Descriptive Geometry

1. Rotational surfaces. Basic constructions on rotational surfaces (plane section, intersections of two rotational surfaces, lighting).
2. Technical lighting of rotational surfaces.
3. Rotational quadratic surfaces. Basic tasks on rotational quadratic surfaces.
4. Helix. Its properties, construction of the tangent, osculating plane and centre of curvature at a given point.
5. Quadratic surfaces, definition, their creation, basic properties. Ruled and non-ruled surfaces, their affine classification (ellipsoids, paraboloids, hyperboloids). Basic tasks on hyperbolic paraboloid and one-sheet hyperboloid. Properties of the rotational one-sheet hyperboloid.
6. Developable ruled surfaces, their determination and use in technical practice. Developable ruled helical surface. Its creation, properties; its development into a plane. Plane sections of the developable ruled helical surface.
7. Non-developable ruled surfaces. Chasles' theorem and its application. (Conoids, helical surfaces.) Tangent planes of non-developable ruled surfaces. Methods of constructing the tangent plane at a point on the surface.
8. Helical surfaces. Ruled and cyclic helical surfaces. Direct helical conoid.
9. Non-ruled surfaces in technical practice (wedge, sum, cyclic). Basic properties and examples of their use.
10. Mathematical representations of curves and surfaces; surface patch, isoparametric lines, surface edges, corner points.
11. Simple arcs: Hermite and Bézier; modelling and evaluation possibilities.
12. Continuity conditions of spline curves.
13. Hermite cubic splines; creation, properties, termination options.
14. Cardinal spline; segment description, shaping parameter, termination options.
15. B-spline curves; knot vector, modelling of B-spline curves.
16. Beta spline curves; continuity conditions, segment properties, curve modelling.
17. Rational curves – construction; rational Bézier curves and their modelling.
18. Surfaces defined by boundaries – Coons patches; creation and mathematical description of ruled, bilinear and bicubic patches.
19. Bézier and B-spline bicubic patches; surface properties, isoparametric lines, boundary lines, corner points.
20. Curve torsion. Frenet formulas.
21. Singular points of planar curves.
22. Envelope of a one-parameter family of planar curves.
23. Developable ruled surfaces, three basic types.
24. First fundamental form of a surface: lengths, angles and areas on the surface.
25. Surface mappings. Isometric, conformal and isareal mappings.
26. Dupin indicatrix and conjugate directions in the tangent plane of the surface.
27. Principal directions and principal curvatures of a surface, Rodrigues' theorem.
28. Principal curvatures of a surface, extrema of normal curvatures (Euler's theorem).
29. Ideals in commutative rings (especially in polynomial rings).
30. Algebraic varieties (affine, projective). Associated ideal of an algebraic variety.
31. Decomposition of an algebraic variety into irreducible components, primary decomposition of an ideal.
32. Dimension of an algebraic variety, dimension of an ideal.
33. Monomial and local orderings in polynomial rings.
34. Gröbner basis of an ideal, application to algebraic varieties.
35. Standard basis of an ideal, application to algebraic varieties.

1. 1. Historical overview and development of projection methods. The relationship of the didactics of descriptive geometry to certain scientific disciplines (selected disciplines from the humanities and natural sciences).
   2. Conic sections (definitions, focal properties, conic as an image of a circle in a collineation).
2. 1. Applications of descriptive geometry and technical drawing in technical practice.
   2. Stereometry at primary and secondary schools – its use in teaching descriptive geometry.
3. 1. Curriculum planning in descriptive geometry, syllabi and thematic plans.
   2. Projection of figures in Monge's projection.
4. 1. Analysis and synthesis in teaching construction tasks in descriptive geometry (analysis – breakdown, synthesis – construction).
   2. Orthogonal axonometry – principle and basic concepts.
5. 1. Concept formation process from the perspective of the theory of didactic situations (with emphasis on the institutionalisation of concepts).
   2. Central projection and linear perspective – principle and basic concepts.
6. 1. Cognitive process from the perspective of the theory of didactic situations (experiment, hypothesis formulation, verification and proof of hypothesis).
   2. Topography in secondary school curriculum.
7. 1. The role of proof in descriptive geometry.
   2. Dimensioned projection and its applications in secondary school curriculum.
8. 1. Didactic principles of teaching descriptive geometry.
   2. Parallel orthographic projection from the perspective of its use in teaching descriptive geometry.
9. 1. Some teaching concepts (the "empty head" concept, the "small steps" concept, the "full head" concept). Students' errors in learning descriptive geometry (their causes, analysis, significance of their existence).
   2. The theorem on the projection of a right angle – its use in projection methods.
10. 1. Types of tasks in descriptive geometry (typical, atypical, proof). Solving a construction task in descriptive geometry.
    2. Division ratio and double ratio – their use in projection methods (properties in parallel and central projection).
11. 1. Overview of projection methods (projections: dimensioned, Monge's, double image, cyclography, oblique projection, axonometry, central projection).
    2. Central collineation and axial affinity.
12. 1. Preparation for a lesson in descriptive geometry, types of lessons, organisation of teaching descriptive geometry.
    2. Rotation of a plane in dimensioned and Monge's projection.
13. 1. Assessment of students in descriptive geometry – grading.
    2. Perpendicularity of lines and planes in orthogonal projection.
14. 1. Use of computers in teaching descriptive geometry.
    2. Transformation of projections (use of 3rd and 4th projections).
15. 1. Basic concepts of the theory of didactic situations in relation to the didactics of descriptive geometry.
    2. Rotation around a line in Monge's projection.
16. 1. Analysis a priori and analysis a posteriori of a given didactic situation.
    2. Projection of prisms and pyramids in orthogonal projection.
17. 1. Induction and deduction in teaching descriptive geometry.
    2. Theoretical solutions of roofs.
18. 1. Didactic transposition.
    2. Intersections of polyhedral bodies.
19. 1. Proof of existence and uniqueness in descriptive geometry.
    2. Intersections of rotational bodies (cylinders and cones).
20. 1. Problem-based teaching, open problems, problem solving.
    2. Projection of cylinders and cones in orthogonal projection, plane sections, intersections of a line with a solid.