Spherical Geometry: Deriving The Formula For The Area Of A Spherical Triangle
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- Опубликовано: 9 сен 2024
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Summary:
► A spherical triangle is a surface area of a sphere bounded by 3 arcs of great circles.
► Any spherical triangle has it's antipodal duplicate
► Each of these spherical triangles are intersections of 3 different spherical lunes.
► Adding up the area of all 6 lunes results in the surface area of a sphere with 4 additional spherical triangles.
►Girard's Theorem: the area (T) of a spherical triangle, with interior angles a,b and c, is given by T = r^2 (a + b + c - pi). The
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5:05 Flex on your friends by telling them that beach balls are actually colored in 6 spherical lunes
mathematically,Its called hosohedron
You can flex even more by calling those lunes spherical digons. That's an actual name, no joke
Vlad Anikin Never understood why they used the prefix di instead of bi which would be easier for people approaching the subject. I think it is long past time we just let digons be bigons.
@@paulfoss5385 story of having the Greeks understand you too lol
@@paulfoss5385 because of consistency. Di- is from Greek and bi- is from Latin so using bi- instead of di- in digon would also make heptagons into septagons. In the end it doesn't matter if you understand how this prefixes work.
Awesome video!
The formula at the end also shows that the angles inside a spherical triangle will always add up to more than π or 180°.
@@cubing7276 I'm not sure if that's what you meant but, on planar euclidean triangles the angles add up to exactly π and for hyperbolic triangles they add up to less then π.
Good insight
The most important trick that goes unstated in this, that took me a good ten minutes to figure out, is how the addition of all six lunes gives you the surface area of the sphere plus four triangles:
1. When you add the two alpha lunes, you get just the area of those lunes.
2. When you add the two beta lunes, the areas of those lunes overlap with the areas of the alpha lunes where the triangles are located. So you add the areas of the beta lunes PLUS two areas of the triangle.
3. Same happens with the gamma lunes. You are adding the areas of the two lunes PLUS two areas of the triangles because of overlap.
And there you have it. Adding the six overlapping lunes gives you the surface area of the sphere plus four triangles. Just substitute α,β,γ into L(θ)=2θr², do a little algebra, and you get r²(α+β+γ-π)=T
Thank you so much!
I would like to add that the area of alpha lunes also includes the area of two spherical triangles. The area of two triangles has to be added once but is added three times -six times the area of the spherical triangle. So four times the area of the triangle has to be subtracted from the area of lunes to ensure that the area is added only once in order to calculate the area of sphere from the total area of the lunes.
basicaly the 4 is 3*2-2
but I dont understand why its 2pi and not 180
@@aaaaaa-rr8xm radians
life saver.
Can’t find this quality anywhere else on RUclips. Keep up the good work 👍
Thanks for watching :)
3blue1brown
H1N1
gotta say it this guy is pretty good, but 3blue1brown is equally as good if not more so!
@@ThinkTwiceLtu What would an angle of zero degrees between two great circles look like then..the same as an angle of 2pi or 360 degrees?..as far as i can tell it would..since either way the circles overlap entirely..
More great geometry. You've a great knack for leaving each scene up for just the right amount of time for my brain to compute what you just emphasised, and that makes all the difference.
Great video! I didn't know this before!
By the way, in these days, no matter where you are, stay healthy.
Thank you:)
I was just wondering if someone has stolen Trevor Cheung's Quora profile-pic but then I realized that this is actually you. :)
This channel is way to underrated, it's much better than other maths channels with millions of subscribers
-presh talwalker is the first name that comes to the mind-
*Fresh Toadwalker
Other than him, there's many good math channel with millon of subscriber that you can try checking out first
I am sharing this video on my facebook page in order to promote it. Hope this will help.
A lot of math channels that really don’t deserve the recognition sure
I cannot believe how much this makes sense.. Thanks for the awesome graphics and the intuitive explanation. Keep up the great work.
Thanks, happy to hear that!
Beautiful proof for a beautiful formula!
Can you make a playlist of all of your videos so i can listen to the amazing selection of the songs you use, pls.
Edit : I'm not saying that i just wanna listen to songs, i love all of your videos, but your selection of songs is just perfect.
Great video, that “Aha” moment was at 4:55 to add up and find the equation. 👏👏👏
Thanks!
