Transactions on Information and Communications Technologies vol 12, 1995 WIT Press, ISSN

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1 Determining algorithms of surface visibility on a solid model and its application T.S. Huang Division of Production Engineering, Machine Design and Automation, Faculty of Engineering, Department of Mechanical Engineering, Katholieke Universiteit Leuven, Celestijnenlaan 300 B, B-3001 Heverlee, Belgium Abstract This paper discusses an algorithm, called three-dimensional convex hull, which is devised to determine the smallest convex polyhedron containing a concave polyhedron given by a solid model beforehand, as a solid model having the same data structure as that of the concave polyhedron, and describes its application to detect potential undercut problem in injection moulds design. Because of some disadvantages of the algorithm on detecting potential undercut, the author proposes a different method to generate the minimal contour solid and extensional solids from a computer solid model. Real undercuts are obtained through a Boolean difference operation on the minimal contour solid by subtracting the extensional solids and the original solid model. This method completely based on solid modelling techniques. 1. Introduction An undercut in a plastic injection mould is any restriction that prevents the moulded plastic product from being demonided from the mould. In other words, undercuts prevent either the core mould half from being extracted after the component has been formed, or the component from being ejected out of the cavity. Designers of plastic injection moulds are frequently confronted with demoulding problems resulting from undercuts and spend lots of time in detecting all possible undercuts from the engineering drawing of the components to be moulded and designing the appropriate lateral sliding mechanisms intended to release the undercut before product ejection. This has led to a strong desire for realizing an automatic undercut detection during mould design. The feature of the algorithm, called three-dimensional convex hull, is the use of the Euler operators to generate the convex hull. This hull must be represented by a solid model having the same data structure as the solid model of the given concave moulding. Then the convex hull can be used on the extraction of

2 22 Visualization and Intelligent Design in Engineering potential undercuts in mould design. One of the difficulties of the algorithm consists of the development of Euler operators for generating a convex hull. It is not so easy, however, to keep the topological information consistent within a CAD data base during the execution of set operations. The development of valid CAD models based on faces, edges, and vertices are carried out in a rigorous way, and the Euler operators must ensure the consistency of the model. In order to create real undercuts, a different manner is proposed to overcome the drawbacks of the algorithm application. This method first determines the outline curves of all visible surfaces of a moulding viewing from the withdrawal direction. It then extrudes the curves along the withdrawal direction to create an extruded solid (hereafter called "extensional solids"). The undercut could be generated by subtracting the extensional solids and the original solid moulding from the minimal contour solid which contains the given moulding. 2. WHAT IS THE THREE-DIMENSIONAL CONVEX HULL? A convex hull is the smallest convex solid containing a concave polyhedron using a solid model having the same data structure as the given part. The convex hull will be suitable for solving interference problems that will be used to generate potential undercuts. In order to make this report self-contained, an introduction to the three-dimensional convex hull algorithm is included [6,8]. A concave polyhedron is represented by using a solid model, called boundary representation (B-rep), which has the data structure shown in Fig. la [3. The boxes and lines represent data and pointers in the database, respectively. The representation of edges is done using the Winged-Edge data structure having pointers to loops (shown in Fig. Ib). For a good exposition on the winged-edge data structure, the reader is referred to reference [3]. The following sections describe the algorithm for defining a convex hull, and interpreting its application by using the moulding shown in Fig. 2 as an example. Geometry Emphasizing surface equation Emphasizing curve equation Point coordinates (a). Data structure for boundary modelling. (h). The w FIGURE 1. Data structure of an object FIGURE 2. Concave polyhedron 2-1. Define "the set of neutral vertices," and "the set of inside vertices" The "set of neutral vertices" is defined as vertices belong to edges, for which the dihedral angle is big than 180 degrees as viewed from the outside of the shape. The number of vertices shown in Fig. 2, (1,2, 3, 4, 5, 6, 7, X, 9, 10, 11, 12), are classified as "the set of neutral vertices." The other vertices of the polyhedron model, which are manifestly contained inside or in the faces of a

