Examples of polar axis in the following topics:
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- Polar coordinates define the location of an object in a plane by using a distance and an angle from a reference point and axis.
- A positive angle is usually measured counterclockwise from the polar axis, and a positive radius is in the same direction as the angle.
- A negative radius would be opposite the direction of the angle and a negative angle would be measured clockwise from the polar axis.
- The polar axis is usually drawn horizontal and pointing to the right .
- Use a polar coordinate to define a point with $r$ (distance from pole), and $\theta$(angle between axis and ray)
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- The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis.
- In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.
- The polar grid is scaled as the unit circle with the positive $x$-axis now viewed as the polar axis and the origin as the pole.
- We move counterclockwise from the polar axis by an angle of $θ$,and measure a directed line segment the length of $r$ in the direction of $θ$.
- Points in the polar coordinate system with pole $0$ and polar axis $L$.
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- Polar coordinates allow conic sections to be expressed in an elegant way.
- We can define any conic in the polar coordinate system in terms of a fixed point, the focus $P(r,θ)$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
- Thus, each conic may be written as a polar equation in terms of $r$ and $\theta$.
- For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
- For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
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- Since the direction of polarization is parallel to the electric field, you can consider the blue arrows to be the direction of polarization.
- What happens to these waves as they pass through the polarizer?
- Lets call the angle between the direction of polarization and the axis of the polarization filter θ.
- If you pass light through two polarizing filters, you will get varied effects of polarization.
- A polarizing filter has a polarization axis that acts as a slit passing through electric fields parallel to its direction.
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- A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.
- For the conversion between cylindrical and Cartesian coordinate co-ordinates, it is convenient to assume that the reference plane of the former is the Cartesian $xy$-plane (with equation $z = 0$), and the cylindrical axis is the Cartesian $z$-axis.
- Then the $z$ coordinate is the same in both systems, and the correspondence between cylindrical $(\rho,\varphi)$ and Cartesian $(x,y)$ are the same as for polar coordinates, namely $x = \rho \cos \varphi; \, y = \rho \sin \varphi$.
- Spherical coordinates ($r$, $\theta$, $\varphi$) as often used in mathematics: radial distance $r$, azimuthal angle $\theta$, and polar angle $\varphi$.
- A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.
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- Each pixel of an LCD consists of a layer of molecules aligned between two transparent electrodes and two polarizing films, and the actual liquid crystals are between these polarizing filters.
- Polarizing filter film with a vertical axis to polarize light as it enters.Glass substrate with ITO electrodes.
- Vertical ridges etched on the surface are smooth.Twisted nematic liquid crystal.Glass substrate with common electrode film (ITO) with horizontal ridges to line up with the horizontal filter.Polarizing filter film with a horizontal axis to block/pass light.Reflective surface to send light back to viewer.
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- If this phase difference is zero, then the wave is linearly polarized (left panel of Fig.2.1) with the polarization vector making an angle $\theta=\tan^{-1}(E_2/E_1)$ with $\epsilon_1$ and a magnitude of $E=\sqrt{E_1^2+E_2^2}.$
- The orientation of the ellipse is characterized by the orientation, tile or azimuth angle $\psi$ which is the angle between the semimajor axis of the ellipse and $s_0,s_1,s_2$.
- One could have defined an alternative representation based on the circular polarizations
- Often it is convenient to use this circular polarization basis rather than the linear polarization basis above (for example, waves traveling through plasma).
- It is possible to recover this polarization information through intensity measurements.
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- Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.
- When given a set of polar coordinates, we may need to convert them to rectangular coordinates.
- Dropping a perpendicular from the point in the plane to the $x$-axis forms a right triangle, as illustrated in Figure below.
- This corresponds to the non-uniqueness of polar coordinates.
- Derive and use the formulae for converting between Polar and Cartesian coordinates
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- Researchers have frequently noted that a single left-right axis is insufficient to describe the existing variation in political beliefs.
- Researchers have frequently noted that a single left-right axis is insufficient in describing the existing variation in political beliefs, and often include other axes.
- Though the descriptive words at polar opposites may vary, often in popular biaxial spectra the axes are split between cultural issues and economic issues, each scaling from some form of individualism (or government for the freedom of the individual) to a form of communitarianism (or government for the welfare of the community).
- The Nolan Chart, created by libertarian David Nolan, shows what he considers as "economic freedom " (issues like taxation, free trade, and free enterprise) on the horizontal axis and what he considers as "personal freedom" (issues like drug legalization, abortion, and the draft) on the vertical axis.
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- The bond in a homonuclear diatomic molecule is non-polar due to the electronegativity difference of zero.