Circle Inversion: The most useful transformation you haven't learned yet (Part 1)

Circle inversion is a very beautiful and interesting technique for problems in geometry. In this video I'll outline some of its main properties and solve a basic problem involving mutually tangent circles and lines.
Part 2 of this series is now live, you can watch it here • ZERO TRIGONOMETRY REQU...
I wrote a blog post about the journey I took through Circle Inversion here
thecalculusofexplanations.com...
This playlist is inspired by the following videos, which you should definitely watch for more information on circle inversion
Epic Circles - Numberphile
• Epic Circles - Numberp...
Problem of Apollonius - what does it teach us about problem solving? - Mathemaniac
• Problem of Apollonius ...
An amazing puzzle involving Circles - Act of Learning
• An amazing puzzle invo...
Check out my related blog for more maths musings below
thecalculusofexplanations.com/
If you'd like to support me making more of this content, consider supporting me on Patreon below
/ thecalculusofexplanations

[@zach5935]

thats so cool. it seems quite abstract and hard to visualise, but your explanation is rather understandable 👍

[@thecalculusofexplanations]

Thank you! It does get easier with practise and solving more problems to visualise where the inverted objects end up

[@invictus327]

I was watching Norman Wildberger's playlist on algebraic topology and this video perfectly illustrated inverse geometry on a circle for me. Thank you.

[@thecalculusofexplanations]

I'm so glad to hear it! I will have a second video up with a slightly harder problem soon

[@p4rk756]

Really really good video on inversions!

[@vahurpaist511]

Totally agree

[@thecalculusofexplanations]

@p4rk756@@vahurpaist511 thank you both

[@thecalculusofexplanations]

Thanks so much, if you enjoyed this make sure you check out part 2!

[@leocrino5144]

Awesome video

[@pierre7770]

Oh no way this is your first video that’s awesome !! Was getting ready to watch part 2 haha. Really beautiful animations, perfect pace, crystal clear explications and fascinating concept. Thank you so much for this video !

[@thecalculusofexplanations]

Thank *you* so much for the kind words :) It's really motivating to hear people are looking forward to part 2, I'm aiming to release it in the next few weeks!

[@thecalculusofexplanations]

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

[@dackid2831]

​@@thecalculusofexplanations You have this video on private.

[@thecalculusofexplanations]

@@dackid2831 I'm pretty sure this is public?

[@dackid2831]

@@thecalculusofexplanations My bad, I am referring to part 2

[@user-po5mv2qi9h]

I love this technique! Thank you for the video! I will wait for the second part.

[@thecalculusofexplanations]

Isn't it awesome? Working on Part 2 as we speak.

[@gonzaaa_t]

ayo loved this video, super clear and concise, great job

[@thecalculusofexplanations]

Thank you so much! Make sure you check out Part 2

[@kenkiarie]

Very interesting. Can't wait for those interesting problems in part 2!

[@thecalculusofexplanations]

Thanks, I appreciate it!

[@thecalculusofexplanations]

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

[@Aequorin628]

I was hoping you were a #SoME2 channel and would have some other videos I could watch, but it's impressive that your first video is this good!

[@thecalculusofexplanations]

Thank you! I actually made this for the first SoME, but I'm getting back into it - look out for Part 2 soon.

[@TheJara123]

brilliant effort man, thanks for the nice video, please keep posting

[@thecalculusofexplanations]

Thanks, that's highly motivating :)

[@40watt53]

Oh my god you don't even have 1000 subscribers and this is incredible dude.

[@thecalculusofexplanations]

Wow thank you so much for the compliment - I feel the same way about many of the newer maths creators, consider watching some #SoME3 videos (including the second part in this series) But I'm definitely looking forward to hitting the 1000 mark, I'd love if you can share these videos with your friends :)

[@tiong2351]

Great complement to this topic in the visual complex analysis textbook :)

[@thecalculusofexplanations]

Do mean Needham? I've heard good things about it but I haven't actually read it, you'd recommend it?

[@youssefdirani]

thanks

[@sungminson3658]

Hi. Thanks for the clear explanation here. Now I know a circle touches origin will be mapped to the line geometrically, but not sure how to specifically prove this in equation form. Is there any reference I can check the proof?

[@adrienrebollo5879]

Excellent video, thanks! Maybe the music is a little bit repetitive, but the math is finely explained.

[@thecalculusofexplanations]

Thanks for the feedback, I'll think about how to adjust the background music!

[@guidosalescalvano9862]

@@thecalculusofexplanations I thought the music was fine. Nice and focussed.

[@guidosalescalvano9862]

This video was so awesome that I wanted to watch the whole series. I then saw the recent date and figured; better subscribe. Then I thought, maybe there are a lot more videos, and realized this is seriously your first post!? Wow...

