38 ms·

The Deceptively Asymmetric Unit Sphere

139 points by ThatGeoGuy 2y ago

ssivark 2y ago
Very nice "visual" introduction a topic that's usually treated very abstractly in math textbooks! If you'd like more of such a visual perspective on differential geometry, I recommend Tristan Needham's book [1]. [1]: https://press.princeton.edu/books/paperback/9780691203706/visual-differential-geometry-and-forms https://press.princeton.edu/books/paperback/9780691203706/vi...
- jumping_frog 2y ago
  Can this playlist based on the book be a good substitute for the book? https://www.youtube.com/watch?v=mKtctCyd0rs&list=PLWEiAJhCw-zRuHqkA0tYZ0_NjxBoZWqPh&index=1 https://www.youtube.com/watch?v=mKtctCyd0rs&list=PLWEiAJhCw-...
  - nyrikki 2y ago
    Depends on how much of the work you do yourself. Math is like skiing or playing the guitar, you don't get better by watching others do it. Personally I find videos useful to augment books, but rarely a substitute for them. But I am bad about pausing, ruminating and practicing, you may be more successful than I. But practice is required IMHO.
    - xanderlewis 2y ago
      Well... there are some things one can get better at by watching — chess, for example. However, of course, you're right: in mathematics (and probably chess?) 90% of the learning has to be done yourself.
- sourcepluck 2y ago
  Ooh thanks, looks really nice!
Animats 2y ago
No mention of quaternions and SLERP?
- ChickenSando 2y ago
  Hey, I'm the author of this post. Quaternions and SLERP are absolutely a fundamental part of 3D vision (and game development too). However, I wanted to focus this post mainly on the question "why is optimizing on the unit sphere difficult?" As the post stands, it's already quite verbose. Maybe I'll find some time to do a deep dive on common Lie Groups used in computer vision e.g. SO(3), SE(3) and Sim(3) and also the common representations used for those groups.
  - VyseofArcadia 2y ago
    I'd love such an article. I wish someone had told me about these computer vision applications of Lie groups when I was writing my dissertation on Lie groups, but better late than never!
  - itishappy 2y ago
    > Maybe I'll find some time to do a deep dive on common Lie Groups used in computer vision e.g. SO(3), SE(3) and Sim(3) and also the common representations used for those groups. +1 Also, great article!
hammock 2y ago
Another awesome mathematics article that loses me about 10-15% of the way in do to my own technical limitations. Any tips from HN on how to improve my ability to get thru, say, 45-50% of these types of articles?? Generally speaking, not specific to the math in OP article
- jtimdwyer 2y ago
  To be clear I am not being sarcastic in saying this but the only method I’ve found to work with any consistency is: Try, try again.
  - Lerc 2y ago
    I find this is what works for me. I seem to be quite a nonlinear learner. I struggle with the methodical x leads to y leads to z approach. I tend to try and take on the whole thing and not really understand it then repeat the process (often from different sources) after a while I just seem to understand more and more
- deleted 2y ago
  [deleted]
- ChickenSando 2y ago
  Author of this post. I have an undergraduate in Applied Mathematics and my training in the "definition -> proposition -> proof" style of mathematics probably comes through in the article more than I wanted it to. That being said, I began studying Differential Geometry and Lie Groups as part of my graduate degree in Electrical Engineering. Engineers think about problems very differently than mathematicians and I've benefited a lot from taking a more geometric-based and visual approach to learning in the years following my undergraduate. So, my prescription would be to play around with math ideas when you see them. Create a script to draw what you are trying to visualize. This was my first time using the `manim` library and I gained a deeper appreciation and intuition for the ideas presented in the article even though I've studied them dozens of times! Overall, learning math is a slow and deliberate exercise. Don't get down on yourself if you don't understand something at first glance. Feel free to pause, verify an idea (either visually or with a formal proof) and then continue on a more firm base of understanding.
red_trumpet 2y ago
Were does the notation T^2 for oriented real projective space come from? That's just bad, because it is not a torus but a sphere, and the two are topologically very different!
- kevinventullo 2y ago
  If you click through the first link there’s an explanation: “In precise mathematical terms, this set of rays is called the oriented real projective plane and is commonly denoted by T^2. If you’ve seen this terminology before, you’ll notice that this is a torus. This is because in real-projective geometry, we also add the points and lines “at infinity”.”
  - lupire 2y ago
    It's incorrect, though. The oriented real projective plane is a sphere, not a torus. The projective points at infinity (one point for every 1-D angle (R mod 2pi)) form the equator of the sphere. The T in T² is for "two-sided" , not Torus. The torus explanation that Tangram gives doesn't make sense. In a pinhole projection, the horizontal and vertical infinites do not "wrap around" to meet. There is no meaningful "horizontal" and "vertical", the system is rotationally symmetric, which forms a hemisphere of curve it to make it compact. (Half sphere because you can only see one half of the space outside a pinhole camera) https://en.m.wikipedia.org/wiki/Oriented_projective_geometry https://en.m.wikipedia.org/wiki/Oriented_projective_geometry
    - ChickenSando 2y ago
      You're right. I've messed up. The notation here is a bit misleading and I didn't dig as deep here as I should have. I'll fix the website soon.
      - erwincoumans 2y ago
        Thanks for sharing such an insightful article of complex material. Some follow-up showing how this helps optimization (gradient descent, Newton solvers?) would be great. >> For convenience of notation, we’ll drop the explicit parametrization of the curves and denote this vector shift as [...] FYI for clarity I wish the explicit parametrization was kept, even though it is more verbose.
bryan0 2y ago
I'm not sure if this is the main point of the article (but it does at least mention it), a good intuitive visualization of how the sphere can be thought of as asymmetric is the Hairy Ball Theorem[0], which basically states: "you can't comb a hairy ball flat without creating a cowlick" [0]: https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem
wcrossbow 2y ago
Great example of a corporate blog! My immediate reaction is I want to work in this team.
ykonstant 2y ago
Something that may be of interest to CS people and reveals the complexity of the unit sphere is the following problem: Find an efficient (class of) algorithm(s) to select a large number N of uniformly distributed points on S^2, where "uniformly distributed" is given in a more flexible sense than the usual one. For instance, you may want to minimize the Weyl discrepancy between average and integral, or you may want to focus more on minimizing the number of ε-clusters of distances. One of the most elegant approaches to this problem is the classic work of Lubotzky, Phillips and Sarnak: Hecke Operators and Distributing Points on the Sphere I and II. They translate the problem to one of generating good sequences of elements of SO(3), which they attack with a combination of harmonic analysis on the semisimple groups, homogeneous dynamics and number theory with Hecke operators as their central tool.

