Thread: An Angle on a Sphere

I am having a hard time trying to come up with a definition of an angle on a sphere. Do the properties, such as for angles in a plane, apply to those in a sphere, and if so how I can prove they do (or do not) apply? Is it possible to create a square on a sphere?

I thought it was possible to create a sphere, just the angles would not have 90 degrees. I understand great circles are the "straight lines" on a sphere. Attempting to understand angles is hard though. Does ASS, SSA and other theorems which do not work in a plane, work on a sphere? Do the regular ones work on a sphere?

And in general on a plane, how do you prove that the opp. angles formed by 2 intersecting strght lines are congruent? What properties of straight lines are being used when trying to prove this? How are we able to check 2 angles are congruent (not using a protractor, ruler, etc, etc of course).

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Some of my thoughts:

If you take a circle (not a great circle) on a sphere, it's analgous to a circle on a plane. Also, every great circle has TWO centers. Further, great circles are the only way to create an angle....

Angles in general have to have straight lines. We can think of angles on a sphere in respect to movement, dynamically, and geometric shape. Locally on a sphere, angles looks almost like a Euclidean space.

Yeah...

This stuff messes with my mind !

Originally Posted by fifthrapiers

I am having a hard time trying to come up with a definition of an angle on a sphere. Do the properties, such as for angles in a plane, apply to those in a sphere, and if so how I can prove they do (or do not) apply? Is it possible to create a square on a sphere?

I thought it was possible to create a sphere, just the angles would not have 90 degrees. I understand great circles are the "straight lines" on a sphere. Attempting to understand angles is hard though. Does ASS, SSA and other theorems which do not work in a plane, work on a sphere? Do the regular ones work on a sphere?

And in general on a plane, how do you prove that the opp. angles formed by 2 intersecting strght lines are congruent? What properties of straight lines are being used when trying to prove this? How are we able to check 2 angles are congruent (not using a protractor, ruler, etc, etc of course).

--------------

Some of my thoughts:

If you take a circle (not a great circle) on a sphere, it's analgous to a circle on a plane. Also, every great circle has TWO centers. Further, great circles are the only way to create an angle....

Angles in general have to have straight lines. We can think of angles on a sphere in respect to movement, dynamically, and geometric shape. Locally on a sphere, angles looks almost like a Euclidean space.

Yeah...

This stuff messes with my mind !

Locally, yes, the properties of the surface will be Euclidean. (Unless the surface has some pathological properties, such as a singularity or a hole.) That's how we can even DO Euclidean geometry on paper, which is on the surface of the Earth, which is curved.

However, plane properties do not hold on a larger scale. Consider, for example, a triangle on the surface of a sphere. It doesn't need to have 180 degrees for the sum of its interior angles. Consider a triangle starting at the pole, go down to the equator, travel the equator for a 1/4 of a circle, then go back to the pole. The sum of the interior angles in this "triangle" is 270 degrees. In a similar manner many other Euclidean properties of figures do not hold.

Check this with the other (more qualified) members of the forum, but I would imagine the best definition of the angle made by two intersecting lines on your surface would be to "zoom in" on the point of intersection to look at things locally, and use the Euclidean definition of an angle. Needless to say, the same angle at two different points on your suface may wind up looking vastly different on a macroscopic scale, so the geometry could include some unusual properties compared to Euclidean geometry.

Also, every great circle has TWO centers.

I've never heard of this and it boggles my vastly limited visualization abilities. Where is the second center??

-Dan

Originally Posted by fifthrapiers

I am having a hard time trying to come up with a definition of an angle on a sphere. Do the properties, such as for angles in a plane, apply to those in a sphere, and if so how I can prove they do (or do not) apply? Is it possible to create a square on a sphere?

I thought it was possible to create a sphere, just the angles would not have 90 degrees. I understand great circles are the "straight lines" on a sphere. Attempting to understand angles is hard though. Does ASS, SSA and other theorems which do not work in a plane, work on a sphere? Do the regular ones work on a sphere?

And in general on a plane, how do you prove that the opp. angles formed by 2 intersecting strght lines are congruent? What properties of straight lines are being used when trying to prove this? How are we able to check 2 angles are congruent (not using a protractor, ruler, etc, etc of course).

--------------

Some of my thoughts:

If you take a circle (not a great circle) on a sphere, it's analgous to a circle on a plane. Also, every great circle has TWO centers. Further, great circles are the only way to create an angle....

Angles in general have to have straight lines. We can think of angles on a sphere in respect to movement, dynamically, and geometric shape. Locally on a sphere, angles looks almost like a Euclidean space.

Yeah...

This stuff messes with my mind !

Have you looked into spherical trigonometry? This Wikipedia page may answer some of your questions about how to define angles on a sphere and what relationships hold.



This picture shows three great circles. Each pair intersects twice, but where are the two centers for each circle?