Circle Inversion: Zero trigonometry required (Part 2)

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The Calculus of Explanations

Күн бұрын

Пікірлер: 36

@windigo77 8 ай бұрын

i want more like this!! Amazing.

@thecalculusofexplanations 8 ай бұрын

Appreciate it!

@angeldude101 Жыл бұрын

One of the neat things about circle inversion is that you can get any 2D conformal (angle-preserving) transformation just by preforming a sequence of circle inversions. In addition, lines are really just circles with infinite radius, and doing a circle inversion across them is the same as a traditional reflection across a mirror. So you can compose these inversions to get things like uniform scaling, translations, rotations around the pair of points where two circles (or lines) intersect, hyperbolic boosts, and much more. And all of this generalizes to arbitrary dimensions. (Technically even 1D where you can generate scalings and boosts by inverting across a dipole.) Of course this also includes 3D inversion across spheres, 4D inversion across hyperspheres, or whatever dimension you want.

@thecalculusofexplanations Жыл бұрын

Really? Wow that’s awesome, I didn’t know that. Do you have any good sources, might include that in a follow up video one day.

@angeldude101 Жыл бұрын

@@thecalculusofexplanations It's mainly from Conformal Geometric Algebra, which as the name suggests can be used to represent arbitrary conformal transformations in a manner similar to ℂomplex numbers and quaternions. These transformations are formed by multiplying some number of vectors together. The vectors in question are often used as mirrors/reflections, and CGA vectors specifically represent circle/sphere inversions. Since the product is associative, the inversions can be composed rather than applied one after the other. Depending on what inversions you composed and in what order, you can form any transformation representable by the algebra, which is any conformal transformation. A more rigorous source? I honestly don't have one. There are some good sources on CGA, but usually in specific contexts, and the interpretation I gave is actually not the one found in most sources. Probably the best source I can think of for CGA would be GA4CS (Geometric Algebra for Computer Science).

@thecalculusofexplanations Жыл бұрын

@@angeldude101 Fascinating stuff, I won't pretend to understand completely but I'm not surprised to hear about another deep connection to another field.

@Xianleft Жыл бұрын

You could apply the Schrödinger equation to that for directional velocity right?

@wargreymon2024 Ай бұрын

That's a deep rabbit hole 💀👍🏻

@marioeraso3674 Жыл бұрын

Awesome! I'm interested in the road coloring throrem. Would love to see your approach to enliving it!

@thecalculusofexplanations Жыл бұрын

Thanks! Looks interesting! I’ll put it on the list of potential topics for a Graph Theory series :)

@RandomKido Ай бұрын

This is absolutely mind boggling!! This is really cool stuff

@thecalculusofexplanations Ай бұрын

thanks so much :)

@RandomKido Ай бұрын

@@thecalculusofexplanations the effort you put into these videos is amazing! Hope to see some new ones if you get the time to 😉

@thecalculusofexplanations Ай бұрын

@@RandomKido I am attempting to install all the necessary tools on a new computer as we speak!

@RandomKido Ай бұрын

@@thecalculusofexplanations Amazing!! I love the topics you're covering

@spiderjerusalem4009 11 ай бұрын

Stewart's theorem does not require trigonometry at all, either you meant it in the sense of its involvement at the midst of solving the problem or its derivation. OP=3-r AP=2+r BP=1+r AO=1, BO=2 (3-r)²=[2(r+2)²+(r+1)²]/3 - 2 3(3-r)²=2(r+2)²+(r+1)²-6 0=2[(r+2)²-(3-r)²]+[(r+1)²-(3-r)²]-6 =10(2r-1)+8(r-1)-6 5(2r-1)+4(r-1)-3=0 14r = 5+4+3=3(4) r = 6/7

@7trdcc110 10 ай бұрын

That was actually very jntestering. Legitimately got me drawing type stuff .Hope u make more vids in the future 👍

@thecalculusofexplanations 10 ай бұрын

Thanks, I really appreciate it. A lot of my early work on this was just drawing a lot of circles and lines in a notebook. Hopefully you saw the first video in the series? I guarantee the next time you see a problem involving circles and lines you'll think about it! I will be making more videos in the future, probably on a different topic, please let me know if you have any suggestions.