3 Visualization and Intelligent Design in Engineering 23 convex hull, are defined as "the set of inside vertices". For creating a three dimensional convex hull, we only need the set of neutral vertices Create afirstconvex polyhedron In order to use Euler operators to define a tetrahedron to be the first convex polyhedron with the same data structure as the concave polyhedron, four points which are not on a plane should be taken from the set of neutral vertices. This procedure is shown in Fig. 3, in which a tetrahedron can be created by using some of the Euler operators such as mvfs, mev, and mef. In Fig. 4, the first tetrahedron (convex polyhedron) of the polyhedral model is represented by a solid model by using points 1,3,7,11 from the set of neutral vertices. l.mvfs(f,,v,) 2. mev (V,,V,e,) 3. mev (V;,V,,e,) 4. mef (V,,V,,e3,fj) mvfs : make Vertex, Face, Solid 5. mev (V,,V,,eJ mev : make Edge, Vertex 6. mef (V,,V,,e,A j. * mff mr,r ;: mane makefrige,face r/u^c, rate 7. mef (V),V,,e&,fj V/ FIGURE 3. Sequence for generating tetrahedron FUJI IRK 4. The first tetrahedron (Convex polyhedron) 2-3. Remove Unfunctional Vertices With the use of the solid model the location of both the inside and neutral vertices is determined relative to the tetrahedron. All vertices lying inside or on the boundary of the tetrahedron are deleted, since they will not contribute to the generation of the new convex hull polyhedron Create a New Convex Polyhedron In order to describe the process of creating a new convex polyhedron, we use an another example shown in Fig. 5 to explan how to create it. First takes a projection of the convex polyhedron using v^ as the pointing reference. VQ, which is one of the set neutral vertices, should be outside the convex polyhedron defined so far. We shall consider the visibility of faces and edges of a convex polyhedron. Let's assume that a convex polyhedron and a specified outside viewpoint are defined in 3-D space and that surface normal are defined as outward normal vectors for all faces of the convex polyhedron. Then the visibility of faces and edges is defined as follows. (Fig. 6) W ty^&7<v "44^ rn% S f, FIGURE 5. Projection drawing,v : 'Hie outward normal vector on the pail's \ui1ace. P '. Ilie view vector trom the suit ace to the viewpom FIGURE 6. Test of face visibility

4 24 Visualization and Intelligent Design in Engineering 1. The face is visible, if the angle a is less than or equal to n/2 (N P > 0). 2. The face is invisible, if the angle a is greater n/2 (N P< 0). 3. The edge is contour line, if one of the adjacent faces is visible and the other invisible. 4. The edge is hidden, if the two faces sharing it are both invisible. 5. The edge is not a contour line but should be drawn because it is visible, when both adjacent faces are visible. So, the shape data for the convex polyhedron is classified into sets of : Fv=(f2, fv fa); visible faces, Fj={fi, fj, fj; invisible faces, Ei={e,, e?, 65,67, 6n, e^}; contour lines, 2=163, 64, e%); hidden lines, 3= {66, eg, em}; other lines (visible but not contour lines) Now a projection drawing of the convex polyhedron is constructed. Second, the shape data for the convex polyhedron projection is classified as following: Fc={f2, fj, W: FC is the set of faces having edges forming contour lines of the projection drawing. Fo={none}: FO is the set of faces that do not belong to F, among the faces drawn in the projection drawing. EC={CI, , e,,, en): E,. is the set of edges forming contour lines. Ev={e6, eg. 610): EV is the set of edges that connect to the vertices of contour lines, but do not belong to E,. Eo={none}; EO is the set of edges which do not belong to E, and E\ among the edges drawn in the projection drawing. YC={VI, Vi, Vi, Vs, Vv, v%}: Vc is the set of vertices included in contour lines. Vo={vJ: Vo is the set of vertices drawn in the projection drawing which do not belong to V^. At this stage, all of the elements of sets F<,, E^, E^, and V^ are deleted from the convex polyhedron by using Euler Operators (kef, kev) [4,5], because they are not necessary to the new convex hull polyhedron. Then a new edge line is first added to the convex polyhedron projection drawing by connecting the viewpoint VQ with any vertex belonging to set V^ using Euler Operators (mev). At the same time, the new face which is defined by the viewpoint v^ and each outline segment is added to the convex polyhedron by using Euler Operators (mef). Now a new convex hull polyhedron is constructed from the convex polyhedron and the viewpoint VQ. Table 1 summarizes the increase and decrease in the number of faces, edges, and vertices produced by the Euler operations shown in Figure 7. It may be seen that those operations respect Eulcr's law. Eulcr's Law states that in any simple polyhedron, the numbers of faces(f), edges(e), and vcrtices(v) must satisfy the equation: v-e+f=2 (1) This rule makes the following assumptions about the polyhedron: (1). All faces are simply-connected, consisting of a single ring of edges and have no holes in them.

