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Lecturer(s)
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Peška Patrik, RNDr. Ph.D.
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Jukl Marek, doc. RNDr. Ph.D.
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Mikeš Josef, prof. RNDr. DrSc.
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Course content
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1. Projective space and its subspaces, intersection and sum of subspaces 2. Arithmetic and geometric base, homogenous coordinate system 3. Analytic expression of subspace 4. Duality in projective spaces 5. Projective extension of affine spaces 6. Complexification of real affine spaces 7. Collineation of projective spaces. Classification of collineations of projective line, plane and 3-space 8. Quadrics on projective space, polar, affine and metric properties of quadrics 9. Vector functions 10. Parametrization of curves. Orientation. Methods of defining of curves 11. Length of curve, natural parameter 12. Tangent, oscillating plane, Frenet frame 13. Frenet formulas, curvature, torsion. Natural equations of a curve 14. Touch of curves, oscillating circle 15. Parametrization of surfaces. Methods of defining of surfaces 16. Tangent, tangent plane and normal of surface. Orientation of surface 17. First and second fundamental form of surface ant their signification 18. Meussnier formulas and theorem 19. Fundamental directions. Normal, geodesic, principal, mean and Gauss curvature. Euler formulas. 20. Gauss and Weiengarten formulas 21. Gauss, Peterson-Codazzi-Mainardi formulas. Christoffel symbols 22. Theorem Egregium. 23. Special curves on a surface 24. Special surfaces (developable surface, surface of constant curvature, surface of revolution) 25. Differentiable manifold, affine connection, Riemannian manifolds
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Learning activities and teaching methods
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unspecified
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Learning outcomes
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Prerequisites
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unspecified
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Assessment methods and criteria
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unspecified
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Recommended literature
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Berger, M. (2004). Geometry I., II. Berlin.
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Bican, L. (2002). Lineární algebra. Praha.
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Budinský B., Kepr B. (1970). Základy diferenciální geometrie s technickými aplikacemi. SNTL Praha.
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Čižmár,J. (1984). Grupy geometrických transformácií. Bratislava.
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Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. VUT Brno.
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Gray, A. (1994). Differential geometry. CRC Press Icn.
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J. Mikeš, E. Stepanova, A. Vanžurová et al. (2015). Differential geometry of special mappings. UP Olomouc.
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J. Mikeš, M. Sochor. (2013). Diferenciální geometrie ploch v úlohách. UP OLomouc.
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Janyška,J.,Sekaninová, A. (2013). Analytická teorie kuželoseček a kvadrik. Brno.
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Metelka, J. (1969). Diferenciální geometrie. SPN Praha.
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Oprea, J. (2007). Differential geometry and its aplications. MAA Pearson Educ.
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Richter-Gebert, J. (2011). Perspectives on Projective Geometry. New York.
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Sekanina, M., Boček, L. (1988). Geometrie II. Praha.
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