Generating Conic Sections with Circles | Part 1. The Ellipse

Learn key problem-solving techniques at:
brilliant.org/ThinkTwice
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Take any circle and pick any one of its interior points. Then the collection of the centers of circles passing through that point and tangent to the initial circle is an ellipse.
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(@thinktwice_ltu)
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[@ThinkTwiceLtu]

Hey! Hope everyone is doing well. This video is part 1 of a small 2/3 episode series I am making on generating conic sections with circles. Hope you enjoy. As always your feedback is highly appreciated as it helps me improve the quality of my future animations. Thanks for the support and I'll see you soon with another video :)

[@eldattackkrossa9886]

nice! what's this animated with?

[@jpalreis]

Great video! Although it might be a little far off your purpose, how about using this geometric intuition to show how it connects to the formula's change of sign when looking at the hyperbolas? I'm referring to the simpler ones, centred at the origin, non rotated. (This could be a bonus episode after the parabola and hyperbola)

[@ThinkTwiceLtu]

@@eldattackkrossa9886 cinema 4d

[@eldattackkrossa9886]

@@ThinkTwiceLtu thanks

[@kkadir]

@@ThinkTwiceLtu this is beautiful!

[@aradziv89]

Fun fact: put the point outside the circle, and it's a hyperbola! Did I spoil the next video?

[@dreznik]

which means if the point is on the circle it will be a parabola?

[@hydraslair4723]

@@dreznik no, if the point A is on the original circle the whole thing collapses to a single point identical to B. This is because the only circle that contains A and is tangent to any other given point of Q is Q itself, so C = B.

[@CardThrower-rb6eg]

@@hydraslair4723 Isn't it a line segment tho? Like imagine the distance from A to B is n. there are infinitely many pairs of positive integers that add up to n, and these distances add up to a line segment when visualized.

[@CardThrower-rb6eg]

Correct me if I'm wrong, but an ellipse with an eccentricity of 1 is just a line.

[@hydraslair4723]

@@CardThrower-rb6eg you are correct in saying that an ellipse with eccentricity 1 degenerates into a line. But here the geometric construction is clear, the constructed circle is unique and is identical to the circle Q. If the result were a line, you would need to have a circle that contains A, is tangent to the original circle, but somehow also travels across the plane along the x coordinate. Try constructing a circle that does that, it's not doable.

[@PowerhouseCell]

As a KZbinr myself, I really appreciate how was much was put into this. Beautifully done! :)

[@ThinkTwiceLtu]

Thank you :)

[@kikivoorburg]

Stuff always becomes so intuitive when you animate it this way! Amazing work

[@ThinkTwiceLtu]

Thanks!

[@beastfromtheeast581]

When I watch these videos they help me understand what my brothers doing on his homework.

[@ourgoalisto6737]

Reminds me of 3B1B ‘s Feynmann’s Lost lecture And why slicing cone gives ellipse great job think twice

[@Listener970]

Was thinking of the same thing. Isn't this used to describe the motion of planets? Nevermind me, not a math student. Just curious

[@darthmath1071]

@@Listener970 It was indeed used by Kepler. Now we have more accurate techniques :)

[@cubicardi8011]

Ahh I remember times years back where you just had a couple hundred subscribers and everyone was complaining how you don't have more.​Beautiful​

[@ThinkTwiceLtu]

Well, I guess it takes some time and support from others to build an audience. I appreciate your support and happy to see you stick around over the years :)

[@quantumcity6679]

Not gonna lie... That animation is actually satisfying 😻 You deserve 1 mil

[@ThinkTwiceLtu]

Thanks a lot

[@Maconhaepaz]

This is KZbin’s best channel by far

[@techboyzeeshan8582]

who says math is boring, we just need to look at it the other way around . Thnx for the amazing visual experience that you have provided , I hope that you reach millions of subscribers :-)

[@samdob8494]

I bless the day 3b1b made me discover this channel. Truly an amazing video, as always. Keep up the great work!

[@jillianonthehudson1739]

I've watched this, well, I've lost track of how many times, but each time brings me to tears. This is the most beautiful proof I've ever seen. Thank you.

[@abhi-wi2mj]

here from 3blue1brown this channel is AWESOME PLEASE NEVER STOP

[@moonwatcher2001]

Art and science blending together. Soothing my soul. Wonderful. Awesome. Namasté. Thank you

[@mohammedal-haddad2652]

Your prolems in the beginning seem impossible but in the end you make the solution so simple. Thank you very much.

