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Lecturer(s)
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Káňa Michal, doc. RNDr. Ph.D.
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Course content
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1. Vector calculus and Euclidean space 2. Projective extension of Euclidean space, homogeneous coordinates 3. Differential geometry of curves - definition and basic concepts, curvature 4. Differential geometry of curves - plane and space curves 5. Bézier curves - Bernstein polynomials, de Casteljau algorithm 6. B-spline curves - B-spline bases, de Boor's algorithm 7. Rational Bézier and B-Spline curves 8. Differential geometry of surfaces - definition and basic concepts, first fundamental form 9. Differential geometry of surfaces - curvature of surfaces, second fundamental form 10. Differential geometry of surfaces - minimal surfaces and geodesics 11. Tensor product surfaces - rectangular Bézier surfaces 12. Triangular Bézier patches 13. Algebraic methods for geometric modeling
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Learning activities and teaching methods
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Lecture, Practicum
- Contact hours
- 39 hours per semester
- Preparation for comprehensive test (10-40)
- 20 hours per semester
- Presentation preparation (report) (1-10)
- 5 hours per semester
- Preparation for an examination (30-60)
- 40 hours per semester
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| prerequisite |
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| Knowledge |
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| We require a basic orientation in the concepts and skills of differential and integral calculus of functions of one or more real variables. Basic orientation in the foundations of linear algebra and analytic geometry is also necessary. |
| Skills |
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| apply skills gained through successful completion of basic mathematics (mathematical analysis) and linear algebra courses |
| work theoretically and practically with the concepts of derivative and integral |
| solve systems of linear algebraic equations |
| work with linear and quadratic objects (equations, common points) |
| work with vectors (linear dependence, vector, scalar, mixed product) |
| Competences |
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| N/A |
| N/A |
| N/A |
| N/A |
| learning outcomes |
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| Knowledge |
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| The student is able to describe curves and surfaces in parametric form (and vice versa, the parametric description to visualize) and from this description to derive important characteristics of the object, especially its curvature. |
| Skills |
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| create parametric descriptions for curves and surfaces |
| visualize the object (curves and surfaces) from the parametric description |
| derive from the description the important characteristics of a curve, in particular its curvature (curvature and torsion) |
| from the description derive important characteristics of the surface (1st and 2nd surface tensor, Gaussian, mean and geodesic curvature) |
| from the identified characteristics to derive properties or type of curves (flatness, envelope curves, tangent properties) |
| from the identified characteristics to derive properties or type of surfaces (developable surfaces, minimal surfaces, envelope surfaces, curves on surfaces) |
| Competences |
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| N/A |
| teaching methods |
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| Knowledge |
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| Lecture supplemented with a discussion |
| Skills |
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| Lecture supplemented with a discussion |
| Competences |
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| Lecture supplemented with a discussion |
| assessment methods |
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| Knowledge |
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| Combined exam |
| Skills |
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| Combined exam |
| Competences |
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| Combined exam |
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Recommended literature
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Oprea, J.: Differential geometry and Its Applications. The Mathematical Association of America, USA 2007..
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Pressley, A.: Elementary differential geometry. Springer, London 2001..
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Budinský, B. Analytická a diferenciální geometrie. 1. vyd. Praha : SNTL, 1983.
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O'Neill, Barrett. Elementary differential geometry. 2nd ed. Amsterdam : Elsevier Academic Press, 2006. ISBN 0-12-088735-5.
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Struik, Dirk Jan. Lectures on classical differential geometry. Second edition. 1988. ISBN 0-486-65609-8.
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Tapp, Kristopher. Differential geometry of curves and surfaces. 2016. ISBN 978-3-319-39798-6.
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