Weeks 2 and 3
From geometry to groups: Distance and motions
I prove that Euclidean distance, as you know it for example
from your "3D Geometry and
Motion" module, is a distance in the mathematical sense
of the word. You will see how the proof shows that collinearity
(the property that three given points can be connected by a line) is a purely
metric property.
Next, we will study the group of motions of Euclidean
space. I will prove that each such motion is affine linear, and you will
learn to distinguish between direct and indirect motions.
Among the
Euclidean motions, there are the orthogonal ones, which we will study
in more detail: We can find a normal form for them. Finally, I will discuss
the peculiarities of special coordinate choices in Euclidean space, that
is Euclidean frames.
Note that this discussion
eventually leads us from a Euclidean point of view to
a Kleinian one.
Helpful literature:
-
M. Reid, Geometry and Topology, chapters of forthcoming
textbook by M. Reid and B. Szendröi, available from General Office:
Sections 2.1-2.9 and 2.14 and A and B.1.
- M. Berger,
Geometry I,
Springer:
Section 8.1.
Note:
Only topics covered in the lectures and exercises
will be asked in the exam.
The list of helpful literature is meant as an aid, also to raise further
interest, but not as an obligation.