[@iceiceisaac]

I’ve been wanting to understand conic sections more deeply, this just blew my mind and heart. Thanks man!

[@HIHIQY1]

These animations are SO smooth. Love it!

[@alexandroskarypidis1891]

I can't resist this off-topic question: What's this heavenly music playing?

[@phiarchitect]

This is exquisitely beautiful. And inspiring. Thank you.

[@hitoshiyamauchi]

Nice explanation. Thank you. Especially, |AC| = |DC| part opened my eyes.

[@AndresFirte]

Beautiful as always!!

[@rafsanzaman2368]

I liked your channel.Please do make more.

[@Name-is2bp]

never stop making these amazing videos please!!!!

[@abcd12221]

love your animations and thought process! thank you.

[@KanoBoom]

Animation is good. Music is good also.

[@pollensalta]

Thank you for all the Vids you put on here. I apreciate very much what you do because it helps me learning!! Great teaching!! Thank you!!

[@sarveshahuja2385]

You and many other such creators are going to make history by inculcating such beauty and love for math in the minds of the generations to come.Thank you 😄 💓🙏🙏🙏

[@cary_domiii]

I remember 3Blue1Brown also creating a video about proof for an ellipse...but he used a cone and 2 spheres....both his and yours are so visual and I understood both proofs really well...

[@Saikrishna.B]

This is really beautiful man Thanks

[@Jop_pop]

Your videos are really going to help the students in the math classes I'll teach this year!

[@ep9196]

Clear and concise math enhanced with great visuals. Nice job Thanks!!!

[@ThinkTwiceLtu]

I'm glad you liked it :)

[@mitranoarthur]

As always amazing and simple video. Very nice.

[@dominiquelaurain6427]

Excellent video with quite relaxing music :-)

[@hameedamathtuber]

Very informative and interesting visual. Thank you for sharing

[@DitDede]

I like all 3. thanks!

[@uctoan4036]

keep up the great work, this is amazing

[@ass09tube]

Very exciting. Great job 👍. Looking forward to watch other members of conic section family. Also epicycloid, astroid n others...

[@Invalid571]

Beautiful animations! Looking forward for part 2! ☺

[@ThinkTwiceLtu]

Thanks ☺️

[@trappist1d741]

I'll never be able to force you to respect my home, respect my privacy and even respect my air conditioner (I get the "air conditioner app ad). But I can dream about a better world and that is something that can't be taken away from me. I have full authority of dreams that I have in which all people respect one another. That's the world I came from.

[@Pedritox0953]

Wonderful video !!

[@albertosanchez7784]

This is so beautiful and Awesome. Thank you by share. Very Clear and Prety

[@Yamun_Sharma]

Beautiful 🏵️

[@omprakashnain6546]

Very very useful video for me. Thank you for such good content.

[@mahmoudalbahar1641]

Many thanks! for making such good video

[@williamolenchenko5772]

Both brilliant and beautiful!

[@mr_saif_i]

Great videos. Visualizing math is the best way for me to appreciate its beauty. Keep up the good work 😊.

[@nahfid2003]

Amazing. This was so awesome :)

[@monoman4083]

Well impressed. never knew this...Thanks..

[@user-in1gn6fw2eab]

Live this style of Video. Keeper it up. So relaxing and informative.

[@ThinkTwiceLtu]

Thank you

[@alpe6127]

This was amazing! What program do you use to create these graphics?

[@swaree]

C4D, as per another comment of theirs.

[@chrisd561]

Awesome video. Thank you.

[@ThinkTwiceLtu]

Thanks for watching:)

[@sudheerthunga2155]

It is amazing , oh I bet if this were to come to our classroom , we would be blindly assigning coordinates and finding the locus , Geometrically it is way beautiful!!!

[@nazishahmad1337]

Can you start a series on how you create these videos I'm too looking for creating such content.

[@ThinkTwiceLtu]

yes, I am thinking of doing that. But I need some time to plan everything out and decide on the format that the tutorial would take.

[@m77dfk]

The animation is very clean and fluid. Well done! However, one observation I made. The proof that this video gave was actually of the fact that 'An ellipse of constant k with foci at A and B, is a SUBSET of the loci traced by centres of all circles passing thru point A, and tangent to Q.' Where's the other part? Left for the viewer eh? 😏😁

[@nothingspecial7399]

My greatest inspiration

[@darkhoodchief]

Cool, would like to see more conics from you.

