(b) Partially obscured surface

This type of surface will be pided into regions that are completely visible or invisible to the current parting direction through an edge-extrusion algorithm, as illustrated in Fig. 9. In the

the red surface as shown.

4.3. Determination of additional parting directions

Assuming that there are some ‘inaccessible’ surfaces invisible from the directions in D, an analysis is performed to identify additional parting directions (APDs) along which such surfaces are visible. APDs found are inserted into the collection D.

The determination of APDs is implemented via the use of G-map and V-map introduced in [24]. A G-map is a map of surface normals onto a unit sphere, where each point on the map represents the in- tersection of the transferred surface normal vector with the surface of the unit sphere. On the other hand, a V-map of a surface is a set of points in the unit sphere whereby every point in the V-map de- viates from the corresponding point in the G-map by an angle less

Fig. 10. Surface regions pided from partially obscured surfaces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 11.    (a) Inaccessible surfaces of a pocket, (b) V-map of the pocket, and (c) APD found from the V-map.

than 90°. To identify the G-map and V-map of each free-form sur- face of the CAD part, we employ Eqs. (6)–(9) in computing the nor- mal vectors ni,j  at the corresponding nodes (i, j) of each  surface.

Fig. 11 illustrates the determination of an APD using a V-map. The pocket feature of the example part in Fig. 11(a) has five inaccessible surfaces, each of whose V-maps is a  hemisphere (Fig. 11(b)). The intersection of these hemispherical V-maps forms the V-map of the pocket. The vector passing through the central point of the V-map and its origin located at the unit sphere is the APD of the pocket (Fig. 11(c)).

In the proposed algorithm, each of APDs is only identified for inaccessible surfaces which are in the same undercut feature. Of course, if the determination process of an APD is applied to inaccessible surfaces belonging to different undercut features, it is possible that no feasible parting direction can be found. Therefore, all inaccessible surfaces must be classified into different groups before the V -map and G-map methods can be used. Each group consists of adjacent inaccessible surfaces connected together to form an undercut feature (Fig. 12).

To classify inaccessible surfaces into different groups, all inac- cessible surfaces are collected and numbered from is1 to isn (where: n is the total number of inaccessible surfaces). Each of inacces- sible surface isi will be examined to find its adjacent surfaces. In the process, if two surfaces have at least one common edge, they are considered as adjacent ones. Once inaccessible surface isj      is

Fig. 12.    Example of groups of adjacent inaccessible surfaces.

determined as the adjacent surface of isi, isj will be inserted into the same group of isi, or in other words, they are in the same un- dercut feature.

Importantly, an APD may be a possible parting direction for the visible-moldable surface set Si identified previously. As shown in Fig. 13, groups s1, s2, and s3 are identified as visible-moldable surfaces associated with parting direction d1 while all surfaces of

the pocket are visible-moldable with the APD d7 determined by the V-maps. However, with d7 , the surface group (s1, s2, s3) and the surfaces of the pocket are all visible-moldable. In such a case, the surfaces are re-classified into the same set associated with APD d7 . Therefore, once an APD of an undercut feature is  identified,