# Complements of Geometry

Keywords |

Classification |
Keyword |

OFICIAL |
Mathematics |

## Instance: 2020/2021 - 1S

### 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, 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

Presencial

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

Prerequisites: Basic euclidean geometry and linear algebra.

### Program

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.

### keywords

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.