# Complements of Geometry

Keywords |

Classification |
Keyword |

OFICIAL |
Mathematics |

## Instance: 2022/2023 - 2S

### Cycles of Study/Courses

### Teaching Staff - Responsibilities

### Teaching language

Suitable for English-speaking students

### Objectives

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

Presencial

### Program

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.

### keywords

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.