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Related Concept Videos

Polar Coordinates: Problem Solving01:27

Polar Coordinates: Problem Solving

Directional radiation patterns are central to antenna analysis, as they illustrate how signal strength varies with direction. These patterns are often modeled using polar plots, where the radial distance from the origin represents signal intensity at a given angle. A commonly used idealized form is the four-lobed rose curve, which captures the concept of directional beams in a simplified mathematical form.The four-lobed rose curve, described by r = cos⁡(2θ), features four symmetric lobes, each...
Spherical Coordinates01:23

Spherical Coordinates

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...
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...
X-ray Crystallography02:18

X-ray Crystallography

The size of the unit cell and the arrangement of atoms in a crystal may be determined from measurements of the diffraction of X-rays by the crystal, termed X-ray crystallography.
Diffraction
Diffraction is the change in the direction of travel experienced by an electromagnetic wave when it encounters a physical barrier whose dimensions are comparable to those of the wavelength of the light. X-rays are electromagnetic radiation with wavelengths about as long as the distance between neighboring...
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.
Beams with Unsymmetric Loadings01:17

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Analyzing a supported beam under unsymmetrical loadings is essential in structural engineering to understand how beams respond to varied force distributions. This analysis involves calculating the deflection and identifying points where the slope of the beam is zero, which are crucial for ensuring structural stability and functionality.
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Measurement of X-ray Beam Coherence along Multiple Directions Using 2-D Checkerboard Phase Grating
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Aberration coefficients and unusual coordinates for specifying rays.

P J Sands

    Applied Optics
    |January 16, 2010
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces a novel method for calculating optical aberration coefficients using non-planar surfaces. This expands the applicability of Buchdahl

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    Measurement of X-ray Beam Coherence along Multiple Directions Using 2-D Checkerboard Phase Grating
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    Area of Science:

    • Optical Engineering
    • Computational Optics
    • Aberration Theory

    Background:

    • Buchdahl's aberration coefficients traditionally rely on ray intersections with planar surfaces in object space.
    • Existing computational schemes are limited by the assumption of planar reference surfaces.

    Purpose of the Study:

    • To present a generalized method for computing aberration coefficients using non-planar surfaces.
    • To adapt existing computational schemes for aberration coefficients to new coordinate systems.

    Main Methods:

    • Development of a method to specify rays by intersection points with arbitrary surfaces.
    • Reinterpretation of intermediate coefficients and modification of identities within iteration equations.
    • Application to object surfaces of revolution, aperture stop coordinates, and W(o) coordinates.

    Main Results:

    • A flexible framework for calculating aberration coefficients with non-planar surfaces.
    • Demonstration of modifications needed for existing computational schemes.
    • Successful application to diverse coordinate systems, including aperture and image space.

    Conclusions:

    • The proposed method significantly broadens the applicability of aberration coefficient computation.
    • This generalization is crucial for analyzing complex optical systems with non-planar elements.