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Hyperbolas

Hyperbolas

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse axis is...

Geometry of Hyperbolas

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A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...

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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...

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Related Experiment Video

Updated: Jul 7, 2026

Demonstration of a Hyperlens-integrated Microscope and Super-resolution Imaging

Demonstration of a Hyperlens-integrated Microscope and Super-resolution Imaging

Published on: September 8, 2017

Hard disks on the hyperbolic plane.

Carl D Modes¹, Randall D Kamien

¹Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USA.

Physical Review Letters

|February 1, 2008

Summary

This summary is machine-generated.

We modeled disordered hard disks in curved space, finding that hyperbolic geometry prevents crystal formation. This work provides a tractable model for studying systems with frustrated global order.

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Published on: February 8, 2014

Area of Science:

Physics
Materials Science
Geometry

Background:

Global crystalline order is typically frustrated in systems with curved geometry.
Hard disk fluids are fundamental models in statistical mechanics.
The hyperbolic plane offers a unique geometric framework for studying frustrated systems.

Purpose of the Study:

To develop a tractable model for disordered monodisperse hard disks on the hyperbolic plane.
To derive the equation of state for this system.
To compare theoretical predictions with simulation results.

Main Methods:

Extending free-area theory and the virial expansion to the hyperbolic plane.
Utilizing computational simulations to model hard disk behavior.
Analyzing systems near isostatic packing.

Main Results:

A tractable model for disordered hard disks on the hyperbolic plane was constructed.
The equation of state for the system was derived.
Theoretical predictions were compared with simulation data.

Conclusions:

The hyperbolic plane naturally frustrates global crystalline order in hard disk systems.
Extended free-area theory and virial expansion provide a valid framework for this system.
The study offers insights into disordered systems in curved spaces.