139 points by ThatGeoGuy 2y ago

Very nice "visual" introduction a topic that's usually treated very abstractly in math textbooks! If you'd like more of such a visual perspective on differential geometry, I recommend Tristan Needham's book [1]. [1]: https://press.princeton.edu/books/paperback/9780691203706/visual-differential-geometry-and-forms https://press.princeton.edu/books/paperback/9780691203706/vi...

jumping_frog 2y ago
Can this playlist based on the book be a good substitute for the book? https://www.youtube.com/watch?v=mKtctCyd0rs&list=PLWEiAJhCw-zRuHqkA0tYZ0_NjxBoZWqPh&index=1 https://www.youtube.com/watch?v=mKtctCyd0rs&list=PLWEiAJhCw-...
- nyrikki 2y ago
  Depends on how much of the work you do yourself. Math is like skiing or playing the guitar, you don't get better by watching others do it. Personally I find videos useful to augment books, but rarely a substitute for them. But I am bad about pausing, ruminating and practicing, you may be more successful than I. But practice is required IMHO.
  - xanderlewis 2y ago
    Well... there are some things one can get better at by watching — chess, for example. However, of course, you're right: in mathematics (and probably chess?) 90% of the learning has to be done yourself.
sourcepluck 2y ago
Ooh thanks, looks really nice!

Animats 2y ago

No mention of quaternions and SLERP?

ChickenSando 2y ago
Hey, I'm the author of this post. Quaternions and SLERP are absolutely a fundamental part of 3D vision (and game development too). However, I wanted to focus this post mainly on the question "why is optimizing on the unit sphere difficult?" As the post stands, it's already quite verbose. Maybe I'll find some time to do a deep dive on common Lie Groups used in computer vision e.g. SO(3), SE(3) and Sim(3) and also the common representations used for those groups.
- VyseofArcadia 2y ago
  I'd love such an article. I wish someone had told me about these computer vision applications of Lie groups when I was writing my dissertation on Lie groups, but better late than never!
- itishappy 2y ago
  > Maybe I'll find some time to do a deep dive on common Lie Groups used in computer vision e.g. SO(3), SE(3) and Sim(3) and also the common representations used for those groups. +1 Also, great article!

