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Polar and Cylindrical Coordinates01:22

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The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
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The polar coordinate system offers an alternative to the Cartesian coordinate system for specifying points in a plane, using a distance and an angle instead of x and y coordinates. This system is particularly advantageous in situations involving circular or rotational symmetry, such as in physics or engineering problems involving waves, oscillations, or orbital paths.Defining Polar CoordinatesIn polar coordinates, a point is represented as P(r, ��), where r is the radial distance...
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In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
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A conic section can be defined in polar coordinates as the set of all points whose distance from a fixed point, known as the focus, bears a constant ratio to their distance from a fixed line, known as the directrix. This constant ratio is called the eccentricity. This definition unifies all types of conic sections—ellipses, parabolas, and hyperbolas—under a single framework. When the focus is positioned at the origin of the polar coordinate system, a single polar equation can...
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Potential Due to a Polarized Object01:29

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A neutral atom consists of a positively charged nucleus surrounded by a negatively charged electron cloud. When placed in an external electric field, the external electric force pulls the electrons and nucleus apart, opposite to the intrinsic attraction between the nucleus and the electrons. The opposing forces balance each other with a slight shift between the center of masses of the nucleus and the electron cloud, resulting in a polarized atom. On the other hand, a few molecules, like water,...
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The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
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Related Experiment Video

Updated: Feb 19, 2026

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Three-dimensional polarization ray tracing calculus for partially polarized light.

Haiyang Zhang, Yi Li, Changxiang Yan

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    |November 3, 2017
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    Summary

    This study introduces a new 3x3 coherency matrix for partially polarized light, enabling calculations for all polarization states in optical systems. The method is validated, offering a significant advancement over existing 2D approaches.

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    Area of Science:

    • Optics and Photonics
    • Mathematical Physics

    Background:

    • Calculating light polarization evolution is crucial for optical systems.
    • Existing methods are limited to completely polarized light.
    • A generalized approach for partially polarized light is needed.

    Purpose of the Study:

    • To present a 3x3 coherency matrix for partially polarized light in global coordinates.
    • To develop a new 3D calculus method for polarization transformation.
    • To validate the proposed method and compare it with 2D calculus.

    Main Methods:

    • Formulation of a 3x3 coherency matrix for partially polarized light.
    • Development of a novel 3D coherency matrix calculus.
    • Experimental validation using a double Gauss optical lens.

    Main Results:

    • The 3x3 coherency matrix successfully describes partially polarized light.
    • The new 3D calculus method accurately calculates polarization transformations.
    • The 3D method demonstrates superior performance compared to 2D calculus.

    Conclusions:

    • The presented 3x3 coherency matrix and 3D calculus offer a comprehensive solution for partially polarized light.
    • This advancement enables precise polarization calculations in complex optical systems.
    • The method provides a robust tool for optical system design and analysis.