Let's begin by saying that this course is not a direct continuation of Math 5335 (Geometry I). There will, however, be some similarities in approach. Thus, skills that continuing students developed in Geometry I will be useful in the present course. On the other hand, students who are entering the course with slightly different preparation [of a comparable level ... ] also will be able to have a good experience.
Please recall that the the parallel axiom in Euclidean geometry can be stated as follows:
In Geometry I, we studied hyperbolic geometry, in which there are at least two lines through a given external point and parallel to a given line. In Geometry II, one of our topics will be projective geometry. In the projective plane, there are no parallel lines: thus, any two distinct lines actually intersect! In higher dimensional projective space, we still don't have parallel lines, but there are pairs of lines that don't intersect -- just as there are pairs of skew lines in Euclidean n-space for n > 3. (As a very particular instance, look around the room where you're sitting as you view this page. Consider the line where the wall in front of you meets the ceiling, and the line where the wall to your right meets the floor. Those two lines certainly do not meet, but we usually do not want to say that they're parallel. Indeed, they're going in very different directions ... )
So, one of the things that we'll do before actually getting into projective geometry is to clarify some concepts from Euclidean geometry (or specifically the analytic version of it) in 3 or more dimensions. Well, really, we'll mostly focus on the case of 3 dimensions, although there are a few interesting tidbits about the case of 4 dimensions that can help us to understand what some 3-dimensional concepts really do [or don't] involve.
Historically, one of the main antecedents of projective geometry was perspective drawing. In such drawings -- and in photographs for that matter -- rectangles usually aren't represented as rectangles. Indeed, two parallel edges often appear to meet at a "vanishing point" -- on the horizon in the case of horizontal edges ... Here's an instance (maybe not a great classic ... ) that I found by doing an internet search on "perspective drawing".
We need to make one small disclaimer here: this material is not entirely the same thing as projective geometry since some pairs of parallel lines actually look like parallel lines in the photograph. For instance, this is true of vertical lines. And, for that matter, if the photographer had stood directly in front of the building [the Calgary armory], then the front surface would have looked reasonably rectangular -- but the photo wouldn't have looked 3-dimensional.
Nonetheless, perspective drawing can give us some good intuition about where projective geometry came from. Without going into the mathematical formalism of how it's actually accomplished, let's just mention that the projective plane has points at finite distance which correspond to the points of the Euclidean plane, and also points at infinity -- one for each equivalence class of parallel lines in the Euclidean plane. We get a line in the projective plane by adjoining the corresponding point at infinity to each Euclidean line, and finally there's the line at infinity: namely the set of all points at infinity. [In some sense, we can think of the line at infinity as a virtual object similar to the horizon in a perspective drawing.]
Another antecedent for projective plane geometry is provided by spherical geometry. The "lines" in that geometry are great circles on the surface of a sphere. One definition of a great circle is that you take a plane that goes through the center of the sphere; the intersection of that plane with the surface of the sphere is a great circle. A practical manifestation of this is that the shortest path between two points on the surface of a sphere is an arc of a great circle. As a particular instance, imagine that you're going to fly from Minneapolis to Venice. Both cities are at a latitude of about 45 degrees north. [So, why our weather is so much colder than theirs is another story ... ] One route would be to follow the 45th parallel. But this isn't the shortest route. Similarly, the plane that contains the 45th parallel is parallel to the plane of the equator and thus doesn't pass through the center of the earth. On the other hand if you stretch a piece of string against a globe, with one end at Minneapolis and the other end at Venice, then you'll approximately see the great circle route -- which passes over the northern part of the province of Quebec. Actually, I found an internet site called " Great Circle Mapper" where you can have such maps interactively drawn. Here's the result, using the default setttings, for the path MSP-VCE:
Now, one "deficiency" of spherical geometry is that two great circles actually intersect at two points, rather than just at one point as is the case with distinct [non-parallel] lines in the plane. But we can remedy this by passing to the projective plane. In fact, there's a specific mathematical transformation under which the points in the northern hemisphere correspond bijectively to points of the Euclidean plane -- or the points at finite distance in the projective plane. Thus, if we're working with the unit sphere x2 + y2 + z2 = 1, then we can make points on the sphere with z > 0 correspond with points of the plane z = 1. Then, the points on the equator (i.e., points with z = 0 will correspond to the points at infinity in the projective plane; however this correspondence turns out not to be bijective but rather 2 to 1. {Namely, there are two opposite points on the equator that map to each point on the projective line at infinity.} But we postpone discussion of the details for a little while anyway. ... [To be continued]
For a slightly more specific list of topics, see the course syllabus.
Text: Geometry II, by B. Fristedt (Available from Alpha Print in Dinkytown)
Other useful books
Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail:
roberts@math.umn.edu
http://www.math.umn.edu/~roberts