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MATH 614 EXAM # 2 REVIEW SHEET
Hi kids, here is the review for your upcoming exam. Enjoy!
(1) Section 5.4: The Fundamental Theorem of Inversive Geometry. We discussed the
manner in which there exists a unique Möbius Transformation that takes any 3
points in Ĉ to any other 3 points in Ĉ, and how such a Möbius transformation is
unique. We closed the section with a discussion as to the manner in which one
would actually construct such a function.
(2) Section 6.1: Hyperbolic Geometry, the disk model. This section simply outlines
what hyperbolic geometry is, and how it initially relates to the inversions we had
done previously. We discussed what sorts of equations would actually give us
d−lines (that is, circles that meet the unit circle at right angles), and discussed the
notion of parallel and ultra-parallel lines. Inversion in a d−lines would necessarily
map D to D, and C to C, and so inversions restricted to D (the hyperbolic disk) are
what call hyperbolic reflections (and compositions of these are hyperbolic transformations). The Origin Lemma proved that any point can be moved to the origin
by a hyperbolic transformation, and a number of important results followed from
that (see pages 351-354).
(3) Section 6.2: Hyperbolic Transformations. We learned that a hyperbolic reflection
had the form as in Lemma 1, which we later used to see that certain Möbius
transformations (az + b)/(b̄z + ā) were compositions of an even number of these.
We saw that Möbius transformations of this form were the composition of two
hyperbolic reflections, and that as a result, one needed no more than 3 hyperbolic
reflections to represent any hyperbolic transformation! We also saw that there is
a certain useful form of a (direct) hyperbolic transformation (see page 363) that
easily describes what point gets sent to the origin, and that this was useful in
determining how to find a hyperbolic transformation moving any one point to any
other.
(4) Section 6.3: Distance in Hyperbolid Geometry. We introduced the distance function on D, and proved that it must satisfy basic properties that distance functions
should. As applications, we saw that we could find the hyperbolic midpoints between two points in D, and also translate the centers and radii from a hyperbolic
circle to the associated Euclidean circle.
(5) Section 6.4: Geometrical Theorems. We proved that the sum of the interior angles
of a d−triangle is less than π, and found that as a result, “larger” triangles have
a smaller angular sum (applying to asymptotic triangles as well). We proved that
if a pair of angles are equal, then the d−triangle must be isosceles, and finally the
surprising result that all similar d−triangles are congruent! We proved at the end of
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MATH 614 EXAM # 2 REVIEW SHEET
this section the analogue to the Pythagorean Theorem for right-angled hyperbolic
triangles.
(6) Section 6.5: Area. We proved that all triply asymptotic triangles are congruent,
and that the area of such a triangle is finite. We also proved the famous result that
the area of a d−triangle is equal to its angular defect.