Basic information about spheres
We will start with the basics. A sphereLines and spheres.
If we take an arbitrary line and a sphere in three space, several things can happen. First, the line and the sphere can miss each other. That case is not very interesting. Secondly, the line can intersect the sphere in only one point. In that case the line is tangentThe case which is most interesting is when the line passes through the center of the sphere. In this case the two points of intersection with the sphere are said to be antipodal
Planes, spheres, circles, and great circles.
Next let's look at a plane and a sphere. Again there are several things that can happen. In the uninteresting case the plane and the sphere miss each other. If they do meet each other there are two possibilities. First they can meet in a single point. In this case the plane is tangentGreat circles become more important when we realize that the shortest distance between two points on the sphere is along the segment of the great circle joining them. On any surface the curves that minimize the distance between points are called geodesics.
A pretty good approximation to a great circle can be drawn through two points on a beach ball by holding a piece of string tight to the ball at the two points in question. The tightness of the string has the effect of minimizing the length of the string, and therefore closely approximating a geodesic.
We now have the beginnings of a geometry on the sphere. In plane geometry the basic concepts are points and lines. On the sphere we have points, of course, but no lines as such. However, since the great circles are geodesics on the sphere, just as lines are in the plane, we should consider the great circles as replacements for lines. We can then compare the two geometries.
Exercise:
Take your favorite geometry book, and find its list of axioms. Which of the axioms are true for the sphere? If a planar axiom is not literally true on the sphere try to rephrase it so it is true. This exercise cannot be completed at this point. It is one that will continue throughout this material. At this point you should check the axioms of incidence. Remember that a great circle is the intersection of the sphere with a plane that passes through the center of the sphere. Almost every fact we will discover about great circles will follow from that fact.
No comments:
Post a Comment