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TECHNICAL PAPERS

Finding Undercut-Free Parting Directions for Polygons with Curved Edges

[+] Author and Article Information
Sara McMains1

Department of Mechanical Engineering, University of California, Berkeley, CA, Berkeleymcmains@me.berkeley.edu

Xiaorui Chen

Department of Mechanical Engineering, University of California, Berkeley, CA, Berkeleyxrchen@me.berkeley.edu

1

Corresponding author.

J. Comput. Inf. Sci. Eng 6(1), 60-68 (Oct 12, 2005) (9 pages) doi:10.1115/1.2164450 History: Received December 01, 2004; Revised October 12, 2005

Abstract

We consider the problem of whether a given geometry can be molded in a two-part, rigid, reusable mold with opposite removal directions. We describe an efficient algorithm for solving the opposite direction moldability problem for a 2D “polygon” bounded by edges that may be either straight or curved. We introduce a structure, the normal graph of the polygon, that represents the range of normals of the polygon’s edges, along with their connectivity. We prove that the normal graph captures the directions of all lines corresponding to feasible parting directions. Rather than building the full normal graph, which could take time $O(nlogn)$ for a polygon bounded by $n$ possibly curved edges, we build a summary structure in $O(n)$ time and space, from which we can determine all feasible parting directions in time $O(n)$.

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Figures

Figure 8

(a) A polygon; (b) its normal graph; (c) its summary normal graph

Figure 9

Intermediate stages of building the summary graph: (a) state after normals 1 and 2 are processed; (b) state after normals 1–3 are processed; (c) state after normals 1–4 are processed; (d) state after normals 1–5 are processed; (e) state after normals 1–6 are processed. (For clarity, this figure shows the state after each normal is processed, but in practice our implementation folds updates of nonturning normals such as 5 and 6 into the processing of turning normals.)

Figure 11

The decomposition of general curved edges into simple curves. The hatched area denotes the interior of the polygon.

Figure 12

(a) Geometry where adjacent segments 1 and 2 have opposite normals. (b) Its corresponding normal graph with the angle of the arc between 1 and 2 equal to 180°. (c) We can imagine inserting an ε-radius semicircle between and tangent to the two adjacent segments with opposite normals.

Figure 13

This part becomes non-2-moldable in direction d⃗ when only its face geometry, but not its face orientation or topology, changes. (The phantom line denotes the alternate, non-2-moldable face geometry.)

Figure 1

(a) Polygon P; (b) 2 monotone chains of ∂P with respect to L

Figure 4

Partition line intersects normal graph (case 1)

Figure 5

Partition line intersects normal graph (case 2)

Figure 6

(a) A polygon with two parting directions to be tested. (b) Its normal graph showing CPLs L1′ and L2′; only L1′ is a valid partition line.

Figure 7

Finding the 2-moldable directions for a polygon by sweeping CPLs

Figure 10

(a) A polygon with a curved edge where the curve is approximated by 7 straight line segment edges, labeled 3 a, b, c, d, e, and 4. (b) The corresponding normal graph. (c) The arcs connecting the normal points for these edges can be simplified to a single oriented arc.

Figure 2

(a) A polygon and a line L; (b) its corresponding normal graph

Figure 3

The projection of line segment edges on a line L

Errata

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