Very interesting video. This is something we never see in high school - formulas in spherical geometry (much less their proof). In fact geometry seems to be somewhat neglected in school in favour of algebra (which may appear to be more pratical and is preparation for calculus). Sure we learned basic facts like the area of a triangle is half the base times the height but in school I always wondered what if we do not know the height? What if we only know the lengths of the three sides (which is much more likely)? It was only many years after high school I discovered Heron's formula; the area of a triangle is the square root of s(s-a)(s-b)(s-c) where a, b, and c are the lengths of the three sides and s = (a + b + c)/2. There are many other fascinating facts about objects as seemingly simple as triangles, such as every triangle has both an incircle (a circle tangent to each side) and a circumcircle (a circle which passes through each vertex). This is not true for polygons in general. Also each of the following three sets of segments in a triangle intersect in a point: 1) the altitudes; 2) the medians; 3) the angle bisectors; 4) the perpendicular bisectors. Then there is Ceva's theorem which shows a connection between the the lengths of the segments cut off on each side by three cevians (a segment drawn from a vertex to an opposite side) passing through a single point. And this is just triangles; there are many other fascinating facts about polygons and geometry, spherical geometry, and three dimensional geometry. Another topic neglected in high school is number theory. There are many fascinating facts about the integers and certain sequences of integers (such as the Fibonacci sequence) and some very interesting but unsolved problems (such as whether there are an infinite number of twin primes and the Collatz conjecture).
Wow, your video quality improved a lot! Feeling so good for you.
Thanks:)
Having to get sit at home and watch your videos. Pure bliss!
Incredible. You are maths' makeup artist
Good 👏 Quarantine👏 Content👏
All jokes aside awesome job :D
Thanks 😊
Beautiful explanation
Thank you:)
All about this was so desperately beautiful, from the formula itself to the yet quite simple proof and the animation - this was both highly interesting and soothing, with a great timing in the video, pausing long enough to ensure understanding but not to much to keep it fluid. Love it!
Thanks a lot:)
I do not speak English (greetings from Russia), if only a little bit, but your videos are understandable, because you speak immediately in 3 truly international languages, these are Music, Beautiful and Clear Visualization and Mathematics! Thank you very much for your wonderful videos!
Спасибо)
i didn't know how much i needed this video. so pure 🥺
:')
I can't think something like this , this so beautiful visualization
This proof is so beautiful and simple resulting in an elegant formula that's also beautiful and simple.
👏Great Animation👏.
Visualization helps a lot.
❤Keep making such quality content ❤
I was trying to visualize this from 2 days but the animations awesome I got that at once. THANKS
Interesting conclusion from that, which is a bit trivial in highnsight, is that every triangle and in extension every polygon on a sphere with non zero area, the sum of it angels is necessary bigger then that kind of polygon on the plane, does this extend to higher dimensions with oclidian spcaes?
euclidian *
This is beautiful!
The proof i knew to this formula uses the fact that the area is proportional to the total curveture inside the spherical triangle.
I'm drooling myself. This is beautiful. Loved it!
Thank you!!
I love these videos, it makes it so much easier to understand.
Great work, keep it up :)
Your videos are pure eyegasm
Incredible Stuff! I loved the presentation.
Thank you
real nice video, nice, smooth transitions, clear, decluttered, followable, i loved it!
Thanks 😊
just like 3B1B you never dissapoint.
:))
That's literally the best video I have ever seen.
Beautifully done, love the animations.
Keep it up man!
Thanks!
Спасибо за видео
It took me less time to understand areas of triangles in spherical geometry from this video than I the time I took to understand why Heron's formula (basic euclidean geometry) works from actual math classes
A very nice presentation on the area of spherical triangle. I learned a lot. Tnx
So well done! Kudos. Keep up the great work!
Great! Now I know how to derive the formula to the area of a spherical triangle!
:)
Very nicely animated!
Very impressive
Beautiful video as usual, thanks for uploading!
I have a question however, the formula for the Area is: T = r² (a + b + c - pi)
The sum of the angles in a triangle equals 180° = pi, so:
a + b + c = pi
Doesn't that mean that the Area is always 0.
(Is it because the sum of the angles of a spherical triangle does not equal pi?)
Yes imagine walking from the equator to the north pole turn 90 degrees and then go down to the equator turn 90 degrees again and go back to the point where you started that triangle has 270 degrees you can have many different numbers bigger or equal to 180 on a sphere
Thanks!
As others have pointed out, this works because the angles of the spherical triangle sum to more than pi.
This means we can do an experiment... Set up a very large triangle where the vertices can all be seen and are visible to an observer located at any vertex. Measure the angles. If the universe is Euclidean, you will always get pi. If space is curved you will only get pi in special cases.
Pretty cool, I thought that in order to understand the proof of this I was going to need very sophisticated mathematics. I am really grateful because you taught me a lot of hidden gems in geometry.
I expected Flat earthers being pissed here, then I realized that Flat earther won't bother to seek knowledge, so they won't be here.
How would this be useful in proving we inhabit a spherical earth? To actually measure spherical lunes (of they were to exist) wouldn’t be an incredible feat (given the presupposed size/shape of the earth). People could try to get a smaller portion, but they’d be basing it on a presupposition of the radius- their confirmation bias.
That was beautiful!!
Such a high quality content, thank you!
Happy to hear that:)
These really are some of the most amazing videos on RUclips
Thank you:)
Your videos are always a joy to watch!
👏 👏 ☺
Keep going, you are an inspiration for all mathematicians/math students!
