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Related Concept Videos

Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a...
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Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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Spherical Coordinates01:23

Spherical Coordinates

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Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
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Euler's Formula for Pin-Ended Columns01:21

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In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
To calculate the critical load, envision...
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Gauss's Law: Planar Symmetry01:27

Gauss's Law: Planar Symmetry

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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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Gauss's Law: Problem-Solving01:10

Gauss's Law: Problem-Solving

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Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector...
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An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production
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Extensible Spherical Fibonacci Grids.

Ricardo Marques, Christian Bouville, Kadi Bouatouch

    IEEE Transactions on Visualization and Computer Graphics
    |November 15, 2019
    PubMed
    Summary
    This summary is machine-generated.

    Researchers developed extensible spherical Fibonacci grids (E-SFG) to allow adding points to uniform spherical grids. This innovation enables refinement of point sets for computer graphics applications, overcoming previous limitations.

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    Area of Science:

    • Computational Geometry
    • Computer Graphics

    Background:

    • Spherical Fibonacci grids (SFGs) offer highly uniform point distributions on spheres, suitable for applications like numerical integration and vector quantization.
    • Existing SFGs lack methods for adding points while preserving uniformity, limiting their use in dynamic or adaptive scenarios.

    Purpose of the Study:

    • To introduce a method for extending existing Spherical Fibonacci Grids (SFGs) while maintaining their uniform distribution properties.
    • To address the limitation of SFGs in applications requiring point set refinement.

    Main Methods:

    • Formal analysis of SFG properties contributing to uniform spherical distribution.
    • Development of the extensible spherical Fibonacci grids (E-SFG) algorithm to add points preserving these properties.
    • Comparative analysis of E-SFG against other extensible spherical point sets.

    Main Results:

    • The proposed E-SFG algorithm successfully extends initial SFGs while preserving near-optimal uniform distribution.
    • E-SFG demonstrated superior performance compared to low discrepancy sequence-based extensible spherical point sets.
    • Quantitative improvements were observed in spherical cap discrepancy and root mean squared error for rendering integrals.

    Conclusions:

    • The E-SFG provides a viable solution for refining uniform point distributions on spheres.
    • This advancement expands the applicability of Fibonacci grids in computer graphics and related fields requiring adaptive point sets.