5 Visualization and Intelligent Design in Engineering 25 (2). The solid body must be simply-connected, with no holes through it. Euler's law given by Eq. (1) are based on excludes some classes of meaningful shapes, such as shapes that contain holes. To allow for shapes, equation (1) must be modified as the following general formula: v-e+f=2(b-h)+l (2) where b, h, and 1 are the number of bodies, through holes, and faces' inner loop respectively [1,2]. For instance, the counts of the various variables of Eq. (2) for the polyhedral model shown in Fig. 2 have the following number of elements: 54 vertices, 83 edges, 29 faces, 1 body, 5 through holes, and 10 inner loo Name Existing polyhedron kef*2 kev inev mef*6 New polyhedron Operation kill edge and face kill edge and vertex make edge and vertex make edge and face (Vertices)-(edges)+(faces)= Table 1. Maintaining internal consistency within the data structure. 0 8 kev 12 _ _ )S. 9 2 mef?0 - ~l Vj > VQ FIGURE 7. Create a new convex polyhedron with v^ as a vertex Acquire Final Convex Hull Polyhedron The step 3 and step 4 are repeated until the set of neutral vertices is empty. As an ultimate result, the convex hull polyhedron is obtained as shown in Fig. X. FIGURE 8. Processing steps leading to the convex hull polyhedron.

6 26 Visualization and Intelligent Design in Engineering 3. APPLICATION OF THE THREE-DIMENSIONAL CONVEX HULL Using the moulding in Fig. 2 as an example, figure 9 shows a procedure for extracting potential undercuts. According to the withdrawal direction D, D as shown in Fig. 10a., potential undercuts No.l, No.3, No.5 and No. 6 become real undercuts: real undercuts interfere with the given part when they move out along withdrawal direction (Fig. lob). (a). Three-dimensional convex hull (h). Mould product (c). Poienlial undercuts FIGURE 9. A procedure for extracting potential-undercuts. Undercut No undercut Undercut Potential undercuts No undercut ggggl Interference between the solid model and potential undercut when the potential undercut moves out along U direction. n Interference between the solid model and potential undercut when the potential undercut moves out along D direction. Note: Real undercut only when interference for both sides. (a). Withdrawal direction (h). Real undercuts FIGURE 10. Real undercuts. Potential undercuts can be automatically extracted during mould design, by representing a convex hull polyhedron by means of a solid model. Besides, real undercuts can also be generated by checking interference between potential undercuts and the given model. 4. CREATE A CONTOUR SOLID TO OBTAIN REAL UNDERCUTS According to the proposed algorithm described above. Fig. 9a represents a convex hull polyhedron for a moulding and Fig. 9c demonstrates potential undercuts extracted by a subtraction set operation on the convex hull polyhedron and the given moulding. This proposed algorithm has a disadvantage when it determines real undercuts by designating a moulding product withdrawal direction. For example, the potential undercuts shown in Fig. 9c are actually a solid part and users must perform a split operation for cutting it on purpose [7]. An attempt was made in this example to split the potential undercut to six separately potential undercuts as shown in Fig. I Ob.

