# Testing the straightness of a line

• Sep 30th 2010, 05:47 AM
statmajor
Testing the straightness of a line
How would you test if a line is straight (on a plane and on a sphere) assuming you dont have tools to do it?

For example, how would a person done it hundreds of years ago?

For straight lines on a plane: I got plumb lines, symmetry/reflection.

But I'm stuck on straight lines on a sphere.

Any suggestions would be greately appreciated (for both sphere and plane).
• Sep 30th 2010, 06:04 AM
Ackbeet
Define "straight line on a sphere". Do you mean great circles? Or just small circles? Or something else?
• Sep 30th 2010, 12:30 PM
statmajor
From what I understand, if you have a line on a sphere, how would you know if it's straight.

For example, if the equator was a line, it would be a straight line, but I have to explain it as if I was talking to someone with no experience about this.

Great circles seem to be the right idea. Do you know of other stuff similiar to this?
• Sep 30th 2010, 03:47 PM
Ackbeet
Quote:

Great circles seem to be the right idea.
Seem? Before you can even tell whether a line is "straight" or not, you must define what a straight line is. On a sphere, this is not going to be the usual intuitive idea of a straight line, because no line that lies on a sphere is straight in the Euclidean sense. You have to tell me what a straight line is. Then we can go about trying to find a test for straightness of lines. Until we've defined our terms, further discussion is not going to mean much. (A lesson for life, perhaps? Hmm.)
• Sep 30th 2010, 03:54 PM
statmajor
I define a straight line as the shorted distant between 2 arbitrary point on a plane.
• Sep 30th 2010, 03:57 PM
Ackbeet
So, on a sphere, straight lines are the shortest path (constrained to be on the sphere) between two points on the sphere?
• Sep 30th 2010, 03:59 PM
statmajor
Probably not. I only defined what a striaght line is on the plane. The problem is that I'm not sure how to define on a sphere.
• Sep 30th 2010, 04:20 PM
Ackbeet
Well, perhaps we could back up a bit. What is the more general problem you're trying to solve?
• Sep 30th 2010, 04:45 PM
statmajor
The problem I'm trying to find is how would you know if a line is straight in the plane and in the sphere.

For a plane I defined a straight line as the shortest distance between two arbitrary points. I also said that if you know a line is straight. For example, a line made from a plumb line (a rope with a weight attached to it) is straight due to the force of gravity acting on it. Thus a line that is either parallel, perpendicular or one that is just translated would also be straight. Also if you choose arbitrary points on the line, and if you were to "walk" on the line without rotating or changing your direction and you managed to hit every singline one of those points than that line is also striaght.

Now I need to figure out how I would define a straight line on a sphere. From the link on great circles you provided me with, it says that "...serve as the analogue of “straight lines” in spherical geometry". I believe that a Great circle is an example of a straight line on a sphere, which also means that there can only be 1 straight line on a sphere as there only exists one great circle per sphere (from what I understand; I could be wrong). It also says that it's the path with the least curvature which could be the key to define straitness on a line.
• Sep 30th 2010, 05:26 PM
Ackbeet
Quote:

...which also means that there can only be 1 straight line on a sphere...
No, there are infinitely many great circles on a sphere. A great circle is any circle drawn on the sphere which, if you were to slice the sphere using the circle as your guide, would slice right through the center of the sphere. Take one great circle on a sphere. It's going to have to opposite points on it that, if you were to draw the straight line through the sphere, would go through the sphere's center. Now rotate the entire great circle through any ol' angle about that diameter, and you'll get another great circle.

So, let's take great circles as straight lines in the spherical geometry. How would you know if a line was a great circle or not? Well, let's assume that you complete the entire line, so it goes all the way around. I'm thinking physically here. Put the sphere on a flat surface such that a point on the line touches the flat surface. Then, if you were to come down with another flat surface parallel to the first flat surface, it should touch the sphere at the opposite point on the great circle. If, instead, you had a small circle (not a great circle), then two points on the opposite sides of the sphere cannot both be on the small circle.

Practically, of course, it would probably be useful if these flat surfaces were glass so you could see through them. And I'm not entirely sure how you'd get them to be parallel, other than, say, having them descend on guiding rods or pistons or something.

That's my idea.
• Sep 30th 2010, 05:28 PM
skeeter
a single great circle connects any two different points on a sphere that are not antipodal, but a sphere does contain an infinite number of great circles.
• Sep 30th 2010, 05:36 PM
statmajor
Quote:

Originally Posted by Ackbeet
So, let's take great circles as straight lines in the spherical geometry. How would you know if a line was a great circle or not? Well, let's assume that you complete the entire line, so it goes all the way around. I'm thinking physically here. Put the sphere on a flat surface such that a point on the line touches the flat surface. Then, if you were to come down with another flat surface parallel to the first flat surface, it should touch the sphere at the opposite point on the great circle. If, instead, you had a small circle (not a great circle), then two points on the opposite sides of the sphere cannot both be on the small circle.

That's a great idea.

Quote:

Originally Posted by Ackbeet
No, there are infinitely many great circles on a sphere

Sorry, I meant to say that there could only be 1 same sized circle per sphere. If you would rotate the circle, it would just be the same circle except rotated.

Thanks again.
• Oct 1st 2010, 05:17 AM
Ackbeet
You mean there's only one size for great circles, right? I'd agree with that.