| Code |
14763
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| Year |
1
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| Semester |
S2
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Solid knowledge of secondary school mathematics.
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Learning outcomes |
On completion of this unit, students should:
1. Acquired an active knowledge and understanding of the basic properties of the main classical geometries: euclidian, affine, projective, spherical and hyperbolic geometries;
2. Compare the euclidian, spherical and hyperbolic geometries in terms of their metric properties, trigonometry and parallels;
3. Classify conics up to euclidian, affine and projective transformations;
4. Recognise the geometrical meaning of different objects in linear algebra and be able to use linear algebraic methods in the resolution of geometric problems;
5. Recognise each geometry as the study of the invariants under the corresponding transformation group.
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Syllabus |
1. Euclidean geometry. Orientation, inner product, cross product and mixed product in R^3. Isometries and homotheties. Conical sections and quadratic surfaces.
2. Affine Geometry. Affine structures and affine transformations. The determinant of a transformation. The Fundamental Theorem of Affine Geometry. Conics under affine transformations.
3. Projective Geometry. The projective plane. Homogeneous coordinates. Projective transformations. The Fundamental Theorem of the Projective Plane. Desargues and Pappus Theorems. Bilinear forms and conics. The principle of Duality.
4. Spherical Geometry. The spherical surface. Relation between the spherical surface and the projective plane. Distance. Isometries. Spherical Triangles. The extended complex plane, stereographic projections and inversions. Möbius transformations.
5. Hyperbolic Geometry. The hyperbolic plane and the axiom of the parallels. The Poincaré model. Distance. Isometries of the hyperbolic plane. Hyperbolic triangles.
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Main Bibliography |
1. D.A. Brannan, M.F. Esplen, J.J. Gray. Geometry, Cambridge University Press, 1999.
2. J. Stillwell, The Four Pillars of Geometry, Undergraduate Texts in Mathematics, Springer, 2005.
3. A. Barros, P. Andrade. Introdução à Geometria Projectiva, Textos Universitários, Sociedade Brasileira de Matemática, 2010.
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Teaching Methodologies and Assessment Criteria |
1. Classroom lessons;
2. Individual work guided by the teacher;
3. Students may be assessed through a final exam or through continuous assessment.
4. Continuous evaluation has the following structure:
i)Lists of exercises performed individually 2 Values
ii)Two tests - 9+9 values.
5. Final exams - 20 values.
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Language |
Portuguese. Tutorial support is available in English.
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