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Related Concept Videos

Hyperbolas01:30

Hyperbolas

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse axis is...
Geometry of Hyperbolas01:30

Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
Kepler's Second Law of Planetary Motion01:29

Kepler's Second Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. His first law states that all planets orbit the Sun in an elliptical orbit, with the Sun at one of the ellipse's foci. Therefore, the distance of a planet from the Sun varies throughout its revolution around the Sun.
While in an elliptical orbit, the total energy of the planet is conserved. Therefore, the planet slows down when it is at apogee and...
Kepler's First Law of Planetary Motion01:10

Kepler's First Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. He formulated his first two laws based on the observations of his forebears, Nikolaus Copernicus and Tycho Brahe.
Polish astronomer Nikolaus Copernicus put forth a theory that stated a heliocentric model for the solar system. According to this heliocentric theory, all the planets, including Earth, orbit the Sun in circular orbits.
On the other hand,...
Ellipses01:30

Ellipses

An ellipse is formed when a right circular cone is intersected by an inclined plane that does not cut through its base. This intersection yields a closed, symmetric curve characterized by distinctive geometric properties. Most notably, an ellipse is defined as the collection of all points in a plane for which the combined distances to two fixed points—called the foci—remain constant.The ellipse features two principal axes: the major and the minor axes. The major axis is the longest diameter,...
Hyperbolic and Inverse Hyperbolic Functions: Problem Solving01:30

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...

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Updated: Jun 15, 2026

Precision Measurements and Parametric Models of Vertebral Endplates
10:35

Precision Measurements and Parametric Models of Vertebral Endplates

Published on: September 17, 2019

Elliptical and hyperbolic zone plates.

C Gomez-Reino, J M Cuadrado, M V Perez

    Applied Optics
    |March 12, 2010
    PubMed
    Summary
    This summary is machine-generated.

    New optical methods created elliptical and hyperbolic zone plates. These zone plates, when reconstructed, show linear foci on both sides due to wave front superposition and recording effects.

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    Last Updated: Jun 15, 2026

    Precision Measurements and Parametric Models of Vertebral Endplates
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    Precision Measurements and Parametric Models of Vertebral Endplates

    Published on: September 17, 2019

    Conducting Elevated Temperature Normal and Combined Pressure-Shear Plate Impact Experiments Via a Breech-end Sabot Heater System
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    Area of Science:

    • Optics and Photonics
    • Holography
    • Diffractive Optics

    Background:

    • Zone plates are diffractive optical elements used for focusing light.
    • Traditional zone plates often have limitations in focal properties and aberrations.
    • The development of novel zone plate geometries is crucial for advanced optical applications.

    Purpose of the Study:

    • To construct elliptical and hyperbolic zone plates using optical methods.
    • To investigate the focusing properties of these novel zone plates.
    • To analyze the impact of linear and nonlinear recording on reconstruction.

    Main Methods:

    • Utilizing optical methods for construction.
    • Recording the superposition of spherical and cylindrical wave fronts on photosensitive material.
    • Investigating cases where wave front foci coincide to form linear zone plates.
    • Analyzing reconstruction effects, including linear and nonlinear recording.

    Main Results:

    • Successful construction of elliptical and hyperbolic zone plates.
    • Observation of linear foci symmetrically positioned on both sides of the zone plates during reconstruction.
    • Demonstration of the influence of recording linearity on focal characteristics.

    Conclusions:

    • Elliptical and hyperbolic zone plates can be effectively constructed using coherent superposition of wave fronts.
    • The reconstruction process reveals predictable linear focal points influenced by recording methods.
    • These findings contribute to the understanding and design of advanced diffractive optical elements.