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Related Concept Videos

Oriented Surfaces01:30

Oriented Surfaces

A surface is called orientable if a consistent choice of unit normal vector can be made at every point on the surface. A thin soap film stretched across a wire loop provides a familiar example. The film separates the air on one side from the air on the other, so one side can be selected as positive and the opposite side as negative. Once this choice is made, a unit normal vector can be assigned smoothly across the entire surface.At each point on the soap film, a unit normal vector points...
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...
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...
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...
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...
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...

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Related Experiment Video

Updated: Jul 7, 2026

Convergent Polishing: A Simple, Rapid, Full Aperture Polishing Process of High Quality Optical Flats & Spheres
13:07

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Null testing convex optical surfaces.

A Szulc

    Applied Optics
    |February 9, 2008
    PubMed
    Summary

    A novel null test accurately measures convex optical surfaces using an auxiliary ellipsoidal mirror. This precise optical testing method ensures high accuracy for optical component manufacturing and quality control.

    Area of Science:

    • Optical Engineering
    • Metrology

    Background:

    • Accurate measurement of convex optical surfaces is crucial for optical system performance.
    • Existing testing methods may lack precision or require complex setups.

    Purpose of the Study:

    • To introduce a new, highly precise null test for convex optical surfaces.
    • To provide an efficient method for testing optical components.

    Main Methods:

    • The test utilizes an auxiliary ellipsoidal mirror with a diameter similar to the convex surface under test.
    • The auxiliary ellipsoid is itself tested using a null method.
    • This approach creates a null test configuration for the primary convex surface.

    Main Results:

    • The described test achieves excellent precision in evaluating convex optical surfaces.

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  • The null testing of the auxiliary ellipsoid contributes to the overall high accuracy of the method.

    • Conclusions:

    • The presented null test offers a precise and effective solution for convex optical surface metrology.
    • This method is suitable for applications demanding high accuracy in optical component characterization.