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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

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...

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,...

Gauss's Law: Cylindrical Symmetry

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

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...

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Related Experiment Video

Updated: Jul 7, 2026

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data

Published on: January 16, 2019

Comments on "A self-organizing network for hyperellipsoidal clustering (HEC)" [and reply].

W Song¹, S Xia, J Mao

¹Dept. of Autom., Tsinghua Univ., Beijing.

IEEE Transactions on Neural Networks

|January 1, 1997

Summary

Hyperellipsoidal clustering using Mahalanobis distance is shown to be impossible as the cost function becomes constant. A modified regularized Mahalanobis distance is key to achieving hyperellipsoidal clusters, confirming the algorithm

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Last Updated: Jul 7, 2026

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data

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Published on: January 16, 2019

The HoneyComb Paradigm for Research on Collective Human Behavior

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Published on: January 19, 2019

Large-scale Reconstructions and Independent, Unbiased Clustering Based on Morphological Metrics to Classify Neurons in Selective Populations

Large-scale Reconstructions and Independent, Unbiased Clustering Based on Morphological Metrics to Classify Neurons in Selective Populations

Published on: February 15, 2017

Area of Science:

Data Mining
Machine Learning
Pattern Recognition

Background:

The Mahalanobis distance was proposed for hyperellipsoidal clustering.
Previous work suggested this method could achieve hyperellipsoidal clusters.

Purpose of the Study:

To analyze the validity of hyperellipsoidal clustering using Mahalanobis distance.
To explain the success of hyperellipsoidal clustering (HEC) algorithms.
To clarify the role of regularized Mahalanobis distance in HEC.

Main Methods:

Mathematical proof demonstrating the clustering cost function becomes constant when using standard Mahalanobis distance.
Analysis of the impact of regularized Mahalanobis distance on clustering outcomes.

Main Results:

Hyperellipsoidal clustering is not achievable with the standard Mahalanobis distance due to a constant cost function.
The use of a regularized Mahalanobis distance is crucial for obtaining hyperellipsoidal clusters.
The HEC algorithm's effectiveness is attributed to its use of regularized Mahalanobis distance.

Conclusions:

The standard Mahalanobis distance does not support hyperellipsoidal clustering.
Regularized Mahalanobis distance is essential for the success of HEC algorithms.
HEC algorithms offer valuable insights into multidimensional data structures.