7 Visualization and Intelligent Design in Engineering 27 So, the user has to develop a split algorithm by using Euler operators to split the potential undercut. The main disabvantage of the algorithm is that it just can be used on a planer solid model. The algorithm of the creating contour solid for extracting real undercuts is shown in Fig. 11. In Fig. 11, the topological and geometrical structure of the moulding and extensional solids are always represented exactly, but the threedimension convex hull is approximated by using polyhedrons. The processing steps of this distinct method are explained in detail below, and use a mouse model as an example. input withdrawal <^sible surfaces^>* > Clip operation direction Top plane Bottom plane FIGURE 11. Hierarchy of extracting undercuts FKUIRF 12. Create a minimal conlo containing the given nu 4-1. Create the Minimal Contour Solid Containing the Moulding For creating a minimal contour solid, first access the edge information of the product from the database to obtain maxi-min z coordinate. From the mouse model created by solid primitives, we can find the A, B, C, and D surfaces are visible as look the model from top view. And then extruded these edges, which belong to visible surfaces, from minimum z position to maximum z position to create a contour solid. At the same time, two planes are defined, the top plane and the bottom plane, as shown in Fig Create Extensional Solids For creating extensional solids select contour lines, which are enclosing a visible surface. We can create an extensional solid by extruding the contour lines along the withdrawal direction until the top or bottom planes defined earlier. This step is repeated until all contour lines of visible surfaces are extruded to become extensional solids. In this example, this operation is performed five times. The A sphere surface will be extruded until its the lowest vertex to the top plane. The edges of B, C, and D face will be extruded along the D direction until the top plane and the inside sphere surface will be extruded along the D direction until

8 28 Visualization and Intelligent Design in Engineering the bottom plane. As a final result, the extensional solids are obtained as shown in Fig. 13. D AD A 5 FIGURE 13. Extensional solids Generate Real Undercuts Real undercuts can be determined by performing a difference set operation to the contour solid, extensional solids, and the given moulding. This procedure that extracts undercuts for specific shapes is shown in Figure 14. Figure 15 gives another example, in which A surface be modified to a completely visible face for creating extentional solid. Contour solid Extentional Solid Extentiona! Solids Solid Model Real Undercuts FIGURE 14. Real undercuts. Contour solid Ixtensional solid A Extensional solid B Rxtensional solid C FIGURE 15. An another example

9 Visualization and Intelligent Design in Engineering Conclusion This report discusses an algorithm for generating the smallest convex hull polyhedron represented as a solid model, having the same data structure as the given moulding. It also describes its applications to extract potential undercuts when designing moulds. The algorithm uses Euler Operations. The proposed algorithm for determining the convex hull polyhedron is composed of the following steps. The first step defines the attributes of the vertices of the given solid moulding. The second step constructs a tetrahedron by selecting four neutral vertices. The third step deletes all of the vertices that are inside or on the boundary of the tetrahedron. The fourth step creates a new polyhedron adding an unused neutral vertex, and using it as "viewpoint" from which one looks to the existing tetrahedron. The third and fourth steps are repeated until a convex polyhedron containing all the vertices defined in the first step is obtained. A new method is proposed for directly extracting real undercuts from a given solid moulding. The method is based on generating a minimal contour solid and extensional solids which are obtained by extruding boundary curves of real visible surfaces of a moulding, viewed from the withdrawal direction. Undercuts can be determined by subtracting the extensional solids and the given moulding from the minimal surrounding box. The advantage of this algorithm is it can directly use on a solid model combined with solid primitives. 6. REFERENCE [1] A. Baer, C. Eastman, M. Henrion, "Geometric modelling: a survey", Institute of Physical Planning, Carnegie-Mellon University, Pittsburgh, USA, Feb [2] H. Chiyokura, "A Method of Representing the Solid Design Process", IEEE CG&A, April 1985, pp [3] I. Zeid, CAD/CAM Theory and Practice, McGraw-Hill, [4] M. Mantyla, R. Sulonen, "GWB: A Solid Modeler with Euler Operators", IEEE CG&A, September 1982, pp [51 M. Martti, "AN INVERSION ALGORITHM FOR GEOMETRIC MODELS", Computer Graphics, Vol. 16, No. 3, July [6] S. Furukawa, "An Efficient Algorithm for Constructing Convex Hulls of 3- D Points", JSPE, Japan, Vol. 50, No. 11, [7] T. Mochizuki, N. Yuhara, "Methods of Extracting Potential Undercut and Determining Optimum Withdrawal Direction for Mold Designing". Int. J. Japan Soc. Prec. Eng., Vol. 26, No. 1 (Mar. 1992). [8] T. Mochizuki, N. Yuhara, "Determining Algorithm of Three-Dimensional Convex Hull Used for Set Operations of Solids and Its Application to Interference Problem", ISCIE, Japan, Vol. 3, No. 11, 1990.