[HN Gopher] The Deceptively Asymmetric Unit Sphere
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The Deceptively Asymmetric Unit Sphere
Author : ThatGeoGuy
Score : 86 points
Date : 2024-11-22 16:00 UTC (7 hours ago)
(HTM) web link (www.tangramvision.com)
(TXT) w3m dump (www.tangramvision.com)
| ssivark wrote:
| 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/vi...
| jumping_frog wrote:
| Can this playlist based on the book be a good substitute for
| the book?
|
| https://www.youtube.com/watch?v=mKtctCyd0rs&list=PLWEiAJhCw-...
| nyrikki wrote:
| 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 wrote:
| 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 wrote:
| Ooh thanks, looks really nice!
| Animats wrote:
| No mention of quaternions and SLERP?
| ChickenSando wrote:
| 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.
| itishappy wrote:
| > 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 wrote:
| 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 wrote:
| 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 wrote:
| 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
| ChickenSando wrote:
| 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.
| griffzhowl wrote:
| I think where you might lose some of the uninitiated in this
| post is in introducing the term "operator" without definition
| or illustration
| Etheryte wrote:
| It's been way too many years since uni for me to do any
| rigorous math, so what tends to help me is to try and get an
| approximate intuition instead. As a specific example, this
| article talks a lot about manifolds. Since I didn't study math
| in English, I don't know what that is, so I go and look it up.
| A simplified, but intuitive model [0] might be:
|
| > A manifold is a space that is locally Euclidean, but globally
| might be complicated, e.g. a torus or sphere, or etc.
|
| Okay, so that's reasonably simple. As an intuition, if we
| imagine a 2D character living on the surface of the sphere, if
| they walk forward, from their perspective they just move
| forward in 2D, but from our outside perspective, they're moving
| in 3D on the curved surface of a sphere.
|
| Once I have this, I try and read until I get lost again. I
| don't try to rigorously solve or follow through with each
| equation, but to rather approximately understand what the idea
| is.
|
| [0] https://math.stackexchange.com/q/1211762/128941
| empath75 wrote:
| You just don't know enough about the foundations the article is
| building off of to follow it. What I do in such a case, is
| forget about the article at hand, and make a list of things
| that I don't understand and then try and learn about them all
| individually. It's okay to just bounce off of a technical
| article and use it as motivation to learn more about the
| subject from other sources.
|
| For example in this article, I got completely stuck on how the
| Exp and Log functions he's talking about relate to the usual
| definitions of those functions, so now I'm going down that
| rabbit hole.
| panic wrote:
| What is it you're trying to get out of reading these types of
| articles? In my experience, it's hard to really retain info
| unless I'm actively working on a related problem, so reading
| something like this out of the blue is mostly just
| entertainment (and a way to check existing knowledge).
| griffzhowl wrote:
| Learning maths is all about having the proper prerequisites
| (and time and effort...). The concepts all build on simpler
| ones in a hierarchy leading down to our basic ideas of numbers
| and space, so if you're missing any of those simpler ones you
| simply won't be able to understand anything more advanced or
| exotic except in a very fragmentary or superficial way.
|
| For differential geometry the prerequisites are linear algebra
| and multi-variable calculus, and the prerequisite for multi-
| variable calculus is single variable calculus, and the
| prerequisites for each of those is basic algebra, trigonometry,
| and elementary geometry.
|
| You don't need to know everything about each of these to get
| things at the next level, but a thorough grounding in the
| basics is essential in my experience. There's a reason every
| STEM field begins with calculus and linear algebra - they're
| used everywhere in anything at a higher level. Once you get
| through those you will find things open up for you.
|
| I don't know your level so it's difficult to make any concrete
| recommendations, but in general I find Lang's books to be clear
| and efficient sources. His Short Calculus covers all the basics
| of single variable calculus in less than 200 pages, instead of
| ~500 pages like many intro to calc books. Similarly his
| Calculus of Several Variables is ~300 pages instead of 500-700.
| Alternatively a mathematical methods for physicists book like
| the one by Riley, Hobson, and Bence might suit you. It's huge
| (~1300 pages), but you can pick out the chapters you want to
| learn from and it builds up from the basics to some quite
| sophisticated mathematics and has references if you want more
| depth on some topic, and great problems.
|
| I find I don't really learn anything from watching videos. They
| can be complementary, but most of the learning with maths comes
| from doing problems after reading through an introduction to
| the concepts
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