How to find areas on a sphere: Girard's theorem

Рет қаралды 14,865

2maniac

Ай бұрын

Taken from the main channel: • Theorema Egregium: why...
This is for reference to a future video.

Пікірлер: 37

@2maniac Ай бұрын

This is purely for a future video on the main channel. Try to guess what it could be!

@leif1075 Ай бұрын

Thanks fpr sharing. Can you share hiw ypu don't get bored and fed up and frustrated with math? Hope to hear from you!

@PMA_ReginaldBoscoG Ай бұрын

Gaussian curvature?

@dirtydanthephysicsman1750 Ай бұрын

No way if you do a video on Gauss Bonnet I’d go crazy! It’s such a beautiful theorem that I’ve thought about how a video on it would go but if I trust anyone on math KZfaq to do it it’d be you!

@S1GMATHS Ай бұрын

Beatiful explanation... It's very clear and easy to understand!!

@txikitofandango Ай бұрын

The area depends on the angles only 😳

@fangjiunnewe3634 Ай бұрын

Well, technically all the angles are multiplied by r^2, but it is typical to use r=1. And yes being independent of the arc length is nice

@anandarunakumar6819 Ай бұрын

Beautifully rendered. Appreciate the wisdom of the principles.

@berkeunal5773 Ай бұрын

Immediately shared, very nice result and the proof is almost trivial!

@eduardoeller183 Ай бұрын

so satisfying! great video!

@languafranter3450 Ай бұрын

Good explanation very satisfying to watch👍

@nanamacapagal8342 Ай бұрын

Equivalent statement, works for non-circle arcs! Imagine a car travelling along the boundary of a spherical triangle. It would have to rotate pi - [angle] at every point. So the rotation should be 3pi - [sum of angles], and the excess is 2pi - [rotation]. This also works on any spherical region, even ones defined by non-great circle curves. Just define rotation as deviation from the great circle, and it should be good

@FlwxX Ай бұрын

wow you explained this really well, good job

@strikerstone Ай бұрын

Best video on this topic

@samueldeandrade8535 Ай бұрын

Oh my Euler! There is a second channel!!!

@duckymomo7935 Ай бұрын

I was studying non Euclidean and was curious about this

@user-mc7bi3mk8l Ай бұрын

Gauss Bonnet for sure 👍🏻

@timefuzzball8097 Ай бұрын

Just a question. The area of this region inside a sphere would logically change if the sphere’s size changed. I don’t see how, only with the angles, you’re able to accurately determine the area for A.

@2maniac Ай бұрын

I would acknowledge that I didn't make it clear that it should be at a unit sphere (I only very briefly mentioned at 0:40 when I mentioned the sphere has surface area 4*pi).

@timefuzzball8097 Ай бұрын

@@2maniac I see, but what if I wanted to measure the are of A in a bigger sphere? Sorry, my math understanding is not very advanced yet.

@pettanshrimpnazunasapostle1992 Ай бұрын

@timefuzzball8097 general formula for surface area of a sphere is 4pi*r^2. So replacing all 4pi with it should work for spheres with larger or smaller radius

@TauGeneration Ай бұрын

so basically spherical integral

@takyc7883 Ай бұрын

how have i never seen this before

@dhruvarai1895 Ай бұрын

hey, great video!!! i was just thinking, how about just extending 2 out of the 3 sides of the "quadrilateral", ig, then then computing the area? kinda like how we can compute the volume of a frustum of a cone, if that makes any sense?

@2maniac Ай бұрын

Not sure what you mean by 3 sides and then a quadrilateral? Do you mean that if we have a quadrilateral, then we can compute the area by "extending" some sides to form a larger triangle, computing the area of the bigger triangle, minus the area of a smaller one? If yes, I think that's possible IF the quadrilateral is made of "great circle" sides, so you can't use it on the quadrilateral-like region in the video for example.

@dhruvarai1895 Ай бұрын

@@2maniac my bad. excuse the typo. And yeah, that's what I meant by extending the sides. Thanks for the clarification 🙏

@ValidatingUsername Ай бұрын

Would be cool if someone came up with an equation for mapping the coastlines of the world with curved vectors between continents that don’t touch so that it was vectorized and open sourced

@ralvarezb78 Ай бұрын

normally I integrate the surface on spherical coordinates

@beaumatthews6411 Ай бұрын

I was going to say straight into the action! I don't mind that at all aha

@janbendrixmalagayo490 Ай бұрын

Why is it a/2π and not 2π/a?

@2maniac Ай бұрын

Out of the full 2 pi possible angle, we take a slice of alpha, so the proportion is alpha/2pi.

@ttmfndng201 Ай бұрын

a is smaller than 2pi, so if it was 2pi/a you would get a number greater than 1. When multiplying by 4pi, you would then get a number greater than 4pi, and find the section of the sphere you're calculating the area of has a greater surface area than the sphere it's on

@mzg147 Ай бұрын

Geogebra 😉

@techgeek7410 Ай бұрын

For a hyperbolic plane just negate this result

@aymanmansoori8936 Ай бұрын

Me not knowing how to measure angles on the surface of a sphere

@gdtargetvn2418 Ай бұрын

The angle between 2 great circles is defined to be the angle between 2 planes that respectively consist of those circles. Equivalently, the angle is also the angle between 2 tangents of those circles