| 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. Affine Geometry.
Affine Subspaces and affine Applications;
2. Euclidean Geometry Euclidean affine subspaces, Reflections, Isometries, Symmetries and Congruences. Fundamental Theorem of Affine Geometry. Conics 3. Projective Geometry.
The projective space. Homogeneous coordinates. Embedded from Affine space into projective space. Duality. Projective transformations. The Fundamental Theorem of the Projective Plane. Desargues and Pappus's theorems.
4. Elliptical Geometry The spherical surface. The Elliptic plane. The Riemann sphere.
5. Hyperbolic Geometry.
The hyperbolic plane. The Poincaré disc. Distance. Isometries in the hyperbolic plane. Hyperbolic triangles.
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Main Bibliography |
1. Leonor Godinho e Margarida Mendes Lopes. Introdução à Geometria. Coleção Ensino da Ciência e da Tecnologia. IST Press. 2024
2. D.A. Brannan, M.F. Esplen, J.J. Gray. Geometry, Cambridge University Press, 1999.
3. J. Stillwell, The Four Pillars of Geometry, Undergraduate Texts in Mathematics, Springer, 2005.
4. 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)Continuous Assessment (CA):
i) Two tests - 8 points each (T1, T2);
ii) Exercise resolution - 4 points (RE);
iii) If T1 + T2 + RE < 9.5 then CA = T1 + T2 + RE;
iv) If T1 + T2 + RE = 9.5, Oral Exam - 20 points (PO). In this case CA = (T1 + T2 + RE + PO)/2;
2) The student will be exempt from the final exam if the final CA grade is greater than or equal to 9.5;
3) Students will be admitted to the final exam if they obtained in CA a grade greater than or equal to 5.5;
4) If the student obtains in CA a grade less than 5.5, they will not be admitted to the final exam;
5) In the exam, all students with a grade greater than or equal to 9.5 will be called for an oral exam - 20 points. The final grade will be the arithmetic mean of the written exam and the oral exam.
6- Students with special status have their own rules defined by the academic regulations.
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Language |
Portuguese. Tutorial support is available in English.
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