Technische Universität München
Zentrum Mathematik
Prof. Dr. Dr. Jürgen Richter-Gebert, Martin von Gagern
Projective Geometry SS 2014
www-m10.ma.tum.de/ProjektiveGeometrieSS14
Worksheet 11 (7. July 2014)
— Classwork—
Question 1. Projective reflection
This task will generalize euclidean reflections to other geometries.
a) On a projective line RP1 there are three given points ∞, a and (a + b). How can you construct the point (a − b),
which is in a certain sense the mirror image of (a + b) in a?
b) The mirror image P 0 of a point P after reflection in a line g can in the Euclidean plane be constructed using the
following steps:
1. Determine a line h which is perpendicular to g and passes through P .
2. Fins the point P 0 on h which lies at the same distance from the intersection of these two lines as P , but on
the opposite side.
Expand these construction steps so it will become clear exactly how each step can be achieved. Restruct your
description to tools which can be transferred to other geometries: meet, join, harmonic set as well as tangents,
polars and poles with respect to a fundamental conic. Don’t refer to the fundamental conic in any way except
for the last three operations just mentioned.
c) Formulate a corresponding construction for point reflections. Again restrict your set of tools like in the previous
subtask.
d) Describe a general set of transformations (which will subsequently be called “projective reflections”) which contains Euclidean reflections both in a line and in a point as special cases. Every element of this class of transformations shall be described using one point and one line. The trnasformations from this class should not refer to
any fundamental conic.
e) Provide a short argument as to why a projective reflection is an involution, i.e. why executing the transformation
twice results in the identity transformation.
f) How many pairs of preimage and image points are required to define such a projective reflection in general?
g) Show that every projective reflection is a projective transformation.
h) Which relation between the defining line and point must hold with respect to some fundamental conic K if the
transformation should describe a reflection in the stricter sense of a given geometry? Choose your condition in
such a way that for the Euclidean plane, reflections in a point or in a line are the only two permissible situations.
To formalize the required relation, you may want to adapt ideas from the definition of primal-dual pairs. A
projective reflection which satisfies this extra condition shall henceforth be called a “K-reflection”.
i) Describe the structure of a K-reflection in hyperbolic geometry. Can you again distinguish between reflections
in a line nd reflections in a point?
j) Also investigate the situation in elliptic geometry. Imagine that geometry on the surface of a sphere. How does
this affect the distinction between reflections in a line and in a point?
k) Show at least for hyperbolic geometry that every K-reflection preserves distances and angles. For which other
geometries can your proof be generalized?
Hl) Show that every involution is a projective reflection.
Hm) Show that every K-reflection in any Cayley-Klein geometry preserves distances and angles.
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— Homework—
Question 2. Angle bisectors
In this task you will show that the angle bisectors in a triangle will meet in a single point in any Cayley-Klein geometry.
You may assume that the triangle is non-degenerate, i.e. that the three corners are not collinear.
Note: This question has also been discussed in the lecture. Try to follow the arguments suggested by the individual
subtasks, supplemented by what you remember from the lecture. Simply reading and reproducing the proof from the
lecture is not the intended way.
a) How many ways are there to construct an angle bisector for a given pair of lines in Euclidean geometry?
b) Draw a sufficiently generic triangle, and draw in all the angle bisectors between its edges. How many points are
there – apart from the triangle corners – where two or more of these bisectors meet? And how many bisectors
meet in these points?
c) Consider two lines which intersect in the center of a circle. Describe how you can construct a (Euclidean) bisector
using only the operations tangent, meet and join. Utilize the symmetry of the situation.
d) Given two lines which intersect in any point inside the fundamental conic. Describe a construction which can be
used to construct a hyerbolic angle bisector for the given pair of lines.
e) Reformulate your construction in such a way that the point of intersection between the two original lines is not
used at all.
f) The circle depicted below, with center M , is to be used as the fundamental conic of hyperbolic geometry. Using a
ruler and possibly a compass, construct the interior angle bisectors of the triangle ABC, i.e. those angle bisectors
which pass through the interior of the triangle.
B
M
C
A
g) Verify that these three angle bisectors indeed meet in a single point.
h) Prove that these bisectors must meet in a single point. Recall a theorem which has six tangents to a conic as
part of its premise. If you fail to recall such a theorem, dualizing the situation might help.
i) How far can this construction of angle bisectors be generalized to other geometries? Does the proof still hold?
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