Given a sphere and a compass, is it possible to draw a great circle on the sphere?

Question

I'm reposting this question in order to continue discussion on this matter. See original posted question:

http://answers.yahoo.com/question/index;_ylt=AqDfOmto4wKkfjyBks2D4Prty6IX?qid=20070716085410AANyufm&show=7#profile-info-uDRfzKDFaa

"Using a compass on a sphere" is the act of drawing a circle on the sphere, given a point on it, and a span. A span shall be defined to be that distance from the circle to the center, which is not the radius of the circle. In order to draw a great circle on the sphere, the correct span has to be found through construction. The problem is finding the exact span, given a sphere of unknown radius.

Practical difficulties of using a real compass on a ball shall be ignored.

Zanti, using a given compass span, a circle can be drawn, and its circumference divided into six parts, but only on a plane, not on a sphere. But anyway, the problem is to find the exact span, so that the great circle may be drawn for any given center. The problem is not to draw the great circle through a pair of points, that's a different problem.

If I pick two points on the sphere and draw two intersecting circles, I now have 4 points in a symmetrical diamond pattern. I can then immediately draw four more circles on the other side of the sphere. The centers of the 2 sets of four points would be antipodes of the sphere.

If the midpoint between 2 points can be found by compass construction on the sphere, then the method of drawing the great circle follows. It's already a tricky enough problem just to be able to do this on the plane, but I've left it open on whether or not this is even possible on the sphere.

Jeffrey, given 4 points in a diamond formation on a sphere, it's trivial to construct 4 more points in a diamond formation on the other side. They're not necessarily congruent, but they would have centers which are antipodes of the sphere.

I should add that this is a MATH problem, not a physical one using a real compass.

Answer 1

To be honest, I read through your proof in the previous question back when you posted it, and I had trouble understanding why it worked. It seemed to me the construction you gave finds four points that form a rectangle easily enough. However, I don't see where the rectangle has any particular orientation, so I don't see where the great circle is derived.

Now, when I thought about the problem, I imagined seven circles all of equal radius, producing intersection points that form a regular hexagon. Using this hexagon, at first I thought any of the three longest diagonals can be extended to form a great circle. But no, that doesn't sound right. Like your rectangles, I don't see where hexagon has any particular orientation.

* * * * *

Wait, suddenly I can visualize this - and now I see why the longest diagonal of my hexagon, along with your diamonds, produce great circles. Here's the visualisation:

Start by drawing any circle on the sphere. Now imagine you slice through the sphere along the perimeter of the circle, producing a sphere with a circular hole. Clearly, if you can connect opposite ends of the hole, i.e. find a diameter, you can extend this line around the sphere, forming a great circle. And, in fact, the longest diagonals of the hexagon do exactly that -- they form a diameter of the circular hole I am imagining. Your diamond pattern works the same way. So, yes, there are ways to do this.

On your post in Additional Details that dividing a circle into six parts works on a plane, but not on a sphere -- why not? Yes, the surface is curved. However, the circles are all the same size, so the curve will be the same for all the circles.

Answer 2

I think you guys are oversimplifying this.

The diamond thing doesn't work. You can't immediately draw a diamond on the other side... you wouldn't know where the other side is. And if you did, then you could just draw a point, and draw the point on the other side right away, and be done with it.

Also, how are you supposed to find the center of the diamond? Magically draw straight lines between the 4 points?

Also, you can't draw the circle going through those points anyways. Put 2 points on a piece of paper. Now try to draw a circle through both of them. It's definitely not easy. You need to figure out how to find the center.

And zanti... you can't actually make your hexagon. It'd be hard enough to draw in the third circle...
And then how did you plan to draw a circle on the diagonal anyways?

I think this construction is either really easy, or impossible.

You could always just start with the compass set on a radius larger than the sphere's, and then shrink the radius until it was actually possible to draw the circle. But I guess that's why we're ignoring physical limitations.

Oh, and also, in order to draw the great circle, we also need a pen or pencil.