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Related Concept Videos

Polar and Cylindrical Coordinates01:22

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The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
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Curvilinear Motion: Polar Coordinates01:27

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In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
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The Frost circle or the inscribed polygon method is a graphical method for determining the relative energies of π molecular orbitals (MOs) for planar, fully conjugated, and monocyclic compounds. This method was first described by A. A. Frost and Boris Musulin in 1953.
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Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
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A neutral atom consists of a positively charged nucleus surrounded by a negatively charged electron cloud. When placed in an external electric field, the external electric force pulls the electrons and nucleus apart, opposite to the intrinsic attraction between the nucleus and the electrons. The opposing forces balance each other with a slight shift between the center of masses of the nucleus and the electron cloud, resulting in a polarized atom. On the other hand, a few molecules, like water,...
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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
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Related Experiment Video

Updated: Jun 3, 2025

Author Spotlight: Non-Invasive Imaging of Complex Bio-Structures Using Polarization-Sensitive Two-Photon Microscopy
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"Pattern regulation in epimorphic fields", aka the polar coordinate model.

Jonathan M W Slack1

  • 1Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.

Developmental Biology
|January 11, 2025
PubMed
Summary

The Polar Coordinate Model (PCM) explains insect and vertebrate appendage regeneration. This 1976 model

Keywords:
CockroachCricketDrosophilaLimb regenerationNewtPolar coordinate modelPositional informationRegeneration

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Area of Science:

  • Developmental biology
  • Regenerative medicine
  • Evolutionary developmental biology

Background:

  • The Polar Coordinate Model (PCM) was published in 1976.
  • It aimed to explain distal regeneration in insect and vertebrate appendages.
  • The model has been influential and frequently cited since its publication.

Purpose of the Study:

  • To detail the historical context and development of the Polar Coordinate Model.
  • To analyze the strengths and limitations of the PCM.
  • To assess the long-term impact and legacy of the PCM in the field of regeneration.

Main Methods:

  • Historical review of scientific literature leading to the PCM.
  • Analysis of the original publications and subsequent research citing the PCM.
  • Critical evaluation of the model's explanatory power and predictive capabilities.

Main Results:

  • The PCM provided a novel framework for understanding appendage regeneration patterns.
  • Key strengths included its ability to account for specific regeneration phenomena.
  • Identified weaknesses and limitations that spurred further research.

Conclusions:

  • The Polar Coordinate Model was a significant contribution to regeneration research.
  • Its impact is evident in its continued citation and influence on subsequent models.
  • The article provides a comprehensive overview of the PCM's scientific journey.