Deriving the area of a spherical triangle from the metric

In summary, the metric for a 2-sphere can be expressed as $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$ and the area of a triangle on the sphere can be calculated by using spherical trigonometry or by using the more familiar metric, $$ds^2 = R^2 ( d^2 \theta + \sin^2(\theta ) d^2 \phi )$$ with ranges of ##0\leq r \leq \pi R/2## for the first term and ##0\leq \theta \leq \pi## and ##0\leq \phi \leq 2\pi## for the second term.
  • #1
Arman777
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The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$

Is there an equation to describe the area of an triangle by using metric.

Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.
 
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  • #2
Hello,
I think that the area of a triangle in metric would be the same as in imperial. It is just a ratio really. 1/2 B * H
would be the same for metric right?

unless you take the sector and neglect the segment.; Triangle = sector - segment. You could probably derive this in radians easily enough.

jeff
 
  • #3
jeff davis said:
Hello,
I think that the area of a triangle in metric would be the same as in imperial. It is just a ratio really. 1/2 B * H
would be the same for metric right?

unless you take the sector and neglect the segment.; Triangle = sector - segment. You could probably derive this in radians easily enough.

jeff
I did not understand what you are trying to mean.
 
  • #5
Well kind of. But I understand the problem ...so I guess there's no need
 
  • #6
Arman777 said:
Is there an equation to describe the area of an triangle by using metric.
It is unclear what you mean. You obviously need some information about the triangle itself. Which information about the triangle do you want to use? Or are you after a more general integral expression for the area of a part of the sphere?

jeff davis said:
I think that the area of a triangle in metric would be the same as in imperial.
This is not about units, it is about the metric tensor used in differential geometry as applied to a curved surface.
 
  • #7
1572603145456.png


Orodruin said:
It is unclear what you mean. You obviously need some information about the triangle itself
Yes I realized that after some time later. First I thought that from the metric we can define a general area equation for any spherical triangle. However as you mentioned that's possible.. Let's suppose we have a triangle like this. N is the north pole and it makes an angle of A. From the metric I find that the angle can be expressed as

$$Area = \int_0^A \int_0^r Rsin(r/R)drd\theta$$
 
Last edited:
  • #8
Your triangle is not well defined, you need to specify three numbers to completely specify the triangle.
 
  • #9
Orodruin said:
Your triangle is not well defined, you need to specify three numbers to completely specify the triangle.
Well yes sorry the other length is also r. So the bottom length becomes ##ARsin(r/R)dr## (from ##ds = Rsin(r/R)d\theta)##

Actually the area is $$Area = \int_0^rARsin(r/R)dr$$ but this is also equal to $$Area = \int_0^A \int_0^r Rsin(r/R)drd\theta$$
 
  • #10
That is not a triangle. A triangle has geodesics as its sides and the line with the bottom length you have specified is not a geodesic unless ##r## is the distance from the pole to the equator.
 
  • #11
Orodruin said:
That is not a triangle. A triangle has geodesics as its sides and the line with the bottom length you have specified is not a geodesic unless ##r## is the distance from the pole to the equator.
Hmm I know that ##r ∈[0, \pi R/2)##. But that's confusing. I mean

1572605300894.png


$$ds^2 = dr^2 + R^2sin(r/R)^2d\theta^2$$ so for ##h##, ##dr = 0 (?)##.

I see you are meaning that if ##r\ne \pi R/2## then ##dr\ne0##

If we know the value of the A is it possible to find the h from sherical trigonometry (without knowing the value of ##r##)? I guess metric is useless to find the ##h## then
 
  • #12
Unless you know three parameters describing the triangle, it is impossible to find the others. This is true also in Euclidean space. If you do not know r, it is impossible to find h just based on the angle.
 
  • #13
Hmm I see. I ll try to look something else for solution. Thanks then
 
  • #14
Orodruin said:
line with the bottom length you have specified is not a geodesic unless rrr is the distance from the pole to the equator.
I ma kind of having trouble to understand this. At post 11 $$h = ARsin(r/R)$$ ? I mean it has to be that way..
 
  • #15
Arman777 said:
I ma kind of having trouble to understand this. At post 11 $$h = ARsin(r/R)$$ ? I mean it has to be that way..
Again, this is not a triangle. It is a circle sector.
 
  • #16
Orodruin said:
Again, this is not a triangle. It is a circle sector.
Okay.. umm then how can I find the area of that thing. What should I do ? Is it better maybe to use spherical coordinates ?
 
  • #17
Of the circle sector or the triangle?
 
  • #18
Orodruin said:
Of the circle sector or the triangle?
Triangle
 
  • #19
Arman777 said:
The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$

Here, by ##r## do you mean ##R \theta##? Then your metric would be the more familiar form.
$$ds^2 = R^2 ( d^2 \theta + \sin^2(\theta ) d^2 \phi )$$

And ##\theta## would have range 0 to ## \pi## and ##\phi## would have range 0 to ##2 \pi##. Then it all becomes very familiar surface-of-a-sphere type geometry. Is that it?
 
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  • #20
Well that could have helped I guess, yes.
But I used the spherical trigonometry to find the area of the triangle.. But the metric you wrote also seems well for my problem as well. Sadly I already handed out the paper...
 

1. What is a spherical triangle?

A spherical triangle is a triangle formed on the surface of a sphere, where the sides of the triangle are arcs of great circles on the sphere.

2. What is the metric of a spherical triangle?

The metric of a spherical triangle is a set of three angles (α, β, γ) and three sides (a, b, c) that describe the shape and size of the triangle on the surface of a sphere.

3. How is the area of a spherical triangle derived from the metric?

The area of a spherical triangle can be derived using the Spherical Law of Cosines, which states that the cosine of a side of a spherical triangle is equal to the cosine of the opposite angle multiplied by the cosine of the adjacent angle. By rearranging this equation, the area of the triangle can be calculated using the given angles and sides.

4. What is the formula for calculating the area of a spherical triangle?

The formula for calculating the area of a spherical triangle is A = (α + β + γ - π)R^2, where α, β, and γ are the angles of the triangle and R is the radius of the sphere.

5. Can the area of a spherical triangle be negative?

No, the area of a spherical triangle cannot be negative. It is always a positive value, as it represents the surface area enclosed by the triangle on the surface of the sphere.

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