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Complements of Geometry

Code: M3004     Acronym: M3004     Level: 300

Classification Keyword
OFICIAL Mathematics

Instance: 2020/2021 - 1S Ícone do Moodle

Active? Yes
Responsible unit: Department of Mathematics
Course/CS Responsible: Bachelor in Mathematics

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
L:B 0 Official Study Plan 3 - 6 56 162
L:CC 1 Plano de estudos a partir de 2014 2 - 6 56 162
L:F 1 Official Study Plan 2 - 6 56 162
L:G 0 study plan from 2017/18 2 - 6 56 162
L:M 51 Official Study Plan 2 - 6 56 162
L:Q 0 study plan from 2016/17 3 - 6 56 162

Teaching Staff - Responsibilities

Teacher Responsibility
Peter Beier Gothen

Teaching - Hours

Theoretical and practical : 4,00
Type Teacher Classes Hour
Theoretical and practical Totals 1 4,00
Peter Beier Gothen 4,00

Teaching language

Suitable for English-speaking students


Enlarge the scope of study of geometry to non-Euclidean geometries, namely spherical, hyperbolic, affine and projective geometry, using mainly, but not exclusively, methods from analytic geometry.

Learning outcomes and competences

On completion of the unit the student should:

(1) Know basic results and properties of spherical, hyperbolic, affine and projective geometry and understand the relation between them.

(2) Understand the similarities and differences between spherical, hyperbolic and euclidean geometry and appreciate the relevance of Euclid's parallel postulate.

(3) Understand the idea of studying a geometry via its transformations and be able to do this, in particular using methods from linear algebra.

Working method


Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Prerequisites: Basic euclidean geometry and linear algebra.


Spherical geometry. Spherical triangles: spherical cosine formula, area. The group O(3) as the isometry group spherical geometry. Hyperbolic geometry: hyperboloid model. Hyperbolic triangles: hyperbolic law of cosines, area. Isometries of the hyperbolic plane as Lorentz transformations.Upper half-plane and Poincaré disc models. Isometries of the hyperbolic plane as as Möbius transformations; classification. Discussion of Euclid's parallel postulate and its independence. Affine geometry: affine coordinates, affine transformations. Projective geometry: projective space, homogeneous coordinates and projective transformations. Projective completion of the affine plane. The fundamental theorem of projective geometry. Projective transformations of the projective line and the cross ratio. Theorems of Desargues and Pappus. Duality. Conics. Euclidean, elliptic and hyperbolic geometry as subgeometries of projective geometry. Additional topics may be covered.

Mandatory literature

Miles Reid and Balazs Szendroi; Geometry and Topology, Cambridge University Press, 2005. ISBN: 978-0-521-61325-5
John Stillwell; Geometry of surfaces. ISBN: 0-387-97743-0

Complementary Bibliography

Birger Iversen; Hyperbolic geometry. ISBN: 0-521-43528-5
V. V. Nikulin; Geometries and groups

Teaching methods and learning activities

Lectures and discussions on the material; discussions of questions and problems; exercise solving.


Physical sciences > Mathematics > Geometry

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation Weight (%)
Teste 100,00
Total: 100,00

Amount of time allocated to each course unit

designation Time (hours)
Estudo autónomo 106,00
Frequência das aulas 56,00
Total: 162,00

Eligibility for exams

No conditions

Calculation formula of final grade

Two tests will take place during the semester. The final mark is the arithmetic mean of the marks of the tests.

Special assessment (TE, DA, ...)

By written and/or oral exam.

Classification improvement

The final exam will have parts corresponding to each of the two tests and individual improvement of the mark of each test is allowed. Improving the mark of the previous academic year should be carried out by taking both tests or, alternatively, the resit exam.
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