j=0 Bi,k(u) · Bj,l(v) · wi,j · Ci,j; its partial

(3) are satisfied only by a group of points on these surfaces (see the

derivative with respect to parameter u is Nmu(u, v) = 

green surfaces in Fig. 7(a)). The second is the ‘partially obscured Bu

surface’ in which only a part of its area is obscured by other surfaces

Bj,l(v) · wi,j  so that its partial derivative with respect to parameter u

Fig. 7. (a) Dual moldable surfaces and (b) partially obscured surface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8.   Surface regions pided from dual moldable surfaces.

is the derivative

of a B-spline basis function and can be expressed by the following

equation:

Bu k − 1

  k − 1

i,k(u) = u

i+k −

Bi,k−1(u) +

ui u

i+k+1 −

ui+1

Bi+1,k−1(u). (7)

Therefore, the partial derivative of S(u, v) with respect to u is:

∂S(u, v)

∂u =

 Nm(u, v) ′

Dn(u, v)

u

Nm′ (u, v) · Dn(u, v) − Dn′ (u, v) · Nm(u, v)

u u

= Dn2(u, v)

Nm′ (u, v) · Dn(u, v) − Dn′ (u, v) · Dn(u, v) · S(u, v)

u u

= Dn2(u, v)

Fig. 9.   Division of a partially obscured surface s1  into two surface regions.

figure, surface s1 is partially obscured by surface s2. Each boundary edge of s2 is used to build an extrusion surface s3 along the current parting direction. The intersection between surface s1 and surface s3 can then be used to pide s1 into an un-obscured region and an

Nm′ (u, v) − Dn′ (u, v) · S(u, v)

u u . (8)

Dn(u, v)

Similarly, the partial derivative of S(u, v) with respect to v is:

obscured region.

Fig. 10 shows an example of the surface regions pided from a partially obscured surface of the part in Fig. 5. The red surface is partially obscured by others when viewed from infinity along

 ∂S(u, v)

∂v =

Nm′ (u, v) Dn′ (u, v) S(u, v)

u u (9)

Dn(u, v)

directions −d3 , −d5 , and −d6 . The edge-extrusion algorithm    is

thus applied, resulting in the eleven surface regions pided from

A silhouette curve can be found by substituting Eqs. (6)–(9) into Eq. (5). In fact, Eq. (5) is a polynomial equation of two variables u and v. In this scenario, there are fewer equality constraints than there are variables. Its zero-solution  set  can  be  computed based on the convex hull and subpision properties of rational spline functions. A set of discrete points approximating the zero set is generated by recursive subpision based on the Newton–Raphson method. The method to solve a general system of m polynomial equations of n variables is described by G. Elber and M.S. Kim [23]. Fig. 8 shows an example of surface regions pided from dual mold- able surfaces of the pedal part in Fig.  5.