Define "straight line on a sphere". Do you mean great circles? Or just small circles? Or something else?
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).
Define "straight line on a sphere". Do you mean great circles? Or just small circles? Or something else?
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?
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.)Great circles seem to be the right idea.
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.
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....which also means that there can only be 1 straight line on a sphere...
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.