10.6 Surface Models

Surface models (Figure 10.2 d) are B-rep models that represent an object as a set of closed connected
higher order surface patches. In majority of CAD/CAM applications, such surface models are required.
They offer better continuity across patch boundaries compared to meshed or faceted models. Further,
contour models become inapplicable when the point cloud is not arranged in the sliced form. Mathematical
forms of surface patches discussed in Chapters 6 and 7 can then be applied here. The degree of