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W polynomials
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
The proof is complicated and uses ideas from intersection theory described in [244]. An alternative way to understand this result is to observe that Y and ν are equivalent. It follows immediately from Definition 26.49 that Y(G;x,t)=∑πmτ(π)(1+t)e(π),
Location Awareness and Navigation in Location-Based Systems
Published in Krzysztof W. Kolodziej, Johan Hjelm, Local Positioning Systems, 2017
Krzysztof W. Kolodziej, Johan Hjelm
Topologic relations structure space. Several topologic categorizations of environments are suggested in the literature. Lynch (1960) mentions five urban design elements to describe the setting of a city. Arthur and Passini (1992) give a typology based on the structuring features of built environments, called circulations. The four-intersection theory of topologic spatial relations between sets (Egenhofer and Franzosa, 1991) defines relations in terms of the intersections of the boundaries and interiors of two sets. Evidence for cognitive hierarchical organization of space was deduced from distance and direction judgments (Hirtle and Jonides, 1985).
Computer Aided 5-Axis Machining
Published in Cornelius Leondes, Computer-Aided Design, Engineering, and Manufacturing, 2019
Andrew Warkentin, Paul Hoskins, Fathy Ismail, Sanjeev Bedi
Intersection theory was used to investigate the nature of contact between the tool and the design surface. The results of the investigation led to the development of an algorithm for determining multipoint tool positioning. This algorithm consists of two parts. In the first part, an approximate tool position and orientation is calculated based on the geometry of the two contact points, CC1 and CC2. The resulting tool configuration places the tool in tangential contact with CC1. However, because the curvature of surface under the tool is not constant, there is no guarantee that there is tangential contact at CC2. The second part of the method refines the initial solution. The tool is then rotated while maintaining tangential contact at CC1 until tangential contact occurs at CC2.
A review on tool orientation planning in multi-axis machining
Published in International Journal of Production Research, 2021
Fusheng Liang, Chengwei Kang, Fengzhou Fang
Two or more CC touch points in the machining process could acquire a larger cutting strip width than single one (Warkentin, Ismail, and Bedi 1998, 2000b; Fan, Ye, Fang, et al. 2013; Fan, Ye, Zhang, et al. 2013; Duvedi et al. 2014, 2015; He and Chen 2014; Duvedi, Bedi, and Mann 2018). Warkentin, Ismail, and Bedi (1998, 2000b) firstly introduced the concept of MPM to increase cutting strip width and the multi-contact points were found based on an intersection theory. Although the machining efficiency was improved according to the MPM method, it was low efficiency and labour intensive to search more than one contact points. Therefore, in multi-axis machining, several studies were carried out on the tool orientation optimisation aiming at efficiently finding multi-point tool positions, such as RCM and IRCM (Fan, Ye, Fang, et al. 2013; Fan, Ye, Zhang, et al. 2013). More typically, He and Chen (2014) improved the MPEC method to make the distribution of residual error symmetric with the minimum inclination angle. The improved MPEC method can acquire two CC points and a larger cutting strip width than traditional MPEC method. By introducing mechanical equilibrium theory into the study of contact status between cutter and desired surface, the tool orientation was optimised to maximise the cutting strip width (Gan et al. 2015). The initial cutter position on designed surface was located by the MPEC method. There were three other rotary freedoms to determine the ultimate cutter position in which the rotation of tool axis can be derived according to the equilibrium state of cutter. The adjustment of tool orientation could keep two contact points on machined surface for a wide cutting strip width.
Feasibility of an alternative method to estimate glenohumeral joint center from videogrammetry measurements and CT/MRI of patients
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2021
Ehsan Sarshari, Matteo Mancuso, Alexandre Terrier, Alain Farron, Philippe Mullhaupt, Dominique Pioletti
It is worth noting that Equation (3) has an intuitive geometrical interpretation. It estimates GH by intersecting four spheres centered at AC, AA, EM, and EL. Their radii can be defined from a single CT/MRI scan of the glenohumeral joint of the subject to be studied. This intersection can be defined using the intersection theory of quadric surfaces (Levin 1979).