Question about area on a sphere

1. Apr 11, 2012

Vorde

For a end project in a math class, I am going to attempt to figure out the laws for spherical trigonometry. Specificially, side lengths and areas for triangles on a sphere.

I am fairly sure I know how to go about side lengths, and I have an idea how to go about the area of a curved triangle, but it's dependent on an assumption I want to make sure is correct.

Without going into details, the assumption is that increasing the interior angle sum (past 180 degrees) will increase the area by an amount dependent only on the angle increase (and the dimensions of the triangle), meaning it doesn't matter which angle is being increased, just that the total angle is being increased.

Is this correct? I would appreciate non specifics if it is wrong- I want to try to figure it out by my self.

Thank you.

2. Apr 12, 2012

vkash

Re: Question about area on a circle

i didn't understand what you say..
How you can increase sum of angles of triangle???

3. Apr 12, 2012

DaveC426913

Re: Question about area on a circle

By putting it on a spherical geometry. It will be greater than 180. More curvature, and that number can go all the way up to 360.

4. Apr 12, 2012

vkash

Re: Question about area on a circle

Oh sorry Vorde it is out of my scope....

5. Apr 12, 2012

DaveC426913

Re: Question about area on a circle

Just doing some sketching. What is the largest possible triangle on a sphere? What does it look like?

6. Apr 12, 2012

Vorde

Re: Question about area on a circle

I think it depends on whether or not you allow overlap (going around more than once), if so infinite, if not then my quick guess would be the maximum angle measure is 540 degrees.

I don't think that effects my question though. Also I apologize, it should be sphere not circle in the title but it's too late to change it.

7. Apr 12, 2012

micromass

Staff Emeritus
Re: Question about area on a circle

I changed it for you.

Anyway, here is a good free book on spherical trig: http://www.gutenberg.org/ebooks/19770

8. Apr 12, 2012

mathwonk

First figure out the relation between angles and areas of "lunes", which is the region between two great circles. Then try to bootstrap up to a triangle, which after all is an intersection of lunes. for simplicity start with a sphere of radius one. you might also stick to triangles lying in one hemisphere. i.e. just as a triangle in the plane is a bounded intersection of three half planes, on the sphere one can define it as the intersection of three hemispheres.

9. Apr 12, 2012

DaveC426913

Re: Question about area on a circle

I would disallow overlap. Likewise, if we allowed overlap when defining regular polyhedra, there could be a lot more than the basic 5 platonic solids.

I figured the largest angle would be 360 degrees. All three vertices would be coincident on the "back" of the sphere, and the sum angle would be 360. i.e. a point - er... - everything except a point.

When I sketch it out, it makes no sense. The largest triangle on a sphere ends up having its 3 vertices coincident, sides of length zero, and area equal to the area of the sphere.

i.e. asking what the largest triangle on a spherical surface could be is equivalent to asking for the inverse of what the smallest area on a plane surface is (which is zero).

See my back-of-napkin sketch.

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Last edited: Apr 12, 2012
10. Apr 12, 2012

Vorde

Re: Question about area on a circle

Thank you on both counts, I am hesitant to follow the link simply because I don't want to accidentally read something I'm trying to figure out on my own, but in the end I might have to.

Good advice, thanks. I actually think I have the beginnings of an idea following this path.

Here is what I was thinking when I said 540 degrees. You could have an isosceles triangle with one point at the pole and two on the equator, the angle value for each point on the equator would be 90, and the angle value at the pole would be dependent on the distance between the two equator points. By varying that distance, and not letting the points overlap, you could get the angle of the pole point up to 360 (or infinitely close to that).

11. Apr 12, 2012

DaveC426913

Re: Question about area on a circle

Huh. You're right.

In fact, it goes even further. You could then move the equatorial line segment south toward the south pole, which would increase the two equatorial angles equally. In fact, you could bring it all the way around until you had the entire sphere inside the triangle except for an infinitesimally small triangle outside it. And that means that the sum of the angles of the triangle are 3 x (360-60) = 900 degrees total!

The only question remaining being this: are we really measuring the interior angles of the triangle anymore?

It is tantamount to counting the exterior angles of a triangle in Cartesian space and claiming the triangle contains the entire universe except what is inside this little area.

Last edited: Apr 12, 2012
12. Apr 12, 2012

Vorde

If you define the inside of the triangle to be any point where a line extending in any direction from that point will intersect one of the sides of the triangle, then yes I would say it is still the interior, although it doesn't seem it anymore.

However I haven't convinced myself that moving the equatorial points towards the south pole will increase the interior angle past 540. I'm having a tough time visualizing this though. It makes sense that a triangle like I described in my last post would account for half the surface area of the sphere, all you have to do to make the triangle as large as possible is to lower the equatorial points in that hemisphere triangle down to the south pole. It seems to me that while doing this the interior angle of the equatorial points wouldn't change, but as I said I can't visualize this well so I may well be mistaken.

13. Apr 12, 2012

mathwonk

if you accept my suggestion restrict to intersections of three hemispheres, you can still get a triangle with all three vertices arbitrarily close to the equator (and the triangle would fill up more and more of a hemisphere). Then what would the three angles approach?

14. Apr 12, 2012

DaveC426913

Not sure what this phrase means. Do you mean only use world lines?

15. Apr 12, 2012

AlephZero

Re: Question about area on a circle

But the surface of a sphere isn't a plane and isn't infinite, so the "common sense" analogies doesn't take the math forward much.

If you look at Euclid's Elements, the Greeks apparently never noticed there was an issue here. Euclid takes the idea of two points being on the same side or on opposite sides of a line as "obvious". There is a fairly well-known "proof" that all triangles are isosceles, based on drawing an "impossible" figure where two lines interect on the wrong side of a third line.

The OP will have to make some definitions about what is "valid", and figure out the consequences. It might be worth a look at the start of Hilbert's book "Foundations of Geometry" where he takes Euclid's axioms apart and puts them back together again, for plane geometry. It's a very readable book - as well as being a great mathematician, Hilbert was pretty good at explaining things. It's on Project Gutenberg, and elswhere on the web in Gutenberg doesn't have the diagrams.

16. Apr 12, 2012

Vorde

Looking back, although my question wasn't directly answered, I think (thanks to you all) that I have a better way of solving my problem now.

Thank you all for your help.

17. Apr 12, 2012

mathwonk

the intersection of hemispheres is always contained in a hemisphere, so is more restrictive than a triangle whose sides are geodesics.

18. Apr 12, 2012

Mholnic-

At risk of butting into a conversation not my own just to satisfy my own curiosity.. wouldn't the above statement be true for ALL triangles existing on a sphere? Including the "infinitesimally small triangle" that DaveC~ mentioned on the 'outside' of your 900~ degree triangle. On an infinite plane you could test whether you were inside or outside the triangle, but on a finite sphere... I don't believe you can. By creating one triangle on a spherical surface you automatically create a second triangle whose angles are complimentary to the other (outside or inside) triangle's angles; given this, wouldn't "inside" and "outside" just be arbitrary definitions based on personal preference?

19. Apr 12, 2012

Vorde

I have to think about this more (referring to Mholnic's conversation- only tangent to the original question), I'm having doubts about some of the triangles discussed before, specifically relating to how great circles only line up to lat/long lines once on a sphere.

Would someone with better knowledge confirm this for me: you can't take a equatorial circle and reduce it in size to a point (at the pole) whilst keeping it a geodesic.

I believe that statement is correct, and if so I believe I was originally correct by saying that the maximum angle measure for a spherical triangle is 540.

20. Apr 13, 2012

Wot he sed.