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The Convex Hull of Two Closed Implicit Surfaces
Published in Wasim Ahmed Khan, Ghulam Abbas, Khalid Rahman, Ghulam Hussain, Cedric Aimal Edwin, Functional Reverse Engineering of Machine Tools, 2019
Algorithms for computing convex hulls of various objects have a broad range of applications in pattern recognition, collision detection [1,2], image processing, statistics, and geographical information system (GIS). Computing the convex hull is to construct a non-ambiguous and efficient representation of the required convex shape. In computational geometry [3], numerous algorithms are proposed for computing the convex hull of a finite set of points, polygons, and polyhedra with various computational complexities [3–5]. For spline surface, we first take the convex hull of its control points as an approximation to its actual convex hull and subdivide the surface into smaller pieces. Then the union created by the control points of these smaller pieces will be a better approximation to its actual convex hull. Repeat the process until the approximating convex hull converges to the exact convex hull within a certain reasonable bound. Seong et al. [6] once presented an algorithm for computing the convex hull of a rational surface without using the subdivisions, usually a troublesome process, in which the problem of computing convex hull is reduced to one of finding the zero-sets of polynomial equations. Considering a special case, Geismann et al. [7] developed an algorithm by reducing the convex hull problem of a set of ellipsoids to that of computing a cell in an arrangement of quadrics.
Hybrid Models and Experimental Design
Published in Jarka Glassey, Moritz von Stosch, Hybrid Modeling in Process Industries, 2018
The convex hull of a set of data points is the smallest convex set that contains all of the points. It can be computed from the input values of the training data using, for example, the Qhull algorithm (Barber et al. 1996). Upon the characterization of the convex hull, the position of new data with respect to the convex hull can be calculated using a set of linear equations (Kahrs and Marquardt 2007). Thus, this criterion can be implemented as a set of linear inequality constraints for process optimization, which is computationally efficient and reduces the search space to the domain to within the convex hull.*
Data-driven analysis
Published in Georgios A. Drosopoulos, Georgios E. Stavroulakis, Nonlinear Mechanics for Composite Heterogeneous Structures, 2022
Georgios A. Drosopoulos, Georgios E. Stavroulakis
This problem can be solved by developing the Voronoi diagram or Dirichlet tessellation [108], for the data set points. When the points of a data set are considered, a Voronoi diagram is built by partitioning a plane into convex polygons, such that each polygon contains exactly one generating point from the data set. Then, every other point located in a polygon, will be closer to the corresponding generating point of this polygon, than to any other point from the data set [230]. A Voronoi diagram created in Matlab for points of a data set is shown in figure 8.6[136].
A new method to optimise finite dimensions thermodynamic models: application to an irreversible Stirling engine
Published in International Journal of Ambient Energy, 2018
F. Lanzetta, A. Vaudrey, P. Baucour
Obtaining the long-awaited ‘power vs. efficiency’ curve, like the one presented in Figure 7, requires computing the convex hull (or convex envelope) of a set of points. The convex hull could be defined as the minimal set of points that will define a polygon which includes all the points of the initial set. With a large number of points (around 2 150 in our case), the convex hull is computed thanks to Graham's algorithm (Graham 1972; de Berg et al. 2000). This algorithm can be described as follows: Find the centre of gravity, G, of the initial set of points.Compute the counter-clockwise angle from G to all other points of the set.Sort the points by angle.Construct the boundary by scanning the points in the sorted order and performing only right turns (trim off left turns).
Railway vehicle bearings risk monitoring based on normal region estimation for no-fault data situations
Published in Journal of Transportation Safety & Security, 2021
Yuan Zhang, Yong Qin, Yan-ping Du, Lei Zhu, Xiu-kun Wei
To illustrate the consistency between the convex hull and the normal region more vividly, Figure 4 shows the schematic of convex hull of the plane point set. The convex hull of the plane point set refers to the smallest simple convex polygon that contains all points in the plane point set and the vertices belong to the plane point set. It can be visualized as a rubber band that just surrounds all points.
Automated post-processing of 3D-printed parts: artificial powdering for deep classification and localisation
Published in Virtual and Physical Prototyping, 2021
Joyce Xin-Yan Lim, Quang-Cuong Pham
Discretise the STL file of the object into a point cloud by using Poisson disk sampling (Yuksel 2015). As mentioned in Section 4.1, this was to create a distribution of points.Obtain the convex hull of the object. The convex hull provides global information on the geometry of the object, including the concave and convex areas.Find the intersection point (Q) on the convex hull, for every point (P) of the point cloud using the Möller-Trumbore ray-triangle intersection algorithm (Möller and Trumbore 1997). The algorithm is a fast method for calculating Q in 3D without having to compute the plane equation of the plane containing the triangle.If P ≠ Q, calculate the length (PQ) and its direction (n). A larger PQ meant that the point of interest was far from the convex hull, which could indicate that it may be a concave point and thus more powder may be found at this point. Thus, the new extended point was , where f was the enlarging factor, while k was a small random value to ensure that with the same f, the powder model generated would be unique for more variability. When f=1, the model would be equivalent to the convex hull.If , this meant that the point lied on the convex hull and could be a convex point. However, since the real objects also could have some powder on convex surfaces, a small random value was drawn from a single distribution and applied along the normal of P.Reconstruct the triangular mesh from the point cloud of new extended points by using the Poisson surface reconstruction method (Kazhdan, Bolitho, and Hoppe 2006) to obtain the powdered model. As mentioned in 4.1, there were some limitations on using the Poisson surface reconstruction method as broken models may be obtained.