@arcofficial516 3 ай бұрын

@@thecalculusofexplanations how about you do a video on barycentric coordinates. Distribution of parts for a single video: Part 1: The problem (to initiate motivation.) Part 2: The theory Part 3: Tackling the Problem Patt 4: Similar problems

@Fractured_Scholar Жыл бұрын

Brilliant! Great work. Did not expect you to reupload the whole video. That said, this is MUCH easier to follow. In response to your request for future content, have you considered connecting what you're showing back to early math? This aids viewers at all levels. Viewers early in their studies get to see things they've recently learned or are learning, while your most advanced viewers get to see how they are simply applying the same concepts learned decades ago and connect them to each other. For example, you casually stated Curvature=Inverse Radius at 2:40. That's a pretty important and succinct statement -- it deserves some weight! 😂 If you were to expand on that section alone, you could talk about how it is geometrically representing the Reciprocal Function, which grade schoolers use to learn Fractions. A mid-skill viewer like myself gets to say: That looks suspiciously similar to a derivative. Is there any relationship there? Then you and your peers (the advanced "viewers"/creators) get to generate more content. Thank you kindly for the update. I look forward to your next video.

@thecalculusofexplanations Жыл бұрын

Thank you so much, I was hoping you would see the improved version, as your feedback was invaluable. I tried to take it all into account and streamline some things, and include the formula for inversion at the start so everyone was on the same page, this necessitated re-recording the audio as well. I'm glad you like it. In terms of explaining things at different levels, that's another great point and something I want to keep in mind. One thing I'm finding is that trying to follow every thought and explain every concept is simply impossible, as the videos become too long, too unfocused and too time consuming to produce. The concept of curvature, for example, deserves its own video (at least) - I can't possibly do it justice within this one, and it wasn't central to explanation. There are also people who've covered it far better than I have on KZbin already. I want to focus on topics I can cover to a decent level in 10 minutes or so, while keeping quality high in both animation and explanation, and if possible highlight areas of maths that I haven't seen a lot of visual explanations for on KZbin. Appreciate the interest, and I'll see you in the next video!

@arcofficial516 3 ай бұрын

@@thecalculusofexplanations Bruv? Why does the distance of a point (a,b) from a line(Ax+By+C=0) in a cartesian plane have formulae |Aa+Bb+C|/√(A²+B²). I do know how to prove the formulae using algebra. But at the same time i can sense the beauty lying in the formulae. Algebra does no good in representing that beauty. Btw awesome video.❤

@comuniunecuosho-campulbudi7611 11 ай бұрын

good content

@thecalculusofexplanations 6 ай бұрын

thank you!

@erawanpencil 4 ай бұрын

@6:40, the image of those Pringle-shaped oblong circles as they undergo the inversion... is that just artistic effect or is that the shape circles actually assume 'during' a conformal transformation? I could be totally wrong but I thought getting skewed circle shapes like that happens during non-conformal skewing/translation actions... is that mathematically part of the process during the transform or just what the graphics show for non-mathematical effect? thanks

@AryanKumar-vo1ic 9 ай бұрын

can you please explain spiral similarity?

@thecalculusofexplanations 9 ай бұрын

Possibly! I'll look into it as I'm not familiar

@mykolanikolayev1714 Жыл бұрын

Please share source code for this animation

@thecalculusofexplanations Жыл бұрын

Hey, thanks for your interest. Unfortunately the code for these is impossibly messy and unreadable, but I will be sharing Github access to code for future videos on my Patreon. I was also thinking about making a video for Patreon supporters about how I make these (the tools and process I use) www.patreon.com/TheCalculusofExplanations/membership

@Xianleft Жыл бұрын

Hey there, would you be interested in how this applies to golf and putting?

@thecalculusofexplanations Жыл бұрын

It does?? I'd be very interested.

@Xianleft Жыл бұрын

@@thecalculusofexplanations consider the tangent of the arc you stand within is relative to the sec and cos of your angles of the follow through as you consider speed and pi. If you could help with the math I could provide more details