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!
Projecting the line y=x onto the Riemann Sphere
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complex-analysis
complex-numbers
stereographic-projections
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0Considering the tag you attached, you want to find the image circle of (the inverse of) the stereographic projection of the line $y=x$, right? – 2017-02-15
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0Sorry 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). – 2017-02-15
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0The line $y=x$ becomes a maximum circle on the Riemann sphere so its radius is the radius of the sphere. This is your question? – 2017-02-15
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0Your 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. – 2017-02-15
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0@EmilioNovati Yes that was my question. I knew that was going to be the projection. But how can I arrive to the solution algebraically? – 2017-02-15
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0@cjackal I was asking what set of points are on the Riemann Sphere after you stereographicaly-project the points on the line y=x to the sphere. – 2017-02-15
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0If you like the answer, would you please accept it? – 2017-02-15
1 Answers
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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:
(The lines from the North pole to the line $y=x$ form a plane perpendicular to the $x$ plane.)
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0Thank you for the great image. – 2017-02-15