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fermatslibrary's profile
Fermat's Library
Fermat's Library
Fermat's Library
@fermatslibrary

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Fermat's Library

@fermatslibrary

A platform for illuminating academic papers. We publish an annotated paper every week. Our chrome extension for arXiv: https://fermatslibrary.com/librarian 

fermatslibrary.com
Joined September 2015

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    Fermat's Library‏ @fermatslibrary Mar 12
    • Report Tweet
    • Report NetzDG Violation

    The way to correctly generate a random point on the surface of a unit sphere is not to pick uniform distributions θ in [0,2π) and φ in [0,π), but instead choose u and v from uniform distributions on [0,1). Then φ = cos⁻¹(2v-1) θ = 2πupic.twitter.com/Y6IX23ZGIv

    6:04 AM - 12 Mar 2020
    • 1,139 Retweets
    • 5,772 Likes
    • Alfons Nilsson Art Vandelay Srta. Mazapán Dmitriy Adrian K. Abhay Hegde Strombavioli Veera “matt” silva
    55 replies 1,139 retweets 5,772 likes
      1. New conversation
      2. Area Man  ✍‏ @CappyAtLast Mar 12
        • Report Tweet
        • Report NetzDG Violation
        Replying to @fermatslibrary

        I didn't understand a single word but I feel much smarter having read this.

        1 reply 2 retweets 63 likes
      3. Kuba Szymanowski 🇵🇱‏ @KKSzymanowski Mar 12
        • Report Tweet
        • Report NetzDG Violation
        Replying to @CappyAtLast @fermatslibrary

        If you pick random latitude and longitude values you won't get a uniform distribution of the points on Earth. You will statistically get more points towards the poles which you can see on the animation.

        2 replies 5 retweets 136 likes
      4. 1 more reply
      1. New conversation
      2. Gregory‏ @ggref Mar 12
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        Replying to @fermatslibrary

        Can someone explain why the first method is incorrect?

        4 replies 0 retweets 6 likes
      3. Hélvio Vairinhos‏ @hvairinhos Mar 12
        • Report Tweet
        • Report NetzDG Violation
        Replying to @ggref @fermatslibrary

        Because latitude circles shrink towards the poles. Uniform selection of φ makes short & large circles equiprobable, so short circles “fill up” faster with random noise. To “fill up” all latitude circles at the same rate, one must select uniformly wrt perimeter, not φ.

        1 reply 1 retweet 43 likes
      4. 1 more reply
      1. New conversation
      2. Binky‏ @MH_Binky Mar 12
        • Report Tweet
        • Report NetzDG Violation
        Replying to @fermatslibrary

        Put simply, the incorrect version: Gets a circle's circumference, then Rotates it 180° in 3D space to make the sphere. Here's the problem: When you rotate the circle, the part of the circle near to axis of rotation does not travel nearly as far as the rest of the circle. And so:

        2 replies 0 retweets 17 likes
      3. Binky‏ @MH_Binky Mar 12
        • Report Tweet
        • Report NetzDG Violation
        Replying to @MH_Binky @fermatslibrary

        If you pick 10 points near the axis of rotation, they will be much closer to each other than the rest of the surface (see the clusters around the two "poles"). The solution: Make the first step generate fewer points near the axis and more elsewhere.

        2 replies 0 retweets 7 likes
      4. 6 more replies
      1. New conversation
      2. Trevor Hawkes‏ @trevorphawkes Mar 12
        • Report Tweet
        • Report NetzDG Violation
        Replying to @fermatslibrary

        A simple and more intuitive approach is to generate a random X, Y, and Z, then scale your random vector to have length equal to the radius of the sphere.

        1 reply 0 retweets 12 likes
      3. Foldster‏ @theFoldster Mar 12
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        • Report NetzDG Violation
        Replying to @trevorphawkes @fermatslibrary

        This is almost correct, but will concentrate points from the corners and edges of the cube more than the sides. You need to throw out any generated random points that lie outside of the sphere.

        2 replies 0 retweets 19 likes
      4. 1 more reply
      1. Roger Sauer‏ @rsauer3473 Mar 12
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        Replying to @fermatslibrary

        This was correctly pointed out in the Star Trek episode featuring the crew’s landing on Talos 4. That Spock was something else.

        0 replies 0 retweets 26 likes
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      1. New conversation
      2. Mauricio 4.6.0‏ @mauricio_agg Mar 12
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        Replying to @fermatslibrary

        2 sqrt(-1)???pic.twitter.com/hBrnad8mwb

        1 reply 0 retweets 3 likes
      3. David Swart‏ @dmswart Mar 12
        • Report Tweet
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        Replying to @mauricio_agg @fermatslibrary

        That's a variable v not a √

        1 reply 0 retweets 8 likes
      4. 1 more reply

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