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Complements of Geometry

Code: M3004     Acronym: M3004     Level: 300

Classification Keyword
OFICIAL Mathematics

Instance: 2022/2023 - 2S 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 study plan from 2021/22 2 - 6 56 162
L:F 2 Official Study Plan 2 - 6 56 162
L:G 0 study plan from 2017/18 2 - 6 56 162
L:M 44 Official Study Plan 2 - 6 56 162
L:Q 0 study plan from 2016/17 3 - 6 56 162

Teaching Staff - Responsibilities

Teacher Responsibility
Maria do Rosário Machado Lema Sinde Pinto

Teaching - Hours

Theoretical and practical : 4,00
Type Teacher Classes Hour
Theoretical and practical Totals 1 4,00
Maria do Rosário Machado Lema Sinde Pinto 4,00

Teaching language

Suitable for English-speaking students


Enlarge the scope of study of geometry to non-Euclidean geometries, namely spherical, hyperbolic and projective geometry.

Learning outcomes and competences

On completion of the unit the student should:

(1) Know basic results and properties of spherical, hyperbolic 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.

Working method



Spherical geometry. Great circles; spherical triangles: spherical cosine formula, area.  Hyperbolic geometry - Discussion of Euclid's parallel postulate and its independence. Upper half-plane and Poincaré disc models;  hyperbolic triangles: hyperbolic law of cosines, area.  Projective geometry: projective space, homogeneous coordinates and projective transformations. 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. 

Mandatory literature

Alekseic4ad Bronislavovich Sosinskic4ad; Geometries. ISBN: 978-0-8218-7571-1

Complementary Bibliography

Marvin Jay Greenberg; Euclidean and non-Euclidean geometries. ISBN: 0-7167-0454-4
John Stillwell; Geometry of surfaces. ISBN: 0-387-97743-0
Birger Iversen; Hyperbolic geometry. ISBN: 0-521-43528-5

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 75,00
Apresentação/discussão de um trabalho científico 25,00
Total: 100,00

Amount of time allocated to each course unit

designation Time (hours)
Estudo autónomo 90,00
Frequência das aulas 56,00
Apresentação/discussão de um trabalho científico 6,00
Trabalho escrito 10,00
Total: 162,00

Eligibility for exams

No conditions

Calculation formula of final grade

Two tests (T1 e T2), each totalizing 7,5 out of 20, will take place during the semester, as well as a written work W and its presentation (5 out of 20). The final mark is NF=T1+T2+W.

The resit exam will have parts corresponding to each of the two tests and the higher of the mark of the test and the mark of the corresponding part of the exam will count for the final mark, the mark obtained for the work ant its presentaton remais the same.

Special assessment (TE, DA, ...)

By written and/or oral exam.

Classification improvement

The resit exam will have parts corresponding to each of the two tests, and improvement of the individual mark of each test is allowed, as well as the re-submission of the work.

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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