Spherical Geometry—A Survey on Width and Thickness of Convex Bodies

“…The convex geometry in the n-dimensional spherical space S n was investigated by many mathematician, such as Robinson, Santaló, Dekster, Leichtweiss [12][13][14][15][16] , Lassak [17] and Guo [10] . In this paper, we introduce the Minkowski measure of asymmetry for spherical convex bodies of constant width in S n .…”

Section: Introductionmentioning

confidence: 99%

The Minkowski Measure of Asymmetry for Spherical Bodies of Constant Width

HOU

Jin

2022

Wuhan Univ. J. Nat. Sci.

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“…Through analysis, it can be found that here the path is generally characterized by the span of the CPG along a certain direction. Therefore, confirming the shortest cutting path can be worked out using the minimum span (i.e., width) of the CPG [24]. In recent years, the width calculation of the CPG has been widely used in collision detection and other calculations [19], but few researches and applications have explored its use in the field of machining.…”

Section: Intersection Cpg Analysismentioning

confidence: 99%

Material Removal Optimization Strategy of 3D Block Cutting Based on Geometric Computation Method

Shao

Liu

2022

Processes

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During the material removal stage in stone rough processing, milling type has been widely explored, which, however, may cause time and material consumption, as well as substantial stress for the environment. To improve the material removal rate and waste reuse rate in the rough processing stage for three-dimensional stone products with a special shape, in this paper, circular saw disc cutting is explored to cut a convex polyhedron out of a blank box, which approaches a target product. Unlike milling optimization, this problem cannot be well solved by mathematical methods, which have to be solved by geometrical methods instead. An automatic block cutting strategy is proposed intuitively by considering a series of geometrical optimization approaches for the first time. To obtain a big removal block, constructing cutting planes based on convex vertices is uniquely proposed. Specifically, the removal vertices (the maximum thickness of material removal) are searched based on the octree algorithm, and the cutting plane is constructed based on this thickness to guarantee a relatively big removal block. Moreover, to minimize the cutting time, the geometrical characteristics of the intersecting convex polygon of the cutting plane with the convex polyhedron are analyzed, accompanied by the constraints of the guillotine cutting mode. The optimization algorithm determining the cutting path is presented with a feed direction accompanied by the shortest cutting stroke, which confirms the shortest cutting time. From the big removal block and shortest cutting time, the suboptimal solution of the average material removal rate (the ratio of material removal volume to cutting time) is generated. Finally, the simulation is carried out on a blank box to approach a bounding sphere both on MATLAB and the Vericut platform. In this case study, for the removal of 85% of material with 19 cuts, the proposed cutting strategy achieves five times higher the average material removal rate than that of one higher milling capacity case.

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Spherical Geometry I

Maehara,

Martini

2024

Birkhäuser Advanced Texts Basler Lehrbücher

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Spherical Geometry—A Survey on Width and Thickness of Convex Bodies

Cited by 8 publications

References 47 publications

The Minkowski Measure of Asymmetry for Spherical Bodies of Constant Width

The Minkowski Measure of Asymmetry for Spherical Bodies of Constant Width

Material Removal Optimization Strategy of 3D Block Cutting Based on Geometric Computation Method

Spherical Geometry I

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