Fig. 4.    (a) Visible-moldable surface and (b) invisible-moldable surface.

Fig. 6. Visible and invisible moldable surfaces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(see the red surface in Fig. 7(b), which is partially obscured by the blue surfaces when viewed from infinity along −d3 ).

(a) Dual moldable surface

A silhouette-detecting algorithm is employed to identify the exact boundary between the visible and invisible regions of a dual moldable surface. First, a dual moldable surface S(u, v) is described by a NURBS equation, which is generally in the following  form:

Fig. 5.    Tentative parting directions. (For interpretation of the references to color in

this figure legend, the reader is referred to the web version of this  article.)

that of [22]). These are the directions from which most surfaces of the part can be accessed. Fig. 5 shows a typical part to be analyzed

during the design process of multi-piece molds. All tentative direc-

tions for accessing the surfaces of the part are found using the prop-

erties of its geometric features; these are presented with different

colored arrows in the figure. Note that for each tentative parting

direction di in D, −di is also included in the collection. The part shown in the figure will be used as a running example for describ-

0,   otherwise.

  v − vj

  vj+l+1  − v

ing steps of the proposed algorithm.

0,  otherwise

The accessibility of all surfaces of a part is analyzed according to the tentative parting directions in collection D. Visible-moldable surfaces for each direction di in D can then be found. The process is performed in the following steps:

Step 1: Identify moldable surfaces for  direction  di based  on Eqs. (1)–(3). All moldable surfaces related to each parting direction di are grouped into a set Si.

Step 2: For each moldable surfaces of set Si, a ray test, as presented in Section 3, is performed to determine whether  this surface is visible in direction −di. Invisible surfaces, which are obscured by other surfaces in Si, are removed from  Si.

As shown in Fig. 6, both the red and blue surfaces of the part are identified as moldable surfaces along parting direction d1 . However, the red surface is completely obscured by other surfaces

where surface S(u, v) has degree k in parameter u and degree l in parameter v, Ci,j is the control point, wi,j is its corresponding weight, n1 and n2 are the number of control points in directions u and v, and Bi,k and Bj,l are the B-spline basis functions defined in u and v. The silhouette of a free-form object is typically defined as the set of points on the object’s surface where the surface normal vector is perpendicular to the vector from the viewpoint. A point on surface S(u, v) with corresponding surface normal vector N(u, v) is a silhouette point if the angle between the parting direction di and N(u, v) is 90°. This means that the following constraint must be  met:

di · N(u, v) = 0 (5)

in which N(u, v) is computed by the following rational equation:

when viewed from infinity along direction −d1 . Thus, the    red

surface is removed from visible-moldable surface set S1.

During the process of identifying visible-moldable   surfaces,

there may be two cases where the surfaces are partially visible. The first is called the ‘dual moldable surface’ where Eqs. (1), (2), and/or

Let Nm(u, v) = n1