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Related Concept Videos

Arc Length Function01:22

Arc Length Function

The arc length function represents the total distance traveled along a smooth curve measured from a fixed starting point to a variable endpoint. For curves that are continuous and differentiable, arc length provides a precise way to quantify distance when straight-line approximations are insufficient.To derive arc length, the curve is divided into many small segments. Each segment is approximated by a straight line whose length depends on the horizontal and vertical changes over that interval.
Arc Length of Space Curves01:21

Arc Length of Space Curves

Arc length represents the total distance traveled along a curve in space. For a moving object such as a helicopter, the path can be modeled by a vector-valued position function\begin{equation*}\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}where t denotes time. Unlike displacement, which measures only the straight-line distance between two points, arc length accounts for every change in direction along the trajectory.To calculate arc length, the interval of motion is divided into...
Arc Length of a Curve01:30

Arc Length of a Curve

In engineering applications like roller coaster design, cable installation, and railway construction, determining the precise length of a curved path is essential. These paths are rarely straight and often follow smooth, continuous curves that require accurate measurement for effective planning.To estimate the length of a curve, the path is initially divided into small segments. Each segment is approximated by a straight line connecting two nearby points on the curve. The sum of these linear...
Integration Applied to Polar Coordinates to Find Arc Lengths01:26

Integration Applied to Polar Coordinates to Find Arc Lengths

In polar coordinates, a plane curve is described by a radial distance r from a fixed point, called the pole, and an angle θ measured from a reference direction. This system is especially useful for paths that naturally involve rotation, such as an expanding spiral followed by a search drone. If the hiker’s last known position is treated as the pole, then the drone’s location at any instant can be represented by the polar equation r = f(θ), where the distance from the pole changes as the drone...
Arc Length of a Curve: Problem Solving01:21

Arc Length of a Curve: Problem Solving

A high-voltage power line spans a 40-meter horizontal distance between two transmission towers, resulting in a 10-meter vertical sag due to the effects of gravity and thermal expansion. The curve formed by the suspended cable is a catenary, which accurately models the behavior of a uniform, flexible cable under its own weight. Unlike a parabolic shape, the catenary is described by the hyperbolic cosine function and offers a precise representation of the cable's form.In this setup, engineers...
Nuclear Fusion02:45

Nuclear Fusion

The process of converting very light nuclei into heavier nuclei is also accompanied by the conversion of mass into large amounts of energy, a process called fusion. The principal source of energy in the sun is a net fusion reaction in which four hydrogen nuclei fuse and ultimately produce one helium nucleus and two positrons.
A helium nucleus has a mass that is 0.7% less than that of four hydrogen nuclei; this lost mass is converted into energy during the fusion. This reaction produces about...

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Related Experiment Video

Updated: Jul 7, 2026

Kinematic History of a Salient-recess Junction Explored through a Combined Approach of Field Data and Analog Sandbox Modeling
06:55

Kinematic History of a Salient-recess Junction Explored through a Combined Approach of Field Data and Analog Sandbox Modeling

Published on: August 5, 2016

Origin of Kern's arc.

Y Takano, K N Liou

    Applied Optics
    |May 20, 1997
    PubMed
    Summary
    This summary is machine-generated.

    Kern's arc is caused by double-plate ice crystals. Computer simulations and ray tracing explain how these ice crystals create this halo phenomenon.

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    Area of Science:

    • Atmospheric optics
    • Crystallography

    Background:

    • Halo phenomena are optical illusions in the atmosphere.
    • Ice crystal halos are often observed but their formation mechanisms require detailed study.

    Purpose of the Study:

    • To explain the formation of Kern's arc.
    • To identify the specific ice crystal type responsible for Kern's arc.
    • To investigate conditions for related halo arcs.

    Main Methods:

    • Computer-simulated halo patterns.
    • Geometric ray tracing.
    • Analysis of light ray paths through ice crystals.

    Main Results:

    • Kern's arc is produced by double-plate ice crystals with a vertical principal axis.
    • Geometric ray tracing confirmed the light paths contributing to Kern's arc.
    • Conditions for the appearance of an arc opposite a circumhorizontal arc were discussed.

    Conclusions:

    • The formation of Kern's arc is definitively linked to specific ice crystal habits and orientations.
    • Understanding these mechanisms enhances our knowledge of atmospheric optical phenomena.