[@thecalculusofexplanations]

Thanks, I really appreciate it - I actually made this one in 2021 with really poor audio (almost inaudible) and re-uploaded it with better sound quality, I am really glad everyone has enjoyed it so much, part 2 is coming soon. Within the next 10 days I would say.

[@guidosalescalvano9862]

@@thecalculusofexplanations Looking forward to it. This is becoming a year of math ^^

[@thecalculusofexplanations]

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

[@Hi-6969]

wait there's a part 2? These videos on inversion really do help. I find inversion one of the funkier oly geo topics to learn, and you're video makes it very clear what it is, and how to to use. Using manim makes these videos even better.

[@thecalculusofexplanations]

Thank you! I appreciate it - look out for Part 2 coming soon.

[@thecalculusofexplanations]

Actually, what do you mean by oly geo? Olympiad geometry type problem? I'm not overly familiar with competition maths, does it appear often as a problem solving technique?

[@thecalculusofexplanations]

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

[@brendawilliams8062]

Maybe the geo topics are the three great spheres? The man who lived kaleidoscopes and saved geometry?

[@Hi-6969]

It appears quite often on harder probs, yes.@@thecalculusofexplanations

[@aik21899]

That was very interesting and I am excited to watch the next one. Maybe you could talk about where this method has been used in proofs in a later video? A "real life application" is always interesting.

[@thecalculusofexplanations]

Thanks for your interest - I'll try to mention some applications in the next video

[@thecalculusofexplanations]

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

[@Fractured_Scholar]

Great basic explanation -- Subscribed and looking forward to part 2! Is it possible to do a real world application of where this is useful?

[@thecalculusofexplanations]

Thanks, it depends what exactly you mean by real world, but I can think about including some remarks on that in Part 2.

[@Fractured_Scholar]

@@thecalculusofexplanations - What I mean is that working equations and graphs can be interesting, but it doesn't necessarily map onto (heh) a viewer's brain or life. Where do we see this behavior in the real world? To me, circle/sphere inversion looks like turning something inside out. But that doesn't fully explain the concept for examples beyond physics or three dimensions. Perhaps the Social Sciences offer practical examples???

[@thecalculusofexplanations]

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

[@Fractured_Scholar]

@@thecalculusofexplanations - Thank you, kindly - I'll give it a watch soon!

[@gdclemo]

Nice video. So if a circle is inside the inversion circle, but goes around the origin, this must map to a circle that entirely surrounds the inversion circle, right?

[@thecalculusofexplanations]

Thanks! Yes, that's correct, I neglected to show that case but the problem in the next video will show an example of exactly that.

[@erawanpencil]

I'm kind of confused on the basics here... what type of 'space' (if that's the right term) does circle inversion take place in? Since the Pythagorean Theorem was used in your example, does that mean circle inversion is a process in regular old flat Euclidean space? I thought 'inversive geometry' (not clear if that's the same as circle inversion) had something to do with hyperbolic spaces, or projective geometry.

[@thecalculusofexplanations]

That's fair, it's not intuitive I suppose - and you're right in that there are many different 'depths' to which you can understand things in maths. At the most basic level I understand inversion as a transformation taking points in R2 to other points in R2, just like rotating, scaling, skewing etc, but whereas those are linear, circle inversion in non-linear. If you watch Part 2, you'll see me talk about how circle inversion correlates precisely with the complex conformal (angle-preserving) map f(z) = 1/z, which is a special case of a "Mobius transformation" - here is the link to stereographic projections / projective geometry.

[@Fractured_Scholar]

Looking at this video again and trying to connect it to Linear Algebra. Three questions: 1) What textbook(s) do you recommend to learn this? 2) Do you know of any examples that directly relate this to Set Theory? 3) Has any class you've taken related this to vectors? (i.e. Consider r to be a vector \vec{r} and OA to be its projection onto another \vec{v}. Given a specific magnitude OA' for \vec{v}, then \vec{r} · \vec{v}=\vec{r}^2.)

[@thecalculusofexplanations]

Hey, glad you're finding it interesting. The connection to linear algebra is a bit difficult, given linear algebra deals with linear transformations, and circle inversion is decidedly non-linear! 1. Not particularly, I think what drew me to the subject was the lack of obvious available resources. If you are wanting to learn the foundations of complex analysis, which is where all this ends up, I've heard good thing's about Needham's "Visual Complex Analysis". There may be some nice books on nonlinear geometric transformations in isolation but I'm not aware of them. 2. I'm not aware of any connection between those concepts 3. Again, the reason I found this interesting was because it's not taught in classes usually. Although it is an interesting though, it's a scaling of the vector from the origin of the circle of inversion, so its a little bit like an eigenvector, but instead of being scaled by a constant its scaled by its own length. I'm not sure if there's anything to that beyond idle speculation, though.

