Join me and a group of fellow learners to learn about the fundamentals of complex analysis using the powerful tools for symbolic computation and visualization in Wolfram Language. Our topics for the study group include elementary complex functions, the Cauchy-Riemann equations, complex integration, Cauchy’s theorem and the residue theorem.
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Please look at the 3rd question in Quiz 1. The question asks whether a given point is stereographically projected into the lower or the upper hemisphere of the Riemann sphere. In fact, the magnitude of the point is exactly 1, so that the given point lies on the equator. But that’s not an available answer.
I find the projection of the Reimann sphere is extremely distorted. I prefer that both sides of the sphere be projected on the inner disk separately so that the two projections have symmetry.
So, I asked this question during the lecture and Marco did not have an answer for it. I had never heard of the stereographic projection of complex numbers. I was wondering if anybody knows of a practical application for it. It seemed to me that it might be used in modulation/coding theory or maybe image recognition problems but I’m just shooting in the dark. Anybody know?
There might be an error in the proof for the formulas of the stereographic projection:
“… And because the triangles (0,z,N) and (z’,z,Overscript[z, ^]) are similar, then |z’|/|z|=1/(1-Z)…”
correct: |z|/|z’|=1/(1-Z) <— This formula was used for the rest of the proof. The formulas itself are valid.
Quiz 1 Question 3: Is the stereo graphic representation of z1=1/2+i*(3^1/2)/2 in the northern (upper?) or southern (lower?) hemisphere?
The text says: “Numbers with absolute value less than 1 are mapped to the southern hemisphere, and 0 to the south pole. Numbers with absolute value greater than 1 are mapped to the northern hemisphere.”
Abs[z1] = 1, but the answer “in the upper hemisphere” comes back wrong. Can you explain?
Can you comment on the significance or the application of the Riemann sphere concept? Why should it matter how lines or circles in the complex plane map to the sphere?
About applications of the Riemann sphere and the stereographic projection, others in this forum have mentioned cartography, but that does not require complex numbers.
The only “practical” application of the Riemann sphere I know of is to Lorentz transformations.
The situation is this: imagine two observers moving at very large relative velocities. By a relativistic effect, the two observers will see the stars in different positions. How to relate these positions? In this problem it can actually be useful to think of the sky as a Riemann sphere for the purpose of calculation. Then, the position of a star will be given by a complex number and the problem becomes how to relate the position of the same star as seen by the two observers, or to relate the two complex numbers.
More precisely, the Lorentz transformations of the “celestial sphere” can be described by conformal transformations of the Riemann sphere. So the theorems about the stereographic projection (circles mapped to circles etc.) become useful to derive properties about Lorentz transformations. For example, a perfectly circular constellation should remain circular as seen by the second observer.
Anyway I am no expert, so I refer you to Chapter 1 of Vol. 1 of “Spinors and Spacetime” by Penrose and Rindler, or perhaps to Chapter 18 of “The road to reality” by Penrose (and I apologize for any wrong statements).
