A Convex Decomposition Using Convex Hulls and Local Cause of Its Non-Convergence

Yong Se Kim; D. J. Wilde

doi:10.1115/1.2926574

A Convex Decomposition Using Convex Hulls and Local Cause of Its Non-Convergence

Journal of Mechanical Design ◽

10.1115/1.2926574 ◽

1992 ◽

Vol 114 (3) ◽

pp. 459-467 ◽

Cited By ~ 18

Author(s):

Yong Se Kim ◽

D. J. Wilde

Keyword(s):

Decomposition Method ◽

Convex Hulls ◽

Boolean Combination ◽

Convex Object ◽

Convex Decomposition ◽

Polyhedral Objects

To exploit convexity, a non-convex object can be represented by a boolean combination of convex components. A convex decomposition method of polyhedral objects uses convex hulls and set difference operations. This decomposition, however, may not converge. In this article, we formalize this decomposition method and find local cause of non-convergence.

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Local Cause of Non-Convergence in a Convex Decomposition Using Convex Hulls

15th Design Automation Conference: Volume 1 — Computer-Aided and Computational Design ◽

10.1115/detc1989-0035 ◽

1989 ◽

Author(s):

Y. S.-Kim ◽

D. J. Wilde

Keyword(s):

Decomposition Method ◽

Convex Hulls ◽

Convex Decomposition ◽

Polyhedral Objects

Abstract A convex decomposition method of polyhedral objects uses convex hulls and set difference operations. This decomposition may not converge. In this article, we formalize this decomposition method and find local cause of non-convergence.

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A Convergent Convex Decomposition of Polyhedral Objects

Journal of Mechanical Design ◽

10.1115/1.2926575 ◽

1992 ◽

Vol 114 (3) ◽

pp. 468-476 ◽

Cited By ~ 32

Author(s):

Yong Se Kim ◽

D. J. Wilde

Keyword(s):

Decomposition Method ◽

Feature Recognition ◽

Current Method ◽

Convex Decomposition ◽

Object Based ◽

Geometric Objects ◽

Polyhedral Objects ◽

Volumetric Representation ◽

Boundary Information ◽

The Given

A convex decomposition method, called Alternating Sum of Volumes (ASV), uses convex hulls and set difference operations. ASV decomposition, however, may not converge, which severely limits the domain of geometric objects that the current method can handle. We investigate the cause of non-convergence and present a remedy; we propose a new convex decomposition called Alternating Sum of Volumes with Partitioning (ASVP) and prove its convergence. ASVP decomposition is a hierarchical volumetric representation which is obtained from the boundary information of the given object based on convexity. As an application, from feature recognition by ASVP decomposition if briefly discussed.

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Form Feature Recognition by Convex Decomposition

ASME 1991 Computers in Engineering Conference: Volume 1 — Artificial Intelligence; Expert Systems; CAD/CAM/CAE; Computational Fluid/Thermal Engineering ◽

10.1115/cie1991-0009 ◽

1991 ◽

Author(s):

Yong Se Kim

Keyword(s):

Decomposition Method ◽

Feature Recognition ◽

Convex Hulls ◽

Convex Decomposition ◽

Geometric Objects ◽

Volumetric Representation ◽

Feature Information ◽

Form Features ◽

Form Feature

Abstract A convex decomposition method, called Alternating Sum of Volumes (ASV), uses convex hulls and set difference operations. ASV decomposition may not converge, which severely limits the domain of geometric objects that can be handled. By combining ASV decomposition and remedial partitioning for the non-convergence, we have proposed a convergent convex decomposition called Alternating Sum of Volumes with Partitioning (ASVP). In this article, we describe how ASVP decomposition is used for recognition of form features. ASVP decomposition can be viewed as a hierarchical volumetric representation of form features. Adjacency and interaction between form features are inherently represented in the decomposition in a hierarchical way. Several methods to enhance the feature information obtained by ASVP decomposition are also discussed.

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Conversions in Form Feature Recognition Using Convex Decomposition

ASME 1992 International Computers in Engineering Conference: Volume 1 — Artificial Intelligence; Expert Systems; CAD/CAM/CAE; Computers in Fluid Mechanics/Thermal Systems ◽

10.1115/cie1992-0029 ◽

1992 ◽

Author(s):

Yong Se Kim ◽

Kenneth D. Roe

Keyword(s):

Decomposition Method ◽

Feature Recognition ◽

Inductive Learning ◽

The Other ◽

New Combination ◽

Convex Decomposition ◽

Feature Information ◽

Form Features ◽

Primal And Dual ◽

Form Feature

Abstract A convergent convex decomposition method called Alternating Sum of Volumes with Partitioning (ASVP) has been used to recognize volumetric form features intrinsic to the product shape. The recognition process is done by converting the ASVP decomposition into a form feature decomposition by successively applying combination operations on ASVP components. In this paper, we describe a method to generate new combination operations through inductive learning from conversion processes of primal and dual ASVP decompositions when one decomposition produces more desirable form feature information than the other.

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Feature Recognition by Volume Decomposition Using Half-Space Partitioning

20th Design Automation Conference: Volume 1 — Dynamic Mechanical Systems; Geometric Modeling and Features; Concurrent Engineering ◽

10.1115/detc1994-0100 ◽

1994 ◽

Author(s):

Yan Shen ◽

Jami J. Shah

Keyword(s):

Half Space ◽

Decomposition Method ◽

Feature Recognition ◽

Boundary Representation ◽

Solid Model ◽

Space Partitioning ◽

Machining Features ◽

Convex Decomposition ◽

Volume Decomposition ◽

Convex Cell

Abstract A volume decomposition method called minimum convex decomposition by half space partitioning has been developed to recognize machining features from the boundary representation of the solid model. First, the total volume to be removed by machining is obtained by subtracting the part from the stock. This volume is decomposed into minimum convex cells by half space partitioning at every concave edge. A method called maximum convex cell composition is developed to generate all alternative volume decompositions. The composing sub volumes are classified based on degree of freedom analysis. This paper focuses on the first part of our system, i.e., the volume decomposition. The other part of the work will be submitted for publication at a leter date.

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On a Problem of Danzer

Combinatorics Probability Computing ◽

10.1017/s0963548318000445 ◽

2018 ◽

Vol 28 (3) ◽

pp. 473-482

Author(s):

NABIL H. MUSTAFA ◽

SAURABH RAY

Keyword(s):

Convex Hull ◽

Recent Work ◽

Polynomial Time ◽

Convex Sets ◽

General Theorem ◽

Convex Hulls ◽

Exponential Bound ◽

Convex Object ◽

Polynomial Bounds ◽

Bounded Convex

Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d).Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

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