Youtor
Stereographic Projection Tutorial Using Similar Triangles
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=r18NKCrUqEI
Hoorah! My second stereographic projection video has now been completed. In my last video, I positioned the plane at the bottom of the unit sphere and used a line parameterization method. In this video, I instead positioned the plane at the equator and found the functions by forming ratios out of similar triangles. • I feel the need to mention that in step 3 where we formed the ratios, instead of looking at the triangles, we could have said that slope is the same at each point on the line for each plane. Therefore, in the xz plane, the slope at points N, P, and Q are the same. Likewise, in the yz plane, the slope at points N, P, and Q are the same. If the slope is the same at those points, you are allowed to take the (difference of rises) / (difference of runs) ratio between any two points on a 2D line and equate it to the analogous ratio of any other two points on the same 2D line. You could also equate the (difference of runs) / (difference of rises). It doesn't matter. Looking at the similar triangles is more fun in my opinion, so that's why I taught it that way. • This type of projection is important to me because it serves as a viewing window into the fourth dimension. In math, we're allowed to stereographically project between n-dimensional spheres and n-dimensional planes. Because of this generalization, the hypersphere of which is naturally embedded in 4D space can be projected down to the entirety of 3D space. • After viewing the stereographic projection of the 3-sphere's Hopf Fibers down to 3-space, it lead me to believe that this projection's greatest purpose lies in its ability to show them. • Timecodes • 0:00 - Listing Givens / General Intro • 3:22 - What We'll be Finding • 4:22 - Part 1 Solution: Step1 • 6:45 - Part 1 Solution: Step 2 • 8:30 - Part 1 Solution: Step 3 • 10:20 - Part 2 Solution: Step 1 • 11:04 - Part 2 Solution: Step 2 • 13:26 - Part 2 Solution: Step 3
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