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TECHNICAL PAPERS

# Mold Accessibility via Gauss Map Analysis

[+] Author and Article Information
Gershon Elber

Department of Computer Science, Technion, Haifa 32000, Israel

Xianming Chen1

School of Computing, University of Utah, Salt Lake City, Utah 84112xchen@cs.utah.edu

Elaine Cohen

School of Computing, University of Utah, Salt Lake City, Utah 84112

1

Readers may contact the authors through Xianming Chen, Department of Computer Science, University of Utah, Salt Lake City, Utah 84112. Telephone: (801)585-6161.

J. Comput. Inf. Sci. Eng 5(2), 79-85 (Dec 24, 2004) (7 pages) doi:10.1115/1.1875572 History: Received September 03, 2004; Revised December 24, 2004

## Abstract

In manufacturing processes like injection molding or die casting, a two-piece mold is required to be separable, that is, be able to have both pieces of the mold removed in opposite directions while interfering neither with the mold nor with each other. The fundamental problem is to find a viewing (i.e., separating) direction, from which a valid partition line (i.e., the contact curves of the two mold pieces) exists. While previous research work on this problem exists for polyhedral models, verifying and finding such a partition line for general freeform shapes, represented by NURBS surfaces, is still an open question. This paper shows that such a valid partition exists for a compact surface of genus $g$, if and only if there is a viewing direction from which the silhouette consists of exactly $g+1$ nonsingular disjoint loops. Hence, the two-piece mold separability problem is essentially reduced to the topological analysis of silhouettes. In addition, we deal with removing almost vertical surface regions from the mold so that the form can more easily be extracted from the mold. It follows that the aspect graph, which gives all topologically distinct silhouettes, allows one to determine the existence of a valid partition as well as to find such a partition when it exists. In this paper, we present an aspect graph computation technique for compact free-form objects represented as NURBS surfaces. All the vision event curves (parabolic curves, flecnodal curves, and bitangency curves) relevant to mold separability are computed by symbolic techniques based on the NURBS representation, combined with numerical processing. An image dilation technique is then used for robust aspect graph cell decomposition on the sphere of viewing directions. Thus, an exact solution to the two-piece mold separability problem is given for such models.

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## Figures

Figure 1

Local vision events under orthographic projection. Row 1 on the left shows a lip event at a parabolic point of the elliptic type. The first and second images are before and after the vision event. The third image is a closeup silhouette corresponding to the second image. Row 2 on the left and row 1 on the right show a beak-to-beak event at a parabolic point of hyperbolic type. Rows 2 and 3 on the right show a swallowtail event at a flecnodal point. From left to right, each of these four rows shows images before, at, and after the vision event, respectively.

Figure 2

Multi-local vision events under orthographic projection. From left to right, Each row shows models before, at, and after the vision event, respectively, followed with their corresponding silhouettes. Row 1 shows a bitangent cross event. Row 2 shows a cusp-cross event. Row 3 shows a triple-point event. The arrows of the last row show where the triple-point event occurs

Figure 3

Mold accessibility example one . Shown on the left is a NURBS model, and on the middle the intersecting curves of the various generating cones with the viewing sphere, as well as the representative viewing directions of the partitioned regions. The curve corresponding to a parabolic developable is in cyan, to a bitangent developable in magenta, and to a flecnodal scroll in yellow. The representative viewing directions are represented as pairs of antipodal small spheres embedded on the transparent viewing sphere. Silhouettes viewed from all these directions are shown on the right. There are four representative nonsingular silhouettes, shown inside boxes with green color, and correspondingly four representative viewing directions (shown in green), from each of which there is a valid 2PM. Note, there are actually only two topologically distinct valid viewing directions. We have two extra since we project the sphere surface onto three faces of the bounding cube, but we apply dilation to them separately as if the three faces are disconnected.

Figure 4

Mold accessibility example two . Shown on the left is a NURBS model, and the middle the intersecting curves of the various generating cones with the viewing sphere, as well as the representative viewing directions of the partitioned regions. The curve corresponding to a parabolic developable is cyan, to a bitangent developable is magenta, and to a flecnodal scroll is yellow. The representative viewing directions are represented as small spheres on the opaque viewing sphere. Silhouettes viewed from all these directions are shown on the right. There is one representative nonsingular silhouette, shown inside a box with green color, and correspondingly one representative viewing direction, colored green, from which there is a valid 2PM

Figure 5

90±10 isoclines (thick black lines) from V for a wine glass surface (a). The silhouettes of the surface from V are shown in a gray color. In (b), the regions with normals that are almost vertical are trimmed away. (c) shows the ruled extension that covers up these gaps at the proper valid slopes. (d) and (e) show the final clipped and ruled extended models of the wine glass from (a) and of the handle of the Utah teapot. Note the sharp, normal discontinuity, along the partitioning line.

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