In class we did some experiments with spheres. This page has some brief notes summarizing the main points of the activity. The geometry of the sphere will be studied carefully in 445, so this was a physical experiment to broaden your intuition about parallels on the plane by contrasting with the geometry of the curved space called a sphere.
Straight paths on the plane are the straight lines. If you place a ribbon flat on a tabletop (modeling a plane), then the ribbon will lie (approximately) along a straight line. If you lay the ribbon down on its edge, then it will curl and lie over a curved path on the plane.
If you place a ribbon flat on a ball (modeling a sphere), then the ribbon will lie (approximately) along a straight path that is a certain kind of circle – a great circle. If you lay the ribbon down on its edge, then it will curl and lie over a curved path on the plane, perhaps a smaller circle.
For any point P on a sphere, the opposite point of P (or the antipodal point of P) is the point Q such that PQ is a diameter of the sphere. In other words, the center O of the sphere is the midpoint of PQ.
A great circle can be recognized in several ways. (1) It is one of the circles of maximum radius on the sphere. (2) For any point on the circle, the opposite point on the sphere is also on the circle. (3) The plane containing the circle also contains the center of the sphere.
The other circles on the sphere are not straight. The distance between two
points on such circles is longer that the distance along a great circle, as
the airlines know well when they fly from
For any two points A and B on a sphere, if the points are not opposite points, then there is exactly one great circle through A and B. This is the circle cut by the plane through A, B and O. If the two points A and B are opposite points, then every circle through A and B is a great circle, and there is an infinite number of such circles. These are the circles cut by the planes containing the line AB.
There are no non-intersecting great circles. Any two great circles intersect in two opposite points. So there are no parallel "lines" (great circles) on a sphere.
On the beach ball example, we saw great circles that look like longitude circles through the north and south poles on a globe. On such a globe the equator is a transversal that intersects the longitude circles at right angles, but in this case having a common perpendicular transversal does not make the great circles parallel.
On the plane a rectangle is a parallelogram with 4 equal angles. Each of the angles is a right angle. On the sphere there are no parallel lines at all, so no parallelograms. A quadrilateral with 4 equal angles exists, but the angles will all turn out to be greater than 90 degrees.
Picture a great circle as the equator on a globe. Then the points at a fixed distance from the equator are a certain number of degrees either north or south of the equator and thus form two circles that are called parallels of latitude.
But wait, did we not say there are no parallels on the sphere? Not quite. We said there are no parallel "lines" or great circles. There is the concept of a parallel curve that consists of the points at a certain distance from a given curve. For example, in the Math 487 lab last week we traced some disks whose boundaries were parallel curves. Thus the parallel curves at distance 1 from a circle of radius 10 are two concentric circles with radii 9 and 11.
The same is true on the sphere. The parallel curves at a certain distance to a great circle are 2 concentric circles, in this case the two parallel circles on the sphere are the same size and of course smaller than the great circle.
On a Mercator projection map of the globe the parallels of latitude are horizontal lines. The region with say north latitude between 40 and 50 degrees and west longitude between 110 and 130 degrees is a rectangle on the Mercator map. The figure on the sphere also has four right angles. But the figure on the sphere has edges on the north and south that are curved, not straight, so it is not a good generalization of a rectangle in the plane.
The concept of "non-intersecting straight paths" and the concept of paths at a constant distance do not coincide except on the plane (i.e., the Euclidean plane that we are studying). They are two different concepts, and a big part of Chapter 2 is that these distinct concepts coincide on our familiar and flat plane.
PS. There will be lots more about the sphere, but the details will be in Math 445.