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MATH 402 Worksheet 10
Friday 11 November, 2016
1. Isometries in the Klein model
We’re going to begin by focusing on reflections in the Klein model, because of the following fact, which we
proved a long time ago using axioms of neutral geometry:
Theorem 1. All isometries can be written as products of one, two, or three reflections.
(If you’ve forgotten this fact, you might want to review how this works for different types of isometries in
Euclidean geometry: translations, rotations, and glide reflections. But do this on your own time, not in
this class.)
Let’s also recall how a reflection r` about a line ` is defined (here ` is a line in the appropriate geometry
and model that you’re working with—the definition works for any of them).
Definition 1. Let P be a point. If P lies on `, then r` (P ) = P . If P does not lie on `, there exists a
unique perpendicular line n from P to `, which intersects ` at a point we’ll call Q. Then r` (P ) = P 0 is the
unique point on n with the property that P Q ' P 0 Q. (This is an equality of line segments in the chosen
geometry and model.)
Finally recall from the homework that in the Poincaré model, we had to deal with two different cases to
define hyperbolic reflections, depending on whether or not ` is a diameter of the Poincaré disk. Since we
know that the Poincaré model and the Klein model are isomorphic, we expect to have a similar situation
here.
1.1 Reflection across a diameter `
Exercise 1. Show that if ` is a diameter, ordinary Euclidean reflection across ` is an isometry of the Klein
disk, and satisfies the definition of Klein reflection.
1.2 Reflection across a chord `
Assume that ` is a Klein line that is not a diameter, and let P be a Klein point not on `. Our goal is to
give a construction of P 0 = r` (P ).
Exercise 2.
(1) Draw the Klein disk, the line `, and the point P . Construct the pole of `, and draw
t, the perpendicular to ` through P . Let Q be the point where t and ` intersect.
(2) Construct the pole of t, and draw a line through P perpendicular to t. Let Ω be the omega-point
of this line on the opposite side of t from its pole.
(3) Draw the Klein line through Ω and Q. Let Ω0 be its second omega point.
(4) Draw the Euclidean line through the pole of t and Ω0 . Let P 0 be the point where it intersects t.
(5) Prove that P 0 is the reflection of P across `. (Hint: use AA-congruence of omega-triangles.)
The technique used in this construction is actually useful for other constructions. For example:
Exercise 3. Give a construction of the perpendicular bisector ` of a Klein line segment P P 0 .
A consequence of the construction of Klein reflections is the following:
Theorem 2. Let ` and m be two Klein lines. Then there is an isometry taking ` to m.
Proof. The proof has three steps (you will fill in the details in the exercise below):
Step 1: We claim that if ` is not a diameter, then there exists some diameter n and an isometry
taking ` to n.
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Step 2: We claim that any two diameters can be mapped to each other by rotations.
Step 3: We claim that this is enough to prove the theorem.
Exercise 4.
(1) Prove the claim in Step 1. (Hint: take P to be any point in `, and consider the Klein
line P O from P to the origin. Let Q be its midpoint.)
(2) Prove the claim in Step 2.
(3) Prove the claim in Step 3.
Exercise 5. Let m and n be two Klein lines which are parallel but not limiting parallel. Devise a
construction for producing the (unique) line ` which is perpendicular to both of them.
Hyperbolic translations
Just as in Euclidean geometry, we can define a translation as a composition of reflections across parallel
lines.
Definition 2.
(1) Define a hyperbolic translation T in the Klein model as the composition of two
reflections r` and rm across lines ` and m with a common perpendicular t.
(2) Define a hyperbolic parallel displacement D in the Klein model as the composition of two reflections
r`0 and rm0 across lines `0 and m0 which are limiting parallel to each other.
Exercise 6.
(1) Show that t is invariant under T .
(2) Show that no Klein point is invariant under D. (Hint: suppose towards a contradiction that P is
fixed by D. Consider the line n joining P to r` (P ).)