Parameterization of Sum of Squares

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Discussion Overview

The discussion revolves around the parameterization of the equation involving sums of squares, specifically exploring how to parameterize multiple equalities such as a^2+b^2=c^2=d^2=e^2+f^2. Participants seek clarification on the methods and steps involved in achieving such parameterizations.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants express uncertainty about how to arrive at parameterizations for the equations a^2+b^2=c^2 and a^2+b^2=c^2=d^2=e^2+f^2.
  • One participant suggests that the approach for the new equality is similar to previous cases and involves adding another angle.
  • Another participant asks for specific steps on how to parameterize the equality a^2+b^2=c^2=d^2=e^2+f^2.
  • There is a request for clarification on the parameterization of the first equality, a^2+b^2=c^2, and its relation to an angle θ.

Areas of Agreement / Disagreement

Participants do not seem to reach a consensus on the specific steps for parameterization, and multiple viewpoints regarding the approach remain present.

Contextual Notes

There are limitations in the discussion regarding the missing assumptions and specific mathematical steps required for the parameterization process.

patrickbotros
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I've seen the parameterization of a^2+b^2=c^2 and also a^2+b^2=c^2+d^2, but I don't know how they arrived at those parameterizations. Would it be possible to parameterize something with two equalities like a^2+b^2=c^2+d^2=e^2+f^2? Any help is appreciated!
 
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Yes, the idea is the same as for the others. Just add another angle.
 
Orodruin said:
Yes, the idea is the same as for the others. Just add another angle.
How? Specifically how to do the a^2+b^2=c^2+d^2=e^2+f^2
 
Start by parametrising the first equality, then the second. How do you parametrise the first?
 
Orodruin said:
Start by parametrising the first equality, then the second. How do you parametrise the first?
I don't know. That's what I'm asking. I just know what the answer ends up benign. Not the steps.
 
Do you understand why ##a^2 + b^2 = c^2## is parametrised by an angle ##\theta##?
 

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