ax · di  = 1 (2)

where ax is the axis of the cylinder. Eq. (2) asserts that a cylindri- cal surface is moldable if its axial vector is parallel to the parting direction di. As for conical surfaces, the following equation is used:

3. Background   information   on   moldable   surfaces   and ray

 ax  · di  ≥ cos   2

testing for visibility

In order to describe the proposed algorithm, the definitions of moldable surfaces, which are extended from those of surface visibility and moldability mentioned in [6], are first presented. This section focuses on the following three types of surfaces: planar surfaces (first-order surfaces), quadric surfaces, and free- form surfaces (third-order surfaces or higher). It is assumed that si is a surface of model M and ni is the normal vector of an arbitrary point on si. Let di be one of the parting directions of model M . Surface si is moldable in direction di if the following condition is met:

ni · di ≥ 0. (1)

The physical meaning of Eq. (1) is that rays from infinity that are parallel to the parting direction di cannot cast any shadow in this direction onto the surface; in other words, the surface is visible in these rays. Equality occurs when the surface is a vertical wall.

For planar and quadric surfaces, the normal vectors of points on these two types of surfaces possess exact directions, thus allowing the fulfillment of Eq. (1) to be easily confirmed (refer to Fig. 2(a) and (b)). However, for a free-form surface, a sub-pision method must be employed: the surface is pided into small regions with equal distances in parametric coordinates u and v, and the normal vector ni,j at a corresponding node (i, j) of the surface is determined. The entire set of normal vectors is then used to confirm the fulfillment of Eq. (1). Only if the equation is met for all nodes can the surface be defined as moldable. Fig. 2(c) shows an example of a moldable free-form surface along parting direction  di.

For surfaces of revolution, a substantially different method must be applied to determine the moldable surfaces: the axis

where ax  is the axis of the cone and α is the angle at the apex.

Eq. (3) shows that a conical surface is moldable if the angle between its axis and parting direction di is smaller than α/2. Moreover, this situation allows removal of the part from the mold without any dif- ficulty caused by these conical surfaces. Fig. 3 shows an example of moldable and unmoldable conical  surfaces.

Once a surface is defined as moldable in a certain direction, its visibility must be examined. Accordingly, all moldable surfaces in each parting direction di are collected and stored in a so-called ‘S- table’. Each moldable surface si is then tested in sequence by rays to determine whether it is obscured by other surfaces included in the S-table. Each ray is originated from the end point or the middle point pi of each edge ei of the test surface si along direction di. If the rays do not intersect with other surfaces in the S-table except the points pi themselves, the test surface si is identified as a visible one; otherwise, si is identified as an invisible surface. Examples of visible and invisible moldable surfaces in a direction di of a model are shown in Fig. 4.

4. Algorithm for automatic parting curve generation of multi- piece molds

4.1. Collection of tentative parting directions

In multi-piece mold design, the parting direction is the direction along which a mold piece can be separated without any obstacles, and thus the determination of feasible parting directions is a prior- ity issue that needs to be addressed. In our approach, a collection D of tentative parting directions is formed based on the geometrical properties of the part’s features by considering the following three types of directions: relative coordinate axes, axes of features of rev- olution, and normal directions of planar surfaces (which follows