Constructive Geometry and Computer Graphics (FSI-SKG)

The aim of the course is to deepen spatial imagination, to familiarize students with the principles of visualization and important properties of some curves and surfaces. The task of the course is to introduce students to the basics of constructive geometry and display techniques in computer graphics. They can apply the acquired knowledge in subsequent specializied subjects.

Constructive geometry and computer graphics course allows students to gain an orientation in basic geometric terms and relationships between them, knowledge of solving spatial problems, properties of curves and surfaces with their visualization issue.

The students have to be familiar with the fundamentals of geometry and mathematics at the secondary school level.

Constructive geometry and computer graphics course summarizes and refines basic geometric concepts, including basic geometric representations, and introduces students to some types of projections, their properties, and applications. Emphasis is placed on rectangular axonometry. The basics of plane kinematic geometry are also presented. The course is dedicated to displaying curves, surfaces and solids with planar sections with an overlap into computer graphics. The theoretical basis is implemented in the algorithms for the visualization of curves and surfaces in the MATLAB system. Practical applications of modeling algorithms are presented in Rhinoceros software.

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

COURSE-UNIT CREDIT REQUIREMENTS:
Participation in exercises, prepare the semester work and sucesufully complete the written test assigned to about the 10th week of classes. Each work must be evaluated at least 50% of the maximal number of points.

FORM OF EXAMINATIONS:
The exam has a practical and a theoretical part. The practical part lasts 90 minutes and contains 3 examples. The evaluation value of practical part is 70 points. Theoretical part is evalueted by a maximum of 30 points.

RULES FOR CLASSIFICATION:

Final classification:
0-49 points: F
50-59 points: E
60-69 points: D
70-79 points: C
80-89 points: B
90-100 points: A

Attendance at seminars is required. The way of compensation for an absence is fully at the discretion of the teacher.

1. Conic sections, focal properties of conic sections, point construction of a conic section, osculating circles, construction of a tangent line from a given point, diameters and center of a conic section, conjugates diameters, pin-and-strip construction of an ellipse, Rytz construction.
2. Central, parallel projection and their properties (point, line, plane, parallel lines, perpendicular lines), collination between planes, central collination, axial affinity, circle in central collination.
3. Axonometry, Pohlke's theorem. Perpendicular axonometry: a line and a point in a plane, lines of a plane. elementary examples.
4. Perpendicular axonometry: metric problems in projection planes, elementary surfaces and solids.
5. Perpendicular axonometry: solids and their slices, intersections with a straight line.
6. Euclidean plane and space. Projective plane and space: eigenpoints (axioms, incidence, Euclid's postulate, projective axiom, geometric model of the projective plane and projective space, homogeneous coordinates of proper and improper points, sum and difference of the points).
7. Representation in Euclidean and projective space. Kinematics, cyclic curves, derivation of parametric equations of kinematic curves in the projective plane, derivation of the parametric equation of a helix in projective space.
8. Analytical surfaces, types of surfaces according to the forming principle and forming curves, isocurves, tangent plane, normal. Cylindrical and screw surfaces, surfaces of revolution.
9. Raster graphics, vector graphics, perception of electromagnetic radiation, color spaces.
10. Algorithms for visualization of curves and visualization of surfaces using u and v curves, triangulation.
11. Algorithms for solving visibility, basic shading and rendering algorithms.
12. 3D visualization, modeling of stereoscopic observation.
13. 3D modeling in the Rhinoceros system.

1. Conic sections: point construction, osculating circles, tangents, pin-and-strip ellipse construction, Rytz construction.
2. Central, parallel projection: collineation between planes, central collination, axial affinity, circle in central collination.
3. Perpendicular axonometry: a line and a point in a plane, lines of a plane. basic position tasks.
4. Perpendicular axonometry: metric problems, elementary surfaces and solids.
5. Perpendicular axonometry: solids and their slices, intersections with a straight line.
6. Introduction to the MATLAB system.
7. Construction of kinematic curves, calculation in MATLAB.
8. Construction of helix, skew surfaces and surface of revolution (part 1).
9. Construction of surfaces of revolution (part 2), calculation in MATLAB.
10. Visualization of curves and surfaces in MATLAB.
11. Surface visualization with solutions for visibility, shading and rendering in MATLAB.
12. Modeling stereoscopic observation in MATLAB.
13. 3D modeling in the Rhinoceros systém.