Hyperbolic Curve

A hyperbolic curve is a mathematical graph in which a variable rises rapidly at first and then approaches a limiting value, making it useful for describing saturation. In pharmacology, drug concentration often relates to response through the rectangular hyperbola E = EmaxC/(EC50 + C), where Emax is the maximum effect, C is concentration, and EC50 is the concentration producing half-maximal response. This pattern reflects progressive receptor occupancy or other capacity-limited processes: increasing concentration produces larger effects initially, but additional drug yields diminishing gains as the system approaches saturation. Hyperbolic curves support potency comparisons, dose selection, and interpretation of concentration-effect experiments.

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Education

JoVE Core - Calculus

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

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2026

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...

Hyperbolic Functions

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2026

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...

Research

JoVE Journal - Environment
Free Sample

Implementation of a Hyperbolic Vortex Plasma Reactor for the Removal of Micropollutants in Water

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2025

This study presents the methodology for the generation of six different types of plasma discharges within a Hyperbolic Vortex Plasma Reactor for the degradation of micropollutants in water, including pharmaceuticals and per- and polyfluoroalkyl substances (PFAS).

Inverse Hyperbolic Functions and Their Derivatives

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2026

The shape of a suspension bridge cable hanging under its own weight is described by a catenary curve, which is modeled using the hyperbolic cosine function. This mathematical model accurately captures the balance between gravity and tension acting along the cable. When a particular vertical position on the cable is known, the corresponding horizontal position can be determined using the inverse hyperbolic cosine function, allowing for a detailed analysis of the cable's geometry.Inverse...

Research

JoVE Journal - Engineering
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Preparation of Free-Surface Hyperbolic Water Vortices

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Cited by 6 •

2023

This paper describes how three different water vortex regimes in a hyperbolic Schauberger funnel can be created, their most important characteristics, and how associated parameters such as the oxygen transfer rates can be calculated.

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