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Related Concept Videos

Cylinders in Three-Dimensional Space01:28

Cylinders in Three-Dimensional Space

A cylindrical surface is generated when a two-dimensional profile curve is translated along a straight line in three-dimensional space. The translated copies of the curve form a surface composed of parallel rulings, each oriented in the same fixed direction. This construction allows many three-dimensional forms to be described using relatively simple planar equations.In Cartesian coordinates, a cylindrical surface is often recognized by an equation that omits one of the three variables. For...
Tangent Planes to Level Surfaces01:31

Tangent Planes to Level Surfaces

A level surface consists of all points in space where a function of three variables takes the same fixed value. If a point lies on this surface, understanding the surface’s geometry there requires more than just knowing the point’s coordinates; it requires describing how the surface is oriented, or how it tilts, near that point.To probe this local geometry, imagine tracing a path that stays entirely on the level surface and passes through the point of interest. This path can be described as a...
Curvilinear Motion: Normal and Tangential Components01:27

Curvilinear Motion: Normal and Tangential Components

When a car traverses a curved road, its motion can be elucidated by breaking it down into tangential and normal components. The car-centric coordinates attached to the vehicle move with it.
The positive direction of the t-axis aligns with the increasing position of the car along the curved path, denoted by the unit vector ut. Simultaneously, the n-axis, perpendicular to the t-axis, dissects the curved path into differential arc segments, each forming the arc of a circle with a radius of...
Gradient Fields01:27

Gradient Fields

A gradient field is a vector field derived from a scalar field. A scalar field assigns a single numerical value to every point in space, such as temperature, pressure, or electric potential. The gradient field describes how that value changes from point to point. It gives both the direction of the fastest increase and the rate of change in that direction.For a scalar field f(x, y), the gradient is written as\begin{equation*}\nabla f=\left\langle \jfrac{\partial f}{\partial x},\jfrac{\partial...
Torsion in Vector Calculus01:20

Torsion in Vector Calculus

A toy train ascending a winding track that curves and tilts offers an intuitive view of torsion, a key geometric concept in the study of space curves. While curvature measures how sharply a path bends, torsion captures how the path twists out of the plane of bending. This twisting behavior is crucial in understanding three-dimensional motion and is precisely described using the Frenet–Serret framework.At each point along a space curve, the Frenet–Serret frame consists of three orthogonal unit...
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Tangent Planes to a Parametric Surface

A tangent plane provides a linear approximation to a curved surface at a specific point, capturing the local behavior of the surface. It can be understood as the plane that just touches the surface at that point and is defined by the tangent directions of curves lying on the surface. These tangent directions arise naturally when the surface is described parametrically, allowing systematic construction of the plane.For a surface expressed in parametric form, the position of any point is...

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Related Experiment Video

Updated: Jun 9, 2026

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow
13:02

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Published on: February 27, 2016

Three-dimensional focal curves generated by one-dimensional gradient-index tapers.

R M González, J Liñares, C Gómez-Reino

    Applied Optics
    |August 25, 2010
    PubMed
    Summary

    Three-dimensional focal curves were created using a gradient-index taper illuminated by an off-axis wave. The study analyzed intensity distribution, considering the taper

    Area of Science:

    • Optics and Photonics
    • Gradient-Index (GRIN) Optics
    • Wave Propagation

    Background:

    • Gradient-index (GRIN) media offer unique light manipulation properties.
    • Understanding focal curve formation in GRIN materials is crucial for optical system design.
    • Off-axis illumination introduces complex wave aberrations.

    Purpose of the Study:

    • To investigate the generation of three-dimensional focal curves.
    • To analyze the intensity distribution along these focal curves.
    • To account for the influence of the exit pupil in GRIN media.

    Main Methods:

    • Utilized a one-dimensional gradient-index taper.
    • Illuminated the taper with an off-axis plane wave.
    • Evaluated intensity distribution considering the exit pupil characteristics.

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    Main Results:

    • Successfully generated three-dimensional focal curves.
    • Observed specific intensity distributions along the focal curves.
    • The exit pupil's form significantly impacted the intensity profiles.

    Conclusions:

    • Gradient-index tapers can produce complex 3D focal curves.
    • Off-axis illumination and pupil effects are critical factors in focal curve analysis.
    • This research provides insights for designing advanced optical elements with GRIN materials.