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Ellipses

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,...

Eccentricity of an Ellipse

Eccentricity of an Ellipse

An ellipse is a fundamental conic section defined by the constant sum of distances from any point on its curve to two fixed points, known as the foci. This geometric property can be physically demonstrated using a pencil, string, and two pins. By anchoring the string at both ends and maintaining it taut with a pencil, one can trace the outline of an ellipse.The shape and extent of the ellipse are determined by its eccentricity, e, defined as the ratio of the distance between the center and a...

Geoid and Ellipsoid

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The Earth's shape is best described as an ellipsoid, a slightly flattened sphere created by rotating an ellipse around its minor axis. This flattening results in the polar axis being about 21 kilometers shorter than the equatorial axis. In contrast, the geoid represents the Earth's gravitational shape and aligns with the mean sea level (MSL). The geoid is an irregular equipotential surface where gravity is perpendicular at every point. Variations in Earth's mass distribution cause geoid...

Geometry of Hyperbolas

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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...

Symmetry

Symmetry

The equation of an ellipse centered at the origin defines all points whose distances from the center maintain a constant ratio between the horizontal and vertical axes. This equation results in a smooth, closed curve that extends further along the x-axis than the y-axis, giving it a horizontal orientation. Such an ellipse demonstrates three kinds of symmetry: across the x-axis, across the y-axis, and about the origin. These symmetries are essential in understanding the graph's structure and...

Adjusting a Traverse

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In the site survey of a four-sided traverse, internal angles are essential to ensure geometric accuracy. The survey revealed that the sum of the measured internal angles was 359 degrees and 48 minutes, which is 12 minutes less than the expected 360 degrees. This discrepancy signals an error likely arising from measurement inaccuracies during the fieldwork.To rectify this error, the adjustment process involved distributing the 12-minute shortfall equally across the four internal angles. By...

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Fabrication of a Low-Cost, Fiber-Coupled, and Air-Spaced Fabry-Pérot Etalon

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Geometrically exact ellipsometer alignment.

D E Aspnes, A A Studna

|January 23, 2010

Summary

This summary is machine-generated.

This study presents a novel ellipsometer alignment procedure using symmetry and four null measurements. This method ensures accurate alignment without being affected by minor optical defects.

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Last Updated: Jun 16, 2026

Fabrication of a Low-Cost, Fiber-Coupled, and Air-Spaced Fabry-Pérot Etalon

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Published on: February 3, 2023

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Using an Automated Hirschberg Test App to Evaluate Ocular Alignment

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Published on: March 24, 2020

Area of Science:

Optical Physics
Metrology

Background:

Accurate ellipsometer alignment is crucial for precise optical measurements.
Traditional alignment methods can be complex and sensitive to instrument imperfections.

Purpose of the Study:

To develop a simplified and robust procedure for ellipsometer alignment.
To utilize the inherent symmetry of the ellipsometer for self-alignment.

Main Methods:

The procedure relies on four null measurements performed on a transparent reflecting surface.
It determines tilt angles of polarizer and analyzer telescope arms.
Reference angles are established for polarization vectors within the plane of incidence.

Main Results:

The alignment procedure is independent of small parasitic ellipticities from polarizer or analyzer defects.
It provides accurate alignment information using only standard operational equipment.
A step-by-step guide for practical implementation is detailed.

Conclusions:

The described method offers an effective and accessible approach to ellipsometer alignment.
This technique enhances the reliability and ease of use for ellipsometric measurements.