Moebius Transformations Revealed
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it whenever someone needs a visual intuition for Moebius transformations, the Riemann sphere, and how they map circles to circles.
Recommended by 14 people across 14 different months, 2011 to 2022.
Why MathOverflow recommends it
MathOverflow users link it as a vivid visual illustration of Moebius transformations and stereographic projection, though one question points out the video's rotation claim is misleading.
Recommended by 1 people across 1 different months, 2010 to 2011.
What it is
A 2.5-minute short film that shows Moebius transformations of the complex plane as simple rigid motions of the Riemann sphere, using stereographic projection to move up a dimension. It was made by Jonathan Rogness with Douglas Arnold and set to Schumann's Kinderszenen.
Cited in 1 Wikipedia article, including August Ferdinand Möbius.
What people said
Strong rec Patrick Da Silva · 2017-09 · Mathematics Stack Exchange · on “What about image of infinity?”
You can deduce the image of $\infty$ by continuity of the application~; there is no need for an extra argument. In other words, if the image of the real line without the point at infinity is a circle minus a point, then the image of the real line with infinity will be a circle with that point. The key argument is really that Moebius transformations are invertible maps of the Riemann sphere preserving those "generalized circles" (which when you see the Riemann sphere as a geometrical sphere embedded in three-dimensional real space, those "generalized circles" are simply hyperplane sections of that sphere). I really recommend this "visualisation" proof [this video] , it is extremely well done. When
Recommends Thierry Zell · 2010-08 · MathOverflow · on “Proofs without words”
In the movie category, I'm surprised that no-one has yet posted a link to Moebius Transformations Revealed [this video] .
Recommends Álvaro Lozano-Robledo · 2011-08 · Mathematics Stack Exchange · on “Other awesome topology related videos like this one?”
Check out this Math Overflow thread [LINK] ! For instance, you may enjoy this video about the Möbius band [this video] .
Recommends John Adamski · 2012-04 · Mathematics Stack Exchange · on “A question about Möbius transformation in Complex Variables”
Here is a link to a very enlightening video. It may help build your intuition about what is behind these mappings. [this video]
Recommends Aaron Golden · 2012-08 · Mathematics Stack Exchange · on “Is there a hyperbolic geometry equivalent to Möbius transformations in spherical geometry?”
There is a sense in which all "interesting" properties of functions in spherical geometry are invariant under conjugation by a Möbius transformation. The reason is that the Möbius transformations correspond to "uninteresting" manipulations of the whole sphere, as illustrated in this video [this video] . Is there an equivalent notion in hyperbolic geometry? In other words, is there a valid statement of the form "All interesting properties of functions in hyperbolic geometry are invariant under conjugation by a $\text{Foo}$ because the $\text{Foo}$s correspond to uninteresting manipulations of the {upper half plane, unit disk, hyperboloid, etc}."?
Recommends KCd · 2013-01 · Mathematics Stack Exchange · on “Whats an extended complex plane?”
For some videos of the geometry behind stereographic projection (turning the complex plane into a sphere without the north pole, and vice versa), see [LINK] starting at 7:18, [LINK] starting at 6:21. The entire video [this video] exhibits the geometric unification that happens with Möbius transformations when you look at them on the Riemann sphere (aka extended complex plane) instead of in the complex plane by itself.
Recommends LucasVB · 2014-07 · Mathematics Stack Exchange · on “General method to "naturally interpolate" to a complex map?”
Given a region of the complex plane and a map $z \to f(z)$, is there a general way to "naturally interpolate" the point $z$ to $f(z)$ in such a way that the movement follows a "natural" smooth path that doesn't generate unnecessary "kinks" and overlaps? Background: I make educational animations [LINK] . A couple of projects I've been playing around with involve complex numbers. I'm trying to figure out a general method to animate complex maps that look good and natural in terms of smooth deformations of the complex plane. Forgive me, but I have no way to confidently and formally state this question at this point. But I can illustrate it. Here's a great video [this video] illustrating Möbius trans
Recommends MvG · 2014-12 · Mathematics Stack Exchange · on “How does the Möbius group act on circlines?”
The video Möbius Transformations Revealed [this video] (also mentioned in this question [LINK] might be a good thing to keep in mind as well.
Recommends Matan L · 2015-02 · Mathematics Stack Exchange · on “Conformal cobblestones”
I am not sure which have been used but you may find these interesting: $f(z) = \sin(z)$ : Vertical and horizontal lines are mapped to hyperbolas and ellipses. Another interesting map is $f(z) = z + \frac{1}{z}$ Here is the source [LINK] to these images which also explain about it in a deeper manner. You may also find interest at Möbius transformations [LINK] , which maps circles and lines to circles and lines: [this video] Another interesting article related: [LINK]
Recommends AdLibitum · 2015-04 · Mathematics Stack Exchange · on “Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions?”
HINT: This is an example of Möbius transformation [LINK] . Try to decompose the transformation $z\mapsto\frac{z+1}{2z+1}$ into simpler transformations as indicated in the linked page and follow the unit circle through. For a nice depiction of Möbius transformations, see here [this video] .