I am trying to project the line "y=x" on the complex plane including the point at infinity to the Riemann Sphere. I know the projection is a circle, but I want to understand how to find the radius of the circle on the sphere. Any hints are appreciated. Thank you!

  • Considering the tag you attached, you want to find the image circle of (the inverse of) the stereographic projection of the line $y=x$, right? – cjackal Feb 15 '17 at 7:52
  • Sorry for not being clear. Here is a better way to phrase the question. Describe the projection on the Riemann Sphere of the following set in the complex plane, the line y=x (including the point at infinity). – GentGjonbalaj Feb 15 '17 at 7:55
  • The line $y=x$ becomes a maximum circle on the Riemann sphere so its radius is the radius of the sphere. This is your question? – Emilio Novati Feb 15 '17 at 7:59
  • Your question is a total nonsense at least in two points: the meaning of the projection is not settled, and there's many(infinitely many) ways to endow a metric structure on the Riemann sphere. But I guess the second ambiguity can be removed when you make it clear by what the projection is. – cjackal Feb 15 '17 at 7:59
  • @EmilioNovati Yes that was my question. I knew that was going to be the projection. But how can I arrive to the solution algebraically? – GentGjonbalaj Feb 15 '17 at 8:09

Why do you need an algebraic proof? The following figure depicts intuitively that the image of the line on the sphere is a great circle: enter image description here

(The lines from the North pole to the line $y=x$ form a plane perpendicular to the $x$ plane.)

  • Thank you for the great image. – GentGjonbalaj Feb 15 '17 at 21:29

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