Circle Inversion: Zero trigonometry required (Part 2)

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This is the second video in our series on Circle Inversion, a beautiful technique that can be used as an alternative to trigonometry to solve geometric problems.
This is a re-uploaded version where I have corrected a small but important error, thanks to one of my viewers for insisting that the video was confusing.
Check out the first video at the link below. It's highly recommended to watch this one first, as we derive the formula for inversion and go through a simple problem showing its use.
• Circle Inversion: The ...
I wrote a blog post about the journey I took through Circle Inversion here
thecalculusofe...
This problem originally came from this Art of Problem Solving page, which shows a variety of different solution techniques.
artofproblemso...
If you’re interested in iterated circle inversion, check out these amazing fractal generating scripts and animations by *[Claudio Esperança](observablehq.c...
observablehq.c...
Möbius Transformations Revealed is a short video by [Douglas Arnold](www.ima.umn.edu...) and [Jonathan Rogness](www.math.umn.ed...) which depicts the beauty of Möbius transformations and shows how moving to a higher dimension reveals their essential unity. It was one of the winners in the [2007 Science and Engineering Visualization Challenge](www.nsf.gov/new....
www-users.cse....
Check out my related blog for more maths musings below
thecalculusofe...
If you'd like to support me making more of this content, consider supporting me on Patreon below
/ thecalculusofexplanations

Опубликовано:

27 сен 2024

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Комментарии : 30

@angeldude101 Год назад

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 Год назад

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 Год назад

@@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 Год назад

@@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 Год назад

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

@marioeraso3674 Год назад

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

@thecalculusofexplanations Год назад

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

@Fractured_Scholar Год назад

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 Год назад

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 RU-vid 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 RU-vid. Appreciate the interest, and I'll see you in the next video!

@arcofficial516 Месяц назад

@@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.❤

@7trdcc110 8 месяцев назад

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

@thecalculusofexplanations 8 месяцев назад

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 Месяц назад

@@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

@spiderjerusalem4009 9 месяцев назад

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

@windigo77 6 месяцев назад

i want more like this!! Amazing.

@thecalculusofexplanations 6 месяцев назад

Appreciate it!

@erawanpencil 3 месяца назад

@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

@Xianleft Год назад

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

@thecalculusofexplanations Год назад

It does?? I'd be very interested.

@Xianleft Год назад

@@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

@themathguy3149 Год назад

Please sir unban trigonometry is kinda hard outhere without it I want my mom 🥲

@thecalculusofexplanations Год назад

Haha, you have my permission.

@fulla1 Год назад

Dude, clear your throat and speak up! The topic was pretty interesting, though.

@thecalculusofexplanations Год назад

Apologies, I was a bit sick before recording this but I couldn't wait any longer!

@AryanKumar-vo1ic 7 месяцев назад

can you please explain spiral similarity?

@thecalculusofexplanations 7 месяцев назад

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

@mykolanikolayev1714 Год назад

Please share source code for this animation

@thecalculusofexplanations Год назад

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