[@thecalculusofexplanations]

P.S. Hopefully you caught Part 2 as well. kzbin.info/www/bejne/ioK3oqWmmrGLiZo&ab_channel=TheCalculusofExplanations

[@AndrewBrownK]

What about projecting a circle through a concentric circle?

[@thecalculusofexplanations]

That should give you another concentric circle on the opposite side of the inversion circle!

[@MilanStojanovic9]

well done, 3b1b would be proud Also very useful for me

[@thecalculusofexplanations]

That's a huge compliment, thank you.

[@thecalculusofexplanations]

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

[@guidosalescalvano9862]

I would love to see how points on circle pairs map to each other. Let's define three circles, the mapping circle, the inner circle and the outer circle. Using the mapping circle the inner is circle inverted to the outer. The inner is inside the mapping circle. Now let's assuming a set of n inner points, spaced at equal inner circle arc distances from each other on the inner circle. The inner points are mapped to outer points. Am I correct that the outer arc distance between the outer points is inversely proportional to the distance of the corresponding inner points to the mapped circle's center?

[@thecalculusofexplanations]

Possibly, if I understand the question correctly. Might be interesting to visualise, I'll consider it if I do a part 3.

[@spiderjerusalem4009]

6:45 that may make sense intuitionally, but if you use the inversion formula, let P=centre of the smallest circle P'=centre of the inverted circle since r=¼, then OP=R-r=1-¼=¾ 1=OP•OP'=¾OP'= OP' = 4/3 therefore, the radius of the inverted circle = OP'-1=4/3-1 = ⅓ ?????

[@spiderjerusalem4009]

Ah, or perhaps because the extended straight line connecting the origin and P does not pass through intersection between the smallest circle and the largest one?

[@sirawatkhongnum5389]

I'm solving a problem that requires this theorem and need a reference book. Could you recommend a book that has the theorem in this video?

[@thecalculusofexplanations]

I haven't read it, but it looks like this one would be a decent reference! en.wikipedia.org/wiki/Geometry_of_Complex_Numbers

[@viktorbergman517]

hey, in that last problem, instead of using the diamenter of the inverted circle plus the radius of the reference circle, i used the radius of both to get the distance of the center of the circle from the origin, so (with C as the center of the circle whos radius we want to find) OC * OC' = 1 (=) OC * (1 + 0.5) = 1 (=) OC = 1/1.5 = 2/3. since the both circles are tangent the radius should be the radius of the big circle minus the distance between both origins, so 1 - 2/3 = 1/3, which isnt what i get when doing it with the whole diameter, did i miss something or am i doing something wrong?

[@viktorbergman517]

ah wait i got it, the center of a circle gets distorted when you do the inversion, its just that the outer edge still forms a circle anyway

[@thecalculusofexplanations]

@@viktorbergman517Well done, this is something that has tripped me up as well, and I animated an explanation of why the inverted center is not the center of the inverted circle (the non-linearity of the tranformation ensures this) - but it ruined the flow of the video, and made it a bit too long so I chose eventually to cut it out. Good work thinking it through.

[@rat_king-]

OK..... but what if a circle has the origin of the inversion circle with its perimeter, or even has the same origin? Isn't that just magnitude... yet what is the area? and what happens when i change the radius?

[@2fifty533]

i think 2 circle inversions gives you a conformal transformation just like how 2 reflections gives a rotation

[@thecalculusofexplanations]

Circle Inversion is actually a conformal map because while distances are changed, angles are preserved!

[@2fifty533]

@@thecalculusofexplanations i mean, if you do 2 successive circle inversions you can get different looking conformal maps that arent the same as a single circle inversion

[@2fifty533]

@@thecalculusofexplanations slightly unrelated, but I've heard of a thing called conformal geometric algebra, and it's like linear algebra except vectors are circles rather than line segments you can get a conformal transformation by taking the geometric product of 2 vectors

[@thecalculusofexplanations]

@2fifty533 ah yes, apologies. It’s an interesting thought! It also ties in with the brief mention of iterated inversions I made at the end of the sequel to this video.

[@thecalculusofexplanations]

@@2fifty533 Sounds fascinating!

[@nsgoneape9899]

bro cooked and then dipped.

[@thecalculusofexplanations]

🙏 more will come - just got busy with work and other projects. I’m not done with the channel yet!

[@sayantanchatterjee2195]

Upload part 2

[@thecalculusofexplanations]

It's almost ready!

[@thecalculusofexplanations]

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

[@LVCID777]

Now imagine a sphere. The inverse of a sphere may be what’s inside a black hole

[@victorhmingthanmawia4773]

Could be another system.

[@richarddizaji7848]

This video’s thumbnail gives a really strange visual effect while scrolling

[@thecalculusofexplanations]

That was somewhat intentional!