hammock 2y ago

Another awesome mathematics article that loses me about 10-15% of the way in do to my own technical limitations. Any tips from HN on how to improve my ability to get thru, say, 45-50% of these types of articles?? Generally speaking, not specific to the math in OP article

jtimdwyer 2y ago
To be clear I am not being sarcastic in saying this but the only method I’ve found to work with any consistency is: Try, try again.
- Lerc 2y ago
  I find this is what works for me. I seem to be quite a nonlinear learner. I struggle with the methodical x leads to y leads to z approach. I tend to try and take on the whole thing and not really understand it then repeat the process (often from different sources) after a while I just seem to understand more and more
deleted 2y ago
[deleted]
ChickenSando 2y ago
Author of this post. I have an undergraduate in Applied Mathematics and my training in the "definition -> proposition -> proof" style of mathematics probably comes through in the article more than I wanted it to. That being said, I began studying Differential Geometry and Lie Groups as part of my graduate degree in Electrical Engineering. Engineers think about problems very differently than mathematicians and I've benefited a lot from taking a more geometric-based and visual approach to learning in the years following my undergraduate. So, my prescription would be to play around with math ideas when you see them. Create a script to draw what you are trying to visualize. This was my first time using the `manim` library and I gained a deeper appreciation and intuition for the ideas presented in the article even though I've studied them dozens of times! Overall, learning math is a slow and deliberate exercise. Don't get down on yourself if you don't understand something at first glance. Feel free to pause, verify an idea (either visually or with a formal proof) and then continue on a more firm base of understanding.

red_trumpet 2y ago

Were does the notation T^2 for oriented real projective space come from? That's just bad, because it is not a torus but a sphere, and the two are topologically very different!

kevinventullo 2y ago
If you click through the first link there’s an explanation: “In precise mathematical terms, this set of rays is called the oriented real projective plane and is commonly denoted by T^2. If you’ve seen this terminology before, you’ll notice that this is a torus. This is because in real-projective geometry, we also add the points and lines “at infinity”.”
- lupire 2y ago
  It's incorrect, though. The oriented real projective plane is a sphere, not a torus. The projective points at infinity (one point for every 1-D angle (R mod 2pi)) form the equator of the sphere. The T in T² is for "two-sided" , not Torus. The torus explanation that Tangram gives doesn't make sense. In a pinhole projection, the horizontal and vertical infinites do not "wrap around" to meet. There is no meaningful "horizontal" and "vertical", the system is rotationally symmetric, which forms a hemisphere of curve it to make it compact. (Half sphere because you can only see one half of the space outside a pinhole camera) https://en.m.wikipedia.org/wiki/Oriented_projective_geometry https://en.m.wikipedia.org/wiki/Oriented_projective_geometry
  - ChickenSando 2y ago
    You're right. I've messed up. The notation here is a bit misleading and I didn't dig as deep here as I should have. I'll fix the website soon.
    - erwincoumans 2y ago
      Thanks for sharing such an insightful article of complex material. Some follow-up showing how this helps optimization (gradient descent, Newton solvers?) would be great. >> For convenience of notation, we’ll drop the explicit parametrization of the curves and denote this vector shift as [...] FYI for clarity I wish the explicit parametrization was kept, even though it is more verbose.

bryan0 2y ago

I'm not sure if this is the main point of the article (but it does at least mention it), a good intuitive visualization of how the sphere can be thought of as asymmetric is the Hairy Ball Theorem[0], which basically states: "you can't comb a hairy ball flat without creating a cowlick" [0]: https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem

wcrossbow 2y ago

Great example of a corporate blog! My immediate reaction is I want to work in this team.

ykonstant 2y ago

Something that may be of interest to CS people and reveals the complexity of the unit sphere is the following problem: Find an efficient (class of) algorithm(s) to select a large number N of uniformly distributed points on S^2, where "uniformly distributed" is given in a more flexible sense than the usual one. For instance, you may want to minimize the Weyl discrepancy between average and integral, or you may want to focus more on minimizing the number of ε-clusters of distances. One of the most elegant approaches to this problem is the classic work of Lubotzky, Phillips and Sarnak: Hecke Operators and Distributing Points on the Sphere I and II. They translate the problem to one of generating good sequences of elements of SO(3), which they attack with a combination of harmonic analysis on the semisimple groups, homogeneous dynamics and number theory with Hecke operators as their central tool.