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The Deceptively Asymmetric Unit Sphere

tangramvision.com

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Re: The Deceptively Asymmetric Unit Sphere

#21

Post by kevinventullo »

red_trumpet wrote: post #20

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!

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”.”

Re: The Deceptively Asymmetric Unit Sphere

#22

Post by School-Cotton »

hammock wrote: post #5

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

Study more math? It’s hard and takes a long time; there isn’t really a royal road.

Re: The Deceptively Asymmetric Unit Sphere

#23

Post by bryan0 »

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

Re: The Deceptively Asymmetric Unit Sphere

#24

Post by lupire »

kevinventullo wrote: post #21
red_trumpet wrote: post #20

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!

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”.”

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

Re: The Deceptively Asymmetric Unit Sphere

#25

Post by keithalewis »

School-Cotton wrote: post #22
hammock wrote: post #5

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

Study more math? It’s hard and takes a long time; there isn’t really a royal road.

Never underestimate the power of das blinkenlights to entertain lazy dumb people.

Re: The Deceptively Asymmetric Unit Sphere

#26

Post by VyseofArcadia »

ChickenSando wrote: post #7
Animats wrote: post #4

No mention of quaternions and SLERP?

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…

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!

Re: The Deceptively Asymmetric Unit Sphere

#27

Post by ChickenSando »

lupire wrote: post #24
kevinventullo wrote: post #21

Earlier quoted context omitted.

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”.”

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 "hor…

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.

Re: The Deceptively Asymmetric Unit Sphere

#28

Post by kevinventullo »

lupire wrote: post #24
kevinventullo wrote: post #21

Earlier quoted context omitted.

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”.”

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 "hor…

Indeed, I should have been a more critical reader!

Re: The Deceptively Asymmetric Unit Sphere

#29

Post by erwincoumans »

ChickenSando wrote: post #27
lupire wrote: post #24

Earlier quoted context omitted.

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 "hor…

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.

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.

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