Some history The the parallel axiom in Euclidean geometry can be stated as follows:
Starting with the ancient Greeks and continuing up to around the year 1800, there was some lingering uneasiness about this axiom. The other axioms of Euclidean geometry could be viewed being self-evident properties of the physical world, but it did not seem to be possible to physically verify the parallel axiom. Indeed, such a verification would seem to involve measurements on a scale that was so large as to be impractical.
Instead, a number of mathematicians tried to show that the parallel axiom could be proved from the other axioms of Euclidean geometry. All of these efforts failed. We now know the reason for this: the other axioms of Euclidean geometry, while consistent with the Euclidean parallel axiom, also are consistent with the hyperbolic parallel axiom, which says that there are at least two lines through a given external point and parallel to a given line.
The coordinate plane R2 studied in analytic geometry classes is the most familiar model of Euclidean geometry. Indeed, it satisfies all of the axioms of Euclidean geometry, including the parallel axiom. But the coordinate plane also provides the materials for constructing a model of hyperbolic geometry. In this course, we'll study the Poincaré half plane model. In this model the points are the usual points of the upper half plane (y-coordinate > 0), and there are two kinds of lines:
Back to the beginning Having established some of the context, it's time to say something more specific about what we're actually going to do in this course. Since we'll study the chapters of the text in order, we'll organize this account similarly.
Basics of vector geometry This is similar to the vector geometry that's used in multivariable calculus, although some of our notation may be new to you. Also, we'll actually prove a few propositions.
Congruence, isometries, and barycentric coordinates By definition, an isometry is a transformation of the plane that preserves distance, i.e., a transformation T: R2 --> R2 such that the distance from T(P) to T(Q) is equal to the distance from P to Q for every pair of points P and Q. At this stage, we'll study some matrix formulas that define isometries.
Triangles and other plane figures We'll use vector geometry methods to study geometric topics like centroid, incenter, and circumcenter for triangles.
Classifying isometries We'll classify isometries of the plane according their geometric types [translations, reflections, and rotations], and we'll study the symmetries of planar figures, i.e., the set of isometries which map a given figure to itself.
Conformal mappings By definition, a conformal mapping is a transformation which preserves angles. This includes isometries, as well as transformations that preserve similarity. Another type of conformal mapping is the circle inversion. Here's what this looks like in the case of the standard unit circle. Thus, if X = (a,b) in R2, let ||X||2 = a2 + b2 be the square of the usual norm of X. We define the circle inversion by mapping X ~~~> (1/ ||X||2)X. Since points on the unit circle satisfy ||X|| = 1, those points are mapped to themselves. Otherwise, the interior of the unit circle [except for the origin ... ] is mapped to the exterior of the unit circle, and the exterior is mapped to the interior. . . .
The axioms of geometry A somewhat systematic study of the axioms of geometry, along with models of various subsets of the axioms.
Introduction to hyperbolic geometry Lines, angles, and area in the Poincaré half plane model. Other topics, including Poincaré isometries and trigonometry, as time permits. We'll use tools from our previous study of conformal mappings.
Text: Geometry I, by B. Fristedt (Available from Alpha Print in Dinkytown)
Other useful books
Prerequisites, etc.
For further information: Please send me an e-mail or call me at the phone number listed below.
Back to the class homepage.
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