Please help. Imposing a metric to preserve distance

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SUMMARY

The discussion focuses on the need to determine the appropriate metric to impose on R² that preserves distances when using the mapping defined by (r, φ) to (x, y) = (2tan(r/2)cos(φ), 2tan(r/2)sin(φ)). The participants seek clarity on the mathematical implications of this mapping and its effects on distance preservation. The inquiry emphasizes the importance of understanding metrics in the context of differential geometry.

PREREQUISITES
  • Understanding of differential geometry concepts
  • Familiarity with polar coordinates and their transformations
  • Knowledge of metric spaces and distance preservation
  • Basic skills in mathematical analysis
NEXT STEPS
  • Research the properties of metrics in R²
  • Study the implications of distance-preserving mappings in differential geometry
  • Explore the concept of geodesics in curved spaces
  • Learn about the role of Jacobians in transformations
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Mathematicians, students of geometry, and anyone interested in the study of metrics and transformations in R².

whattttt
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If we use the mapping
(r,phi)---->(x,y)=
(2tan(r/2)cos(phi),
2tan(r/2)sin(phi))
Which metric do we have to impose on R^2 in order that the mapping preserves distance. Any help would be greatly appreciated. Thank you very much
 
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hi whattttt! :smile:

(have a phi: φ :wink:)

tell us what you think, and we'll comment :smile:
 
Nice
 

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