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Related Concept Videos

Polar Coordinates01:24

Polar Coordinates

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 from a fixed...
Graphs of Polar Equations01:17

Graphs of Polar Equations

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...
Polar and Cylindrical Coordinates01:22

Polar and Cylindrical Coordinates

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.
Fischer Projections02:18

Fischer Projections

Learning to draw Fischer projections of molecules and understanding their relevance plays a crucial role in the visual depiction of organic molecules. A Fischer projection is a two-dimensional projection on a planar surface to simplify the three-dimensional wedge–dash representation of molecules. This is especially helpful in the case of molecules with multiple chiral centers that can be difficult to draw. Here, all the bonds of interest are represented as horizontal or vertical lines. While...
Potential Due to a Polarized Object01:29

Potential Due to a Polarized Object

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,...
Polar Equations of Conics01:29

Polar Equations of Conics

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 describe any conic...

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Geometric representation for partial polarization in three dimensions.

Colin J R Sheppard1

  • 1Optical Bioimaging Laboratory, Division of Bioengineering, National University of Singapore, Singapore 117574. colinjrsheppard@gmail.com

Optics Letters
|July 25, 2012
PubMed
Summary
This summary is machine-generated.

We developed a geometric method to find coherency matrix eigenvalues. This leads to a triangular plot representing the 3D degree of polarization, offering a new way to visualize polarization measures.

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Area of Science:

  • Optics and Photonics
  • Mathematical Physics

Background:

  • Coherency matrices are fundamental in describing polarized light.
  • Previous methods for visualizing polarization measures, like spherical plots, have limitations.

Purpose of the Study:

  • To present a novel geometric construction for coherency matrix eigenvalues.
  • To introduce a triangular composition plot for the three-dimensional degree of polarization.
  • To offer an alternative visualization to spherical plots for polarization measures.

Main Methods:

  • Geometric construction for coherency matrix eigenvalues.
  • Representation of polarization measures using matrix invariants.
  • Development of a triangular composition plot.

Main Results:

  • The geometric construction directly yields eigenvalues.
  • The triangular plot simplifies the representation of different polarization measures.
  • Polarization measures exhibit simple properties and are expressible via matrix invariants on this plot.

Conclusions:

  • The triangular composition plot provides an effective alternative to spherical plots for visualizing 3D polarization.
  • This geometric approach simplifies the analysis and understanding of polarization properties.