Edit:
I especially like the chill lofi music. ☺
Thank you:)
I usually don't watch your vids if I don't know what you're exposing, but I'm gonna watch it all now bc you deserve all the attention you receive and so much more, thanks for this I really appreciate it
Thanks for the support:)
Ahhh, and on a plane, alpha+beta+gamma would go to pi, making the term inside the brackets go to 0, bit simultaneously, r would approach Infinity!
Well pointed
Well, this reflects that if we place a planar triangle on a sphere, only 1 point touches the sphere; so we have no area.
Awesome animation! Thanks a lot 🙏
A bit complicated, but an amazing video well explained!
Thank you:-)
You're work is excellent..... Good job
😂......ohh... I didn't see that....thanks for *tailling* me
the final formula is... beautiful!
Great Video! Keep up the good work. Wonderful proof thanks to the animations
Thanks:)
Interesting to note that the sum of the internal angles of a spherical triangle is NOT pi radians as in the case of the good old plane triangle. The area of the spherical triangle would be zero, according to the equation obtained in this video.
Captivating and Beautiful, thank you!
Thanks for watching!
Superb quality great video. Top math visualisation channel on RUclips.
Thanks for watching 😊
OMG THAT WAS FUCKING AMAZING PLEASE KEEP MAKING VIDEOS
Nice video, thanks for posting 😊
Seriously epic, mate. It's a great channel
Thanks!!
Very nice video!
I am curious though, in the limit of large radius (very little curvature) we should expect that even though the triangle would get very big, does the formula reduce back to good ol bh/2?
Technically a straight line can be a considered as an arc of a circle of radius infinity . So in that way, every 2d triangle is a spherical triangle with the sides being arcs of great circles of a sphere with radius infinity
The proof come so smooth ❤
GREAT VISUALS! SUBSCRIBED
Thanks for this helpful video
Excelent video (once again). One point remains, for me at least: It's no trivial the angle between 2 great circles is the same as the one formed by the the arcs of the spherical triangle. Ie, the "inner angle" is equal the "tangent angle".
I don't understand your question but isn't the tangent angle itself is the definition of the great circle angle?
@@hamiltonianpathondodecahed5236 The question would be: is the inner angle (the one "touching" the circunference's center) the same as the "tangent angle" (the one tangent to the circunference's surface)? For me, it seems to be, but it's not trivial...
Now, THIS. THIS IS IT.
Man, these videos are amazing! Could you animate a proof of L'Hopital's rule? I'm sure there has to be a neat visual proof
Angular excess times radius squared. Very nice explanation!
Thank you 😊
great video and beautiful animations!
Thank you :,-)
3blue1brown supports Think Twice, I think that's adorable.
Excellent you saved my neck thanks...
I came into this knowing nothing and feel like I just learned a new language
Great quality. Keep it up.
Thanks!
Why are the sides of triangle only "Great Circle"?
Can the sides be any smaller circles?
It can but it won't be considered a proper triangle anymore, since the only geodesics on a sphere are the great circles. Geodesics are lines which minimize distance, and are defined by the local geometry; in the plane they are straight lines for example.
Excellent video as always.
Thank you!!
This is nice. But how many degrees of freedom are there among the three angles? Does the third one depend on the other two? Hmm maybe not
beautiful!
Galing Naman daming matutunang Tayo dito.. thank you for sharinng
This is amazing. May I know which software you use for your animation?:)
Processing and c4d:)
Truly brilliant. When it is explained that the some of areas of 3 lunes are equal to 2 times area of sphere and T. It needs little bit more explanation or I have to watch that part again.
First add the two red lunes. Then add the blue ones... They each overlap one of the red lunes exactly over one of the two green triangles, so we've overcounted by two triangles. Repeat with the yellow lunes, which overlap the red and blue exactly at the two triangles. We have thus overcounted by 4.
Also, it's not that the sum of 3 lines is 2 times the area of sphere + 4T; you have it backwards. It's that 2 times the area of the three distinct lunes is the area of *one* sphere + 4T
Wonderful as always👌
Thanks!
Please explain visually the infinite sum of natural numbers. Can you do it .
Thanks, i love it
Very nice content like it.keep it up
Very cool!!
Great, thanks so much. How duid u do this animation?
Brilliant indeed!
Dot comes to existence,
dot moves -> line,
line moves -> circle,
circle moves -> sphere.
*happiness noises*
yeah! but this assumes that area varies linearly with the angle which we don't now if its true
This is brilliant
Can you do the same for hyperbolic surfaces?
Beautiful!
Thanks:)
Beautiful 🤩
It's beatiful. Which software do you use, please?
Processing and c4d
@@ThinkTwiceLtu Thank you.
Beautiful.
:-)
Change the music, and it was already perfect
Bruv uplooaaaaddd dudeeee
1:33 why is 4 pi r^2 / 2 pi = L(a) / a ???
Great video!
Thank you 😊
For a flat plane, r is infinite and alpha+beta+gamma-pi=0 so the area can be anything