A similar statement can be made about a y' axis and the parallel centroidal y axis. Or, in general, any centroidal B axis and a parallel B' axis.
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In geometry, a centroidal Voronoi tessellation (CVT) is a special type of Voronoi tessellation or Voronoi diagram. A Voronoi tessellation is called centroidal when the generating point of each Voronoi cell is also its centroid, i.e. the arithmetic mean or center of mass. It can be viewed as an optimal partition corresponding to an optimal distribution of generators.
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Hongkai Dai, Andres Valenzuela, and Russ Tedrake. "Whole-body motion planning with Centroidal Dynamics and Full Kinematics" International Conference on Humanoid Robots, IEEE 2014.
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A number of algorithms can be used to generate centroidal Voronoi tessellations, including Lloyd's algorithm for K-means clustering or Quasi-Newton methods like BFGS.
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Many patterns seen in nature are closely approximated by a centroidal Voronoi tessellation. Examples of this include the Giant's Causeway, the cells of the cornea, and the breeding pits of the male tilapia.
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Maria Emelianenko is a Russian-American applied mathematician and materials scientist known for her work in numerical algorithms, scientific computing, grain growth, and centroidal Voronoi tessellations. She is a professor of mathematical sciences at George Mason University.
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The first is an orthogonal line passing through the corresponding Euler point to the chosen face. The second is an orthogonal line passing through the centroid of the chosen face. This orthogonal line through the twelve-point center lies midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelve-point center lies at the midpoint of the corresponding Euler point and the orthocenter for that face.
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These methods alternate between steps in which one constructs the Voronoi diagram for a set of seed points, and steps in which the seed points are moved to new locations that are more central within their cells. These methods can be used in spaces of arbitrary dimension to iteratively converge towards a specialized form of the Voronoi diagram, called a Centroidal Voronoi tessellation, where the sites have been moved to points that are also the geometric centers of their cells.
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The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis.H. Williams, Measuring the inertia tensor, presented at the IMA Mathematics 2007 Conference. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.
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The method is quite versatile and can be used to analyse frames of any number of storeys or floors. The position of the centroidal axis (the center of gravity line for the frame) is determined by using the areas of the end columns and interior columns. The cantilever method is considered as one of the two primary approximate methods (the other being the portal method) for indeterminate structural analysis of frames for lateral loads. Its use is recommended for frames that are taller than they are wide, and therefore behave similar to a beam cantilevered up from the ground.
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The cantilever method is an approximate method for calculating shear forces and moments developed in beams and columns of a frame or structure due to lateral loads. The applied lateral loads typically include wind loads and earthquake loads, which must be taken into consideration while designing buildings. The assumptions used in this method are that the points of contraflexure (or points of inflection of the moment diagram) in both the vertical and horizontal members are located at the midpoint of the member, and that the direct stresses in the columns are proportional to their distances from the centroidal axis of the frame. The frame is analysed in step-wise (iterative) fashion, and the results can then be described by force diagrams drawn up at the end of the process.
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