[@2false637]

I woke up and saw this. I am now inspired to grab a paper and do some math! Thank you!

[@aiksi5605]

Thank you, this was chilling

[@ThinkTwiceLtu]

Thanks for watching

[@rubenayla]

So elegant animations

[@pfizroternelson5355]

That's brilliant! That auxiliary line explains so intuitively that even me as a 9th grader understands the whole process when it appears😆 Btw would you please tell me what's the name of that amazing backgroud music? :)

[@calcoholics1956]

Great video... You are such an inspiration to small KZbin channels like us ❤️❤️❤️

[@ThinkTwiceLtu]

Thanks for the kind words

[@ashutoshmahapatra537]

U r Brilliant! And Ur sponsor is Brilliant too, Awesome Video ;)

[@phugoidoscillations]

Fantastic video. Seeing this makes me beg you to consider doing a video on the subject of eccentric anomaly and time of flight for elliptical orbits, think of it as a practical application of this material.

[@10v-hlk6]

I like the sound in this video!

[@azfarahsan]

the legend is back

[@profesordoctorkanjerman3953]

Truly beautiful

[@pilucapiluca9735]

Anybody knows the song? Thanks in advance!

[@anushrao882]

This channel is pure wizardry.

[@ThinkTwiceLtu]

[@arvindp551]

Love!

[@hedgehogchaser2494]

Beautiful video! What music did you use? And what software for making the animations?

[@zanfur]

Awesome video, but I think there is a logic error (though the conclusion is correct). The validation portion seems backwards. BC and DC are equal because they are two radii of the same constructed circle, and this is why AC + BC = AC + DC, and because BC + DC = k from construction of the larger circle, then so does AC + BC. The video has this logic backwards, assuming that it is an ellipse and trying to show that it's a circle. Showing that if A then B does *not* generally show that if B then A, so you have to either go directly from your construction to the definition of an ellipse, or show that assuming not an ellipse leads to a contradiction in the construction. Going from a definition of an ellipse to a construction only shows that *some* such constructions are an ellipse, but doesn't show that all of them are. Similarly, the second circle is tangent to the first by construction, in that we made it share a point on the line between the two centers. It's not proved by the fact that A and D are equidistant from C, that just shows that C is the center of a circle containing A and D. You would need to additionally show that for every C, the tangent line at the point of overlap of the circles is the same for both circles (which it is, you just would need to show it). All the right ideas and concepts for the proof are there and it's super easy to follow in this beautiful format, but I thought you might like the logic error pointed out. Or maybe I missed something and this is all nonsense!

[@timh.6872]

I concur, the argument presented appears to be backwards.

[@cmarley314]

In your first paragraph, I think you confused some letters. Anyways, we start with the assumption of an arbitrary ellipse, which already makes AC + BC = k. Next, a circle is constructed in a way such that BC + CD = BD = k, where BD is the radius of Q. BC != CD. As to the logic, the video starts out with using circles to draw something we think is an ellipse. Then it's proven by *starting* with an ellipse and showing how the circles came to it. I'm pretty sure starting with an arbitrary ellipse and making no assumptions specific to that initial condition in the proof, would prove it for all ellipses. For the second paragraph, we're using the fact AC=DC to construct a circle of radii DC, which must be tangent to Q because the circle C touches D at only one place, and radii are perpendicular to tangent lines. I'm not sure what you mean by "the tangent line at the point of overlap of the circles."

[@zanfur]

@Christopher Marley "A implies B" does not follow from "B implies A". Try this on for size to get the main point: "Starting from an arbitrary square and making no assumptions specific to that initial condition and showing that it is a rectangle proves that all rectangles are squares". It doesn't follow.

[@cmarley314]

@@zanfur I see your point, it does seem to go backwards. So I proved it for myself starting from the circles, rather than making circles from an ellipse. Not too hard. Then, I wonder why you use "implies." I could consider this a sort of "if and only if," meaning A and B are corollaries (for lack of better word). For example, if X carries an umbrella, it is raining, *and* if it is raining, then X carries an umbrella. This relies on the fact that "X carries an umbrella *if and only if* it is raining."

