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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...

Residuals and Least-Squares Property

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...

Geometry of Hyperbolas

Geometry of Hyperbolas

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...

Optimization Problems

Optimization Problems

Optimization problems often involve identifying maximum or minimum values under specific constraints. A well-known example is determining the longest horizontal pipe that can be moved around a right-angled corner, where a 3-meter-wide hallway meets a 2-meter-wide hallway. This scenario, common in architectural design and industrial transport, can be understood conceptually through geometric and trigonometric reasoning.To visualize the problem, consider the pipe as a straight line that touches...

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

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...

Calibration Curves: Linear Least Squares

Calibration Curves: Linear Least Squares

A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
For data that follow a straight line, the standard method for fitting is the linear...

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Related Experiment Video

Updated: May 18, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

The L1-norm best-fit hyperplane problem.

J P Brooks¹, J H Dulá

¹Virginia Commonwealth University, 1015 Floyd Avenue, P.O. Box 843083, Richmond, VA 23284.

Applied Mathematics Letters

|October 2, 2012

Summary

This summary is machine-generated.

We present a new algorithm for the L(1)-norm best-fit hyperplane problem. This method uses geometric insights and linear programming for global optimality, applicable to computer vision and statistics.

Related Experiment Videos

Last Updated: May 18, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

Area of Science:

Multivariate Statistics
Computer Vision
Operations Research

Background:

The L(1)-norm best-fit hyperplane problem is crucial in various data analysis fields.
Existing methods may lack guaranteed global optimality or computational efficiency.

Purpose of the Study:

To formalize a novel algorithm for solving the L(1)-norm best-fit hyperplane problem.
To provide a method with a new proof of global optimality.

Main Methods:

Derivation using first principles and geometric insights.
Leveraging concepts of L(1) projection and L(1) regression.
Solving a small number of linear programs.

Main Results:

A formalized algorithm for the L(1)-norm best-fit hyperplane problem.
Demonstration of global optimality through a new proof.
Implementation for validation and testing.

Conclusions:

The developed procedure is computationally accessible and validated.
The algorithm has broad applicability in location theory, computer vision, and multivariate statistics.