Angles within a spherical triangle

[blondii]

Hi Guys,

need some assistance. I am sure what I am asking is trivial but i still need help. How could i find the angle within a spherical triangle (triangle formed on a sphere). Now this triangle has equal lengths on all 3 sides.

Pleas help!

 

[Studiot]

What you are asking is not trivial.
But unfortunately neither is it possible without further information.

Consider the following:

Take a globe with lines of latitude and longitude marked.

Take any pair of lines of longitude (containing some angle alpha) and follow the triangle from formed the pole to a parallel of latitude where the intersecting length of the parallel of latitude is the same as the length of both lines of longitude (which are obviously the same and meet the parallel of latitude at right angles)

 

[blondii]

Thanks studiot for your explanation. Well i do have other information about the triangle. I just realized the triangle is not a spherical one because two of its sides are made up of two arcs from a great circle and the last side is made up of an arc from a small circle. Given i know the lengths of the three sides and one angle.

Say for the triangle ABC on the sphere, side AB and BC are made up of arcs from great circles whereas side AC is made of an arc from a smaller circle. Angle between side AB and BC is also known. Is there a special formula or a way to find the area for this triangle.

Please help.

 

[holy_toaster]

There is an explicit formula for the SUM of angles in a spherical triangle. The sum is of course larger than Pi (=180 degrees). If I remember correctly there is an additional summand in the formula depending on the area A of the triangle and the curvature radius R of the sphere. Something like A/R^2. Just look it up on Wikipedia. If you then have additional information on the length of sides you should be able to determine individual angles.

 

[blondii]

Thanks holy toaster. I know the explicit formula and surface area formula but they all relate to a spherical triangle. As you may have seen in my question, the triangle is not a spherical one as only two of its sides are part of an arc from a great circle. Will the same formulas applied to spherical triangles also be applied to non spherical ones or is there a technique I'm missing? I believe the formulas are incompatible because everywhere I'm reading from only talks about applying the formulas for spherical triangles (only) but not non-spherical triangles.

 

[holy_toaster]

As far as I know, you have a spherical triangle if it lives on the surface of a sphere. It does not matter if the sides are (parts of) great circles or not. The formula for the sum of angles is valid for all spherical triangles, but it depends on the area.

 

[blondii]

Well I've read quite abit before posting in this forum and it all says that it is not a spherical triangle even though it is on the surface of the sphere.

1. A spherical triangle is a 'triangle' on the surface of a sphere whose three sides are arcs of great circles.

2. The length of each side is the length of the arc, and is measured in degrees, this being the angle which the points at the ends of the arc make at the centre of the sphere.

3. There are three angles between these three sides. The sum of these three angles does not make 180 degrees. Instead, the total is always greater than 180 degrees.

4. When a spherical triangle is illustrated, each of the three sides is drawn as an arc.

I'm sure the formulas were formulated with respect to the definition of an ideal spherical triangle. The question is how can we relate these formulas to non-spherical ones. Is there a site i can be forwarded to that can tell me otherwise because every online page basically talks about spherical triangles but don't specifically talk about the triangles that lie on the sphere but a not spherical (by definition).

 

[Studiot]

Post a diagram of your problem.

 

[holy_toaster]

I see. Indeed this holds only for spherical triangles consisting of arcs. But in this case there is indeed more detailed information on your problem necessary...

 

[Studiot]

OK I think I understand your difficulty.

Take any two points P and Q on the surface of a sphere.

You can always draw an arc of a great circle between them.
You find the plane containing this arc by drawing radii from P and Q to the centre O.

P and Q may also be connected by other curves on the surface, for example the circular curve in a plane at right angles to the axis is a lesser arc unless the plane is diametral. (when it is a great arc).

I have shown the difference in the attachment.

So you can always create a spherical triangle between any three points on the surface of a sphere. You may have to do some additional plane trigonometry however.

go well

 

[blondii]

Well guys i have attached the picture so that you guys know what I am talking about. Now i want to find the surface area of triangle ABC while maintaining the current definition of the triangle in the diagram (i.e, AB is arc from a great circle, BC is an arc from a great circle and AC an arc from a small circle). Now I am trying to find the area that is shaded (ABC). Thanks for the explanation studiot, but can i still implement that method to finding the area of the shaded region in the figure attached?

 

[blondii]

Well guys i have attached the picture so that you guys know what I am talking about. Now i want to find the surface area of triangle ABC while maintaining the current definition of the triangle in the diagram (i.e, AB is arc from a great circle, BC is an arc from a great circle and AC an arc from a small circle). Now I am trying to find the area that is shaded (ABC). Thanks for the explanation studiot, but can i still implement that method to finding the area of the shaded region in the figure attached?

Note that i would like to find the area of ABC that includes the area when AC is an arc of a small circle and NOT when we convert the arc between the two points AC to be of from a great circle bacause then the area would be different.

 

[Studiot]

There seems to be some mixup between your description in words and your diagram.

As drawn

B is the pole.

BC, BA are arcs of great circles

AB is an arc of a parallel small circle or an arc of a great circle set at some angle to the pole.

What you need to understand is that these two arcs AB in fact trace out the smae path on the surface of the sphere.

They have different radii because their respective radii are measured in different planes. This is possible in 3dimensions, but not, of course possible in 2D

 

[blondii]

Thanks guys I have finally figured it out. Let's say the surface area above the region where points A and C are on is ω. Let's also assume that the angle subtended by the line both points A and B are on (as shown in my previous attached diagrams) is θ with regards to the base of the partial sphere and not the hemisphere. Thus the area ABC is equal to (θ/2π)*ω.

Where:
ω = π(c^2 + H^2) : - Formula for area of partial sphere.
-c -> radius of the base of the partial sphere.
-h -> height from the base of partial sphere to the top of the sphere.
π = constant pi

Thanks guys for all your assistance again.

Cheers

 

[felper]

the angle you seek, is the angle between the tangent vectors of the end of each side of the triangle

 

[camel_jockey]

If I draw a random triangle on the plane, angles a,b and c, and then I "lift" this into the sphere above (using mapping x,y \mapsto (x,y,sqrt( 1- x^2 - y^2)) , how do i find the new angles if we are using a non-specified metric g??

I am hoping someone can answer this in the language of manifold mappings, pushforwards, metrics, inner products etc...