[@zanfur]

@@cmarley314 Yeah like I said your conclusion is correct, and all the bits are in your animations to easily show it. Just the direction was backwards, and it happens to go both ways with this one. You just didn't prove it. :-) "A implies B" is a common synonym for "if A then B", used in logic textbooks. You can't just say it's an iff (common abbreviation for "if and only if") relationship, though -- to show an iff relationship, you have to show both directions of the if. You only need too prove one direction to make your point, though, which is that the circles construction does actually construct an ellipse. Whether you can construct those circles given an ellipse is irrelevant, from a logicsl proof standpoint. Just to be clear, I loved the video. I'm only being pedantic here because your stuff is such high quality I thought you might want to know it had a logic error in it.

[@Thats_A_Dummy_Name]

Great visualization as usual. I'm wondering how the shape of the ellipse is changing when the metrik for distance calculations changes. Instead of L^2-Norm one could use any L^P-Norm. What happens in case of the choice of cosine-distance?

[@abhi-wi2mj]

my 8th standard's sir once showed us how to make an elipse with 2 nails and 1 long tied string it's amazing how construction of something can give us a much better understanding somehow

[@ostanin_vadym]

Thank you for content

[@ThinkTwiceLtu]

Thanks for watching :)

[@Roaxial]

What happens if you generate an ellipse with 3 points, Where AD + BD + CD = k?

[@rbettsx]

I really prefer the palette and style of your 'shaded colour out of colour' animations to your 'emissive color out of black' animations, which IMO, get lost in among other (some very notable) math visualisations out there on KZbin. It was your aesthetic which first drew me to your videos. If you got some stages of those others, without annotations, made up as high-quality silk-screen prints by a master, on good paper, with a nice thick layer of pigment, almost in shallow relief, I think there's a good chance you could sell them well. Is there anybody you know who could help you out in that direction?

[@fabiangn8022]

Gracias

[@dog8484]

best channel

[@ThinkTwiceLtu]

Thank you very much!

[@dragonsdream4236]

Wow incredible! What do you use to make these beautiful videos?

[@ThinkTwiceLtu]

thanks! I use cinema 4d.

[@JubilantJerry]

Is there actually a straightedge and compass construction of an arbitrary point on this ellipse? It doesn't seem easy to construct C with its constraints.

[@sarojpandeya7883]

Amazing

[@dreznik]

may i ask you what software you use to produce your geometric animations?

[@sebastianmorataboada9795]

Me, opening my long forgotten installation of Geogebra: this is where the fun begins

[@spamtongspamton9900]

“i have no idea what any of this means but the colors are pretty”

[@abhisheksamal1970]

Best 👍💯 bro ..... Can you do something on vectors and phasors?? What is the difference between them please😫🙏🙏💓

[@ThinkTwiceLtu]

Hey, thanks for the suggestion. It's a great topic for a potential video.

[@abhisheksamal1970]

@@ThinkTwiceLtu thanks bro.... Waiting for your video

[@abhisheksamal1970]

@@ThinkTwiceLtu also on harmonics....everyone wants to learn math from you please provide maximum videos

[@RSPikachuAlpha]

What’s the name of the music? I love it

[@omibuddyy]

Which tool you use?

[@ryanbell3704]

subtle ACDC meme at 4:07 lol

[@elnurbda]

спасибо за видео ролик

[@hallvard3739]

What is the background music called?

[@quantumgaming9180]

How were you able to create these animations?

[@gamedevrony158]

Which software or python library do you use ? Manim??

[@vuvffufg]

Beautiful

[@ThinkTwiceLtu]

thanks

[@SupriyoChowdhury5201]

Nice

[@shreyas3852]

Incredible

[@ThinkTwiceLtu]

:)

[@benheideveld4617]

Post Trump I can’t stand the word “incredible” anymore... It is also perused to get people to watch clickbait video’s. You won’t believe what...

[@Gamikane]

I don't know anything about math or geometry but this is really interesting

[@ThinkTwiceLtu]

Happy to hear that :')

[@nothingspecial7399]

Dudeee 🤣🤣

[@krumpy8259]

That's beautiful, but I don't want to even think of how elusive the mathematical jargon would be regarding the relationship between these objects. Visualizations FTW (and maybe then cubersome Notation :D )

[@harikrishna2k]

Wow proved the statement by just extending a line segment

[@RnnR72954]

Great

[@victr3d]

I would never have thought Cinema 4D would be used here, but it's a great idea (albeit it would destroy my 2011 computer)

[@ThinkTwiceLtu]

Yes, c4d gets heavy on my laptop pretty fast, but it's not too bad for simple scenes like this with low geometry.

[@victr3d]

@@ThinkTwiceLtu indeed. But even when I try to do a 2d blender animation my computer crashes (༎ຶ ෴ ༎ຶ)