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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...
Quadric Surfaces01:28

Quadric Surfaces

Quadric surfaces are three-dimensional surfaces characterized by second-degree equations in the variables x, y, and z. These surfaces are smooth and continuous, and specific combinations of squared and linear terms define their shapes. The main types of quadric surfaces include ellipsoids, cones, paraboloids, and hyperboloids. Each type exhibits distinct geometric features depending on how the variables are arranged and related within the equation.Ellipsoids are closed surfaces formed when all...
Bending of Curved Members - Neutral Surface01:16

Bending of Curved Members - Neutral Surface

In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.
Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within the...
Parametric Surfaces01:30

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...
Tangent Planes to a Parametric Surface01:22

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...
Tangent Planes to Surfaces01:19

Tangent Planes to Surfaces

In multivariable calculus, the concept of a tangent plane plays a central role in approximating curved surfaces. When dealing with a surface defined by a function of two variables, such as z = f(x, y), the tangent plane at a given point provides the best linear approximation to the surface near that point. This local linearization allows complex, nonlinear geometries to be treated using simpler, planar models.The construction of the tangent plane involves taking vertical slices of the surface...

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Related Experiment Video

Updated: Jun 16, 2026

Microfabrication of Implantable Optics Integrated in a Microstructured Imaging Window for Advanced In Vivo Imaging
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Published on: April 11, 2025

Three-dimensional curved surface for integrated optics.

S K Sheem

    Applied Optics
    |February 16, 2010
    PubMed
    Summary

    Curved dielectric surfaces like cones and spheres guide light waves. Experiments show light continuously deflects toward larger cross-sections, with applications in integrated optics.

    Area of Science:

    • Optics
    • Wave phenomena
    • Integrated optics

    Background:

    • Previous studies explored homogeneous dielectric cylinders and curved boundaries for optical waveguides.
    • Thin dielectric films are established as substrates for guiding light waves.

    Purpose of the Study:

    • Investigate wave behavior guided by conic and spherical surfaces.
    • Analyze light deflection phenomena on these curved structures.
    • Explore potential applications in integrated optical devices.

    Main Methods:

    • Experimental observation of light propagation on conic and spherical surfaces.
    • Theoretical analysis using geometric optics principles.
    • Theoretical analysis using wave equations.

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    Main Results:

    • Conic and spherical surfaces continuously deflect guided light towards regions of larger cross-section.
    • Observed deflection is consistent with predictions from both geometric optics and wave equations.

    Conclusions:

    • Conic and spherical surfaces effectively guide and manipulate light waves.
    • The observed light deflection phenomenon has potential applications in integrated optics design.
    • Further research can explore advanced applications based on these guiding principles.