Math 5-335 Fall 2004
Hints for the December 2 and
and December 9 homework
Hint for exercise #9 in §7.9 I'll give a re-wording of the problem, which may help to reduce some of the confusion related to understanding what the problem is all about. In doing this, I'll start by introducing some terminology. Thus, if we're given an incidence geometry, i.e., a model of the indicence axioms I1, I2, I3 we'll define the "dual geometry" as follows:
Now, here's the promised re-wording:
Show that if we start with the 7-point geometry from §7.2, then
the resulting dual geometry satisfies the axioms I1, I2, I3.
Well, that's the main point, actually. But if we want to expand on it slightly, we can say that the following statements need to be proved:
As noted in the text, you're allowed to use the result of exercise #8 without giving a proof. This could be useful in connection with proving I1 for the dual geometry . . .
Hint for exercise #28 in §8.6 By definition, the hyperbolic area (or Poincaré area) of a region in the upper half plane is . This doesn't correspond directly to Euclidean area in any straightforward way. To have a model of something with physical connotations, think of a distribution of mass (or static electrical charge, or whatever) that's proportional to 1/y2, i.e., inversely proportional to the square of the distance from the x-axis. So, there's a lot of mass close to the x-axis, but very little mass per unit area far away from the x-axis.
Generally, integrals like this are done as iterated integrals, with the order of integration depending on the shape of the region. Thus, if the region looks like this:
On the other hand, if the region looks like this:
Finally, of course, if a region is of neither of these types, then we decompose it so that each of the pieces is of one type or the other.
Oh yes, one other comment (slightly vague, but specific examples will be
given in class. It's that some regions that have infinite Euclidean area
will have finite hyperbolic area (as in this problem, actually, as well as
for one type of asymptotic triangle). The opposite situation is
possible too. The second situation is fairly common for regions near
the x-axis -- but asymptotic triangles, where the bottom end
is tightly pinched, are a noteworthy exception.
Hint for exercise #31 in §8.6
First of all, I just want to affirm that the suggestion in the text is not only
a valid method, but also a lot easier than calculating an
integral. Just look for the appropriate formula in the text, where hyperbolic
area is expressed in terms of angular measures.
Another point worth mentioning is that (,0) is an asymptotic vertex of this triangle.
We could get a more intuitive picture of this by calling the asymptotic vertex
(0,). Whether you
call this point by its official name or by some unofficial name as I'm doing
here, the main consequence is that two of the sides of the triangle are
vertical (and infinite at the top end), while the third side is a circular arc.
Comments and questions top;
roberts@math.umn.edu
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