Solving a Geometric Problem with Friends

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SUMMARY

The discussion centers on proving that a conformal bijective map between two rectangles in the complex plane is linear. Participants emphasize that the map is uniquely defined by its action on three points, suggesting a manual verification of various cases to identify geometric intuitions. The conclusion drawn is that examining specific configurations will reveal the linear nature of the mapping.

PREREQUISITES
  • Understanding of conformal mappings in complex analysis
  • Familiarity with bijective functions and their properties
  • Knowledge of geometric transformations in the complex plane
  • Basic skills in manual verification of mathematical cases
NEXT STEPS
  • Research the properties of conformal maps in complex analysis
  • Study the implications of bijective functions in geometric contexts
  • Explore examples of linear transformations in the complex plane
  • Investigate manual case verification techniques in mathematical proofs
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in geometric transformations and their properties in the complex plane.

henry1964
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A couple friends worked on this problem (for a week now...)

Trying to show that a conformal bijective map that sends vertices of one rectangle to vertices of another rectangle on the complex plane has to be linear.
I would appreciate any help, Thank you.
 
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Well, the map is completely determined by its action on three points, right? Just check every possible case by hand. One or more should be "obvious", geometrically speaking.
 

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