A01 Surfaces of revolution. Basic constructions on surfaces of revolution (intersection with a plane, intersection of two surfaces of revolution, illumination).

A02 Technical illumination of surfaces of revolution.

A03 Quadratic surfaces of revolution. Basic problems on quadratic surfaces of revolution.

A04 Helix and its properties. Construction of the tangent, osculation plane and center of curvature in methods of projections.

A05 Quadratic surfaces - definition, constructions, basic properties. Ruled surfaces, non-ruled surfaces, their affine classification (ellipsoids, paraboloids, hyperboloids). Basic problems on a hyperbolic paraboloid and one-sheeted hyperboloid. Properties of one-sheeted hyperboloid of revolution.

A06 Developable surfaces, constructions and use in technical practice. Helicoid surface as a developable surface. Construction, properties and developing into a plane. Plane intersection of helicoid.

A07 Undevelopable ruled surfaces. Chasles theorem and its use (conoids, helicoid). Tangent plane of undevelopable ruled surfaces. Tangent plane construction methods at a point of a surface.

A08 Helicoid surfaces. Linear and cyclic helicoid surfaces. A helicoid as a conoid.

A09 Non-ruled surfaces of technical practice (wedge, sum, cyclical). Basic properties and examples of their use.

A10 Mathematical representations of curves and surfaces, surface patches, isoparametric curves, surface edges, corner points.

A11 Hermit and Bézier arc, their properties and evaluation algorithms.

A12 Continuity conditions in spline curves.

A13 Hermit cubic splines - construction, properties, examples of endpoint conditions.

A14 Cardinal splines, representation of a segment, shaping parameter and examples of enpoint conditions.

A15 B-spline curves, knot sequence, construction, modeling of B-spline curves.

A16 Beta-spline curves, continuity conditions, properties of a segment, modeling of a curve.

A17 Construction of rational curves, rational Bézier curves and their modeling.

A18 Surfaces determined by the edge conditions - Coons patches. Construction and mathematical description of ruled, bilinear and bicubic patches.

A19 Bézier and B-spline bicubic patches, surface properties, isoparametric curves, border curves, corner points.

A20 Torsion of a curve. Frenet's formulas.

A21 Singular points of planar curves.

A22 The first fundamental form of a surface and computation of the lengths, angles and area on a surface.

A23 Dupin's indikatrix and conjugate directions in the tangent plane of a surface.

A24 Principal directions and curvatures of a surface, Weingarten mapping.

A25 Gaussian curvature of surfaces.

A26 Ideals in commutative rings (particularly in rings of polynomials).

A27 Affine and projective algebraic varieties. Associated ideal of an algebraic variety.

A28 Radical ideal. Hilbert's nullstellensatz.

A29 Zariski's topology. Decomposition of algebraic variety into irreducible components.

A30 Coordinate ring variety. Morphisms and rational varieties.

A31 Ordering of monomes in polynomial rings. Algorithm of division, Gröbner's base ideal.

A32 Calculations in algebraic geometry, applications of Gröbner base and resultants (comparison of varieties, elimination ideal).

B01 Applying didactic principles in teaching of descriptive geometry.

B02 Organization and creation of curriculum plans in descriptive geometry.

B03 Applications of educational methods in descriptive geometry.

B04 Specifics of problem solutions in descriptive geometry (complete solution, construction problems).

B05 Steps and tools of developing of spatial imagination.

B06 Development of logical reasoning. Complete sorting with demos in descriptive geometry.

B07 Conceptual process in descriptive geometry (axioms, definitions, theorems).

B08 Functions and techniques of proving in descriptive geometry.

B09 Organizational forms of education in descriptive geometry at schools.

B10 Descriptive geometry and modern means of education.

Oral exam.

English

The questions on the state exam summarize the knowledge from subjects Didactics of Deskriptive Geometry, Surfaces of Technical Practice, Algebraic Geometry, Differential Geometry and Computer Geometry.

0/100

The student shows overview over the knowledge gained in the block Theoretical base of descriptive geometry and applications.