Differential Geom: Showing Hyperbolic Circle from Euclidean Circle

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SUMMARY

This discussion focuses on demonstrating that a Euclidean circle centered at the origin can be represented as a hyperbolic circle using the metric defined by E = G = 4/(1-u²-v²)² and F = 0. The participants explore the integration of the expression √(E(u')² + 2Fu'v' + G(v')²) to derive the hyperbolic radius from a given Euclidean radius r. The discussion emphasizes the importance of using appropriate parametrized curves to facilitate this integration, specifically in the context of hyperbolic geometry.

PREREQUISITES
  • Understanding of hyperbolic geometry concepts
  • Familiarity with Euclidean geometry and circles
  • Knowledge of differential calculus and integration techniques
  • Proficiency in using metrics in geometric contexts
NEXT STEPS
  • Research the properties of hyperbolic circles in hyperbolic geometry
  • Learn about parametrization techniques for curves in differential geometry
  • Study the integration of metric expressions in geometric contexts
  • Explore the implications of metric transformations in geometry
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Mathematicians, geometry enthusiasts, and students studying differential geometry or hyperbolic geometry who seek to understand the relationship between Euclidean and hyperbolic circles.

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Given \{(u,v)\inR^2:u^2+v^2<1\} with metric E = G =\frac{4}{(1-u^2-v^2)^2} and F = 0. How can I show that a Euclidean circle centered at the origin is a hyperbolic circle?
 
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Note: rotations about the origin preserve this metric.
 
Given \{(u,v)\inR^2:u^2+v^2<1\} with metric E = G =\frac{4}{(1-u^2-v^2)^2} and F = 0.
With a Euclidean circle centered at the origin with radius r, how can I find the hyperbolic radius by integrating \sqrt(E(u')^2+2Fu'v'+G(v')^2), what parametrized curve should I use?
 

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