Thread: Straight on a Sphere

What is Straight on a Sphere?

a.) Imagine yourself to be a bug crawling around on a sphere. (This bug can neither fly nor burrow into the sphere). Thr bug's universe is just the surface; it never leaves it. What is "straight" for this bug? What will the bug see or experience as straight? How can you convince yourself of this? Use the properties of striaghtness.

b.) Show (that is, convince yourslef, and give an argument to convince others) that the great circles on a sphere are straight with repsect to the sphere, and that no other circles on the sphere are straight with repsect to the spehere.


Any thoughts that can be helpful?

Originally Posted by matgrl

What is Straight on a Sphere?

a.) Imagine yourself to be a bug crawling around on a sphere. (This bug can neither fly nor burrow into the sphere). Thr bug's universe is just the surface; it never leaves it. What is "straight" for this bug? What will the bug see or experience as straight? How can you convince yourself of this? Use the properties of striaghtness.

b.) Show (that is, convince yourslef, and give an argument to convince others) that the great circles on a sphere are straight with repsect to the sphere, and that no other circles on the sphere are straight with repsect to the spehere.


Any thoughts that can be helpful?

You can think of the shortest distance between two points on the surface of a sphere. The curve will be always be a segment of a great circle.

Thank you very much, "You can think of the shortest distance between two points on the surface of a sphere. The curve will be always be a segment of a great circle", this is what I have learned in class.

Is there any way you could prove this fact? How do we know this for sure?

Originally Posted by matgrl

Thank you very much, "You can think of the shortest distance between two points on the surface of a sphere. The curve will be always be a segment of a great circle", this is what I have learned in class.

Is there any way you could prove this fact? How do we know this for sure?

Proofs can get into some heavy-duty maths.

Geodesics on spheres are great circles - MathOverflow

Great Circle -- from Wolfram MathWorld

Since the question doesn't ask for a formal proof, maybe you could use a watered-down argument like: Suppose you're in the Northern hemisphere and want to go to the north pole (assume it's just a point, and ignore the difference between magnetic/geomagnetic north poles, etc.). Then to minimise distance you would follow your compass in such a way that you're always going exactly North. ("Why," you ask? Because that's what our notion of "straight" and "distance" is on Earth! This isn't a formal proof.) So your path would be part of a meridian. If you're in the Southern hemisphere then you would expect North to be the exact opposite of South, thus you would again trace out part of a meridian to get to the North pole. Meridians are of course just great circles.

This is awesome! Thank you very much for your help! I have one question though: How would you show the distance with your compass exactly? Is there any diagrams I could look at for doing this?

Originally Posted by matgrl

This is awesome! Thank you very much for your help! I have one question though: How would you show the distance with your compass exactly? Is there any diagrams I could look at for doing this?

Not sure what you mean. Compass only gives direction, not distance. If you know your location in longitude and latitude, then you can find your distance to the North pole using some basic geometry. In fact longitude isn't even needed. Suppose you are at location X, the centre of Earth is O, and the north pole is N. You use latitude to find angle XON, plus you must know the radius of the Earth.