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Geometry II:
Discrete Differential Geometry
(Summer 2018)


This is a basic course of the Berlin Mathematical School
and lectures will be held in English. The tutorials will be held
in English or German, depending on the students attending each
one.
News
[28.9.]
Oral exams
will be offered 912h on Thu. 8 Nov.
[23.7.]
One source
for the Delaunay triangulation algorithms is these
lecture notes from ETH Zürich; see Chapters 6 and 7.
[2.7.]
There will
be no lecture on Wednesday, 4.7.
Isabella's office
hours this week are by appointment only (not on Thursday as
usual), please write an email.
Here are the
references mentioned in today's tutorial:
Medkova,
"The Laplace Equation" (Smooth theory of Dirchlet boundary
value problems, Hölder continuity for domains explained in 1.17),
Pinkall,
Polthier, "Computing discrete minimal surfaces and their
conjugates" (Reference [cotanminimal] for the simplicial
minimal surface algoithm in the script)
and
Wardetzky
et al., "Discrete Laplace operators: No free lunch"
We will talk about
aspects of the second reference in next week's tutorial!
Isabella is
taking questions for the last tutorial on July 16th. Please send
an email with topics you would like to have covered or questions
about any of the topics from this semester.
[26.6.]
Here are
some further references mentioned in the tutorial (not relevant
for the exam):
Izmestiev et al., "There
is no triangulation of the torus with vertex degrees 5,6,... "
and
Günther et al., "Smooth
polyhedral surfaces" (proof spherical area of Gauss map
equals discrete Gaussian curvature)
[7.6.]
You can
find Conway's Zip Proof here.
[29.5.]
You are
allowed to hand in the following exercises after the due dates:
Ex. 5.1 > June
4th
Ex. 6.2 >
June 11th
We will cover the
topic "elastic rods" in the tutorial on Monday.
[23.5.]
Isabella's
office hours tomorrow will be from 10:1512.
 [15.5.]
For the
derivation of the EulerLagrange eq.I recommend to have a look into
V.I.Arnold, Mathematical Methods of Classical Mechanics, Springer,
2nd ed., p. 55 ff. (or take a copy in the lecture tomorrow)
 [11.4.]
 There is no lecture on Wed 18 Apr; the first lecture will be
on Thursday 19 April.
 There are no tutorials on Mon 16 Apr; tutorials and office
hours will start in the second week.
Contents
Classical differential geometry studies smooth curves and
surfaces, for instance in terms of their curvatures. This course
will consider analogous notions and results for discrete (usually
polygonal) curves and discrete (usually polyhedral) surfaces.
Specific topics will include:
 Curvatures of discrete curves, discrete elastic curves
 Curvatures of discrete surfaces, special parametrizations and
classes of surfaces
 Discrete conformal and harmonic maps, the discrete Laplace
operator, Delaunay triangulations, circle packings, discrete
conformal equivalence, discrete Riemann surfaces, variational
principles
References
 Alexander I. Bobenko, Lecture
notes from the Summer 2014 version of this course.
 John M. Sullivan, Curves of Finite
Total Curvature, in Discrete Differential Geometry,
Birkhäuser, 2008, pp. 137–161.
 John M. Sullivan, Curvatures
of Smooth and Discrete Surfaces, in Discrete
Differential Geometry, Birkhäuser, 2008, pp. 175–188.
Exercise sheets
 Sheet 1,
due 30 April
 Sheet 2,
due 7 May
 Sheet 3,
due 14 May
 Sheet 4,
due 23 May
 Sheet 5,
due 28 May
 Sheet 6,
due 4 June
 Sheet 7,
due 11 June
 Sheet 8,
due 18 June
 Sheet 9,
due 25 June
 Sheet 10,
due 2 July
 Sheet 11,
due 9 July
Homework policy and exam information
 To get a certificate for the tutorial and thus qualify to
take the oral exam, you need to satisfactorily complete 60% of
the homework assignments.
 The exercises are to be done in groups of two people and
handed in at one of the tutorials each Monday.
 Oral exams
will be offered 1012h on 25 July and 27 July.
It will also be possible to take the exam in late September or in
early November.
Office Hours
