Czech

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Institute of Mathematics and Descriptive Geometry (MAT)

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Basics of planar geometry and 3D geometry as taught at secondary schools.

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Requirements for successful completion of the subject are specified by guarantor’s regulation updated for every academic year.

Lectures

1. Central and parallel projection. Perspective collineation and affinity. Curve affine to a circle. Triangle and trimmel construction of an ellipse, Rytz´s construction.
2. Construction Euclidean problems. System of basic problems. Monge´s projection – position problems.
3. Monge´s projection – metric problems. Introduction further projection planes. Displaying bodies.
4. Cutting prism, pyramid, cylinder, and conic surfaces. Intersections of the straight line with prism, pyramid, cylinder, and conic surfaces.
5. Sphere and its section, intersection with a straight line. Axonometry – classification.
6. Orthogonal axonometry – position problems, metric problems in co-ordinate planes. Method of Skuhersky.
7. Basics of the theory of curves and surfaces. Helix, its properties and construction.
8. Right closed and opened ruled helicoidal surfaces, their sections and tangent to the section curve.
9. Developable surfaces and their developments.
10. Developable surfaces given by two curves in a plane.
11. Surfaces of revolution and their sections. Tangent to the section curve.
12. Intersections of two surfaces of revolution I. Tangent to the intersection curve.
13. Intersections of two surfaces of revolution II.

Seminars

1. Focal properties of conics. Tangents from a given point and parallel to a given direction to conics. Constructions using focal properties.
2. Perspective collineation and perspective affinity. Construction of an ellipse affine to a circle. Tangents of an ellipse using affinity.
3. 3D Euclidean constructions using basic problems of descriptive geometry. Constructive problems in Monge´s projection.
4. Constructive problems in Monge´s projection. Displaying bodies.
5. Sections of prisms, pyramids, cylinders and cones in Monge´s projection.
6. Section of a sphere. Intersections of a straight line with the above bodies.
7. Orthogonal axonometry – constructive problems in coordinate planes.
8. Method of cuts. Some metric methods solved by Skuhersky´s method. Helix.
9. Helix in axonometry. Helicoidal conoid.
10. Developable surfaces – complanation of a cone, cylinder. Developable surfaces given by two planar curves.
11. Developable surfaces – continuation. Sections of two surfaces of revolution with a tangent to the section curve.
12. Intersections of two surfaces of revolution , tangent line to teh intersection curve.
13. Credit.

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Know how to construct conics from the properties of their foci. Understand and apply the pinciples of perspective collineation and perspective affinity. Understand the basics of Monge´s projection and orthogonal axonometry. Know how to solve simple 3D problems. Display basic geometric bodies in each projection, its cut by a plane or intersected by a straight line. Know selected facts on the theory of curves and surfaces, construct a helix given by its elements and a right ruled helicoidal surface. Know the properties of developable surfaces, construct the development of a cylinder or a cone, inflex tangent of a base edge, radius of curvature at a point of a developed base edge. Know the properties of the surfaces of revolution, construct a section of a surface of revolution and a tangent to the section curve at the section point, construct intersection curve of two surfaces of revolution and a tangent to the intersection curve.

Extent and forms are specified by guarantor’s regulation updated for every academic year.

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Not applicable.

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