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Eccentricity of an Ellipse01:27

Eccentricity of an Ellipse

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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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Ellipses01:30

Ellipses

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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...
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The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
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In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
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Polar Equations of Conics01:29

Polar Equations of Conics

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A conic section can be defined in polar coordinates as the set of all points whose distance from a fixed point, known as the focus, bears a constant ratio to their distance from a fixed line, known as the directrix. This constant ratio is called the eccentricity. This definition unifies all types of conic sections—ellipses, parabolas, and hyperbolas—under a single framework. When the focus is positioned at the origin of the polar coordinate system, a single polar equation can describe any...
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Euler's Formula to Columns with Other End Conditions01:15

Euler's Formula to Columns with Other End Conditions

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Euler's formula is very important in the field of structural engineering, providing a foundation for understanding the critical loading conditions of pin-ended columns. This formula links the modulus of elasticity, the moment of inertia of the cross-section, and the column's length, offering a precise calculation of the critical load at which a column is prone to buckling.
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Related Experiment Video

Updated: Nov 8, 2025

Generating Strictly Controlled Stimuli for Figure Recognition Experiments
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Euler diagrams drawn with ellipses area-proportionally (Edeap).

Michael Wybrow1, Peter Rodgers2, Fadi K Dib3

  • 1Faculty of Information Technology, Monash University, 3800, Clayton, Australia. Michael.Wybrow@monash.edu.

BMC Bioinformatics
|April 27, 2021
PubMed
Summary
This summary is machine-generated.

Edeap offers a scalable method for drawing area-proportional Euler diagrams using ellipses, balancing accuracy and usability. While eulerr is most accurate, Edeap provides a strong alternative with better scalability than venneuler.

Keywords:
Area proportionalDiagram generationEllipsesEuler diagramsMulti-criteria optimization

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Area of Science:

  • Data Visualization
  • Computational Biology
  • Network Analysis

Background:

  • Area-proportional Euler diagrams are vital for visualizing complex datasets in fields like microarray analysis and biosciences.
  • These diagrams are increasingly applied across diverse domains, including social networks and general data representation.

Purpose of the Study:

  • To introduce Edeap, a novel, simple, and scalable method for generating area-proportional Euler diagrams utilizing ellipses.
  • To provide a comprehensive evaluation of Edeap against existing state-of-the-art methods for Euler diagram generation.

Main Methods:

  • Edeap employs a search-based technique to optimize a multi-criteria objective function for area accuracy and usability.
  • The evaluation compares Edeap with circle-based (veneuler) and ellipse-based (eulerr) methods using real-world and synthetic datasets.

Main Results:

  • In terms of accuracy, the ordering from best to worst is eulerr, Edeap, and venneuler.
  • Edeap offers a balance between accuracy and runtime, outperforming eulerr in speed and venneuler in scalability.
  • Edeap and venneuler can handle up to 20 sets, while eulerr is limited to 11 sets.

Conclusions:

  • Edeap presents a scalable and effective solution for creating area-proportional Euler diagrams, particularly for larger datasets.
  • The choice between Edeap, eulerr, and venneuler depends on specific project needs regarding accuracy, runtime, and scalability.
  • Edeap is available as open-source software and a web tool, promoting accessibility for researchers.