Stereographic Projection Tutorial Using Line Parametrization

[@adityabaghel1270]

Wow every little detail explained so wonderfully I'm amazed! Looking forward for new videos. Great job

[@manuspillai2389]

You should get more views. Great content!

[@curtischee2532]

Very good presentation. I am doing self study on manifolds. S^2 is an example of a manifold, so your video was very helpful. I hope you do videos on manifolds.

[@aaronTNGDS9]

Great presentation. Clear and concise.

[@Apubams]

Thank you a lot, I've been looking for these explanation for long time, you made an excellent work. Amazing!

[@sweardog]

Glad I could help you out!

[@erikross-rnnow5517]

Yo! Thank you so much - this was an amazing introduction to this topic for me. I was really struggling with this in my textbook's exercises and you made it sound so simple! YOU A REAL ONEEEE

[@sweardog]

Lol I really like this comment. Glad I could help!

[@PSP1S0]

Nice was able to apply this for a game using a semisphere and projecting a map.

[@sweardog]

Awesome! Is there anything that I can see yet?

[@PSP1S0]

@@sweardog Yes, though mostly developing this to support projects for someone else. Tried to reply a few times, but either my comments are held for review or deleted cause they contain links. To describe the current implementation. Debug visualization from and to P and Q results. draw a plane using its verts as Q input and move vertices of said plane to P based on a t value. Texture wrapping is arbitrary. Still working on depth mapping to create a 3d earth. Calculations and stuff is all still done by the CPU, will transfer most of the code to the GPU. This to speed up mesh generation above 100 subdivisions (the point where it shows slow downs). And apply depth with moving height maps, use these to translate the P vectors.

[@adrian-x5v]

nice video!

[@kylepls]

Great video, thanks! :)

[@swearTV]

You're welcome! Thanks for leaving a comment!

[@danielbiernat3299]

That was an amazing explanation, thanks so much!

[@sweardog]

You're welcome! I'm glad I could help.

[@jonbovi-69]

Subscribed! Thank you so much. This is very helpful and was a clear and great explanation.

[@sweardog]

You're Welcome! Glad I could help. New video on stereographic projection dropping soon!

[@hi_im_buggy]

Thank you, this was very helpful!

[@sweardog]

You're welcome! I'm glad I could help!

[@Gerardog]

Clear explanation. Great animation. Well done! 🔥

[@sweardog]

Thanks!

[@איתיימין-נ6מ]

Great!

[@jason_liam]

Great video . How can we do this for a circle and a line? Can you create a video for that. Thanks. Your channel deserves more subscribers and views.

[@sweardog]

I'll at least tell you and maybe form it into a video later. First, draw a centered-at-origin unit circle in the R^2. Next, draw a line in R^2 at height y = -1. This set-up is the same as my set-up in the video, except that it's in R^2 instead of R^3. Also, in this circe-to-line set up that I'm suggesting, giving the three points of the diagram the same names, this would mean that N = (0, 1), P = (p1, p2), and Q = (q1, -1). If you follow the same steps as in this video with these points, you should be able to find the forward function: f(p1, p2) = (2p1/(1-p2), -1) and the inverse function: g(q1, -1) = (4q1/(q1^2 + 4), (q1^2 - 4)/(q1^2 + 4)). I've been thinking about remaking this video in manim - the 3b1b math animation python library -and maybe further explaining a few things such as how it's done from a circle to line, sphere to plane, hypersphere to R3, and ultimately an n-sphere to n-space.

[@jason_liam]

@@sweardog Ok thanks man. Yeah i have used manim myself, it is good for animation..

[@ebshadow7974]

Hi! first of all thank you for this video its great. I have a question can we use Spherical coordinate system paramets for 'p' and still is it Possible to make Stereographic Projection for that?

[@sweardog]

Thanks! Yes, we can redefine these functions using 2-dimensions. For the forward function, simply replace p1, p2, and p3 with their spherical representations. I like theta as polar angle and beta as planar angle, so p1 = cos(theta)cos(beta), p2 = cos(theta)sin(beta), p3 = sin(theta). This gives the function f(P) = (2cos(theta)cos(beta)/(1-sin(theta)), 2cos(theta)sin(beta)/(1-sin(theta)), -1) for the forward function. Going the reverse direction with polar coordinates, q1 = rcos(theta), q2 = rsin(theta) and so g(Q) = ((4rcos(theta)/(r^2 + 4), (4rsin(theta)/(r^2 + 4), (r^2-4)/(r^2+4)), remembering that cos^2(theta) + sin^2(theta) = 1

[@uptodate3563]

Please this is important for me!! Let P= [0.7 0.6, 0.5] Then, by applying Qx = 2*px/1-pz, and Qy= 2*py/1-pz: Qx= 2.8, Qy= 2.4 On reverse (obtaing px, py) Px = 4*Qx/qx² + Qy² + 4 = 0.6363 (which is not the real px above), similarly Py= 4*Qy/Qx² + Qy² + 4 = 0.5454 ?! Am I substituting wrongly in the equations ?!

[@sweardog]

You started off with P =[0.7, 0.6, 0.5], which is an element of R^3 and not an element of a centered-at-origin-unit-sphere. Function g from this video takes a point in the plane and then sends it back to the sphere. g wont send a point in the plane back to a point that's not of the sphere. If you normalize P = [0.7, 0.6, 0.5] to P_normalized = [.6674, .5721, .4767], you should find that function g sends it back to where it came from.

[@uptodate3563]

@@sweardog thank you so much for your anticipation, I was so tired that I didn't see this is not even on a unit sphere. Again thanks for your respectful answer

[@sweardog]

@@uptodate3563 Yeah, the same happens to me sometimes when I'm tired. You're welcome! Glad I could help.

[@demon6582]

Hehe ;)

[@MGoebel-c8e]

Try making pauses in between your sentences. Your audience might feel as if you are slapping them. Maybe cut out a little bit of generality instead to keep the video short, guessing that was the goal for the haste:)