Computer Geometry and Graphics
FSI-1PG-KAcad. year: 2019/2020
Computer geometry and Graphics introduces basic knowledge of projective geometry and computer graphics which is used in CAD systems and graphics modelers. The base of the subject is in connection of theoretical knowledge with the work in graphics modelers. Synthetic and analytic constritions of basic plane and spatial figures and methods of their mapping and software representation are the course substantiality.
Learning outcomes of the course unit
Students will obtain the wide overview about the computer graphics and also about the parts of geometry which support the technical display systems (solids projection, curves and surfaces of technical practice, lighting). The student will be able to work in graphic studio and use the fundamental functions of the software with new theoretical knowledge. This approach learn him how to work with other graphic systems in short time.
Knowledge of mathematics in secondary school level, especially geometry and descriptive geometry (two-plane (Monge's) projection, axonometry).
For students who did not attend the descriptive geometry on secondary school there is a possibility to attend the course Selected Chapters from Descriptive Geometry.
Recommended optional programme components
Recommended or required reading
Borecká, K. a kol.: Konstruktivní geometrie, CERM, s.r..o. Brno, 2002
Urban, A.:: Deskriptivní geometrie, díl 1. - 2., , 0
Seichter, L.:: Konstruktivní geometrie, , 0
Medek, V. - Zámožík, J.: Konštruktívna geometria pre technikov,
Martišek, D.: Počítačová geometrie a grafika, VUTIUM, Brno 2000
Borecká, K. a kol.: Konstruktivní geometrie, , 0
Paré, Loving, Hill Descriptive Geometry New York 1972
Steve M. Slaby Fundamentals of Three-Dimensional Descriptive Geometry New York 1976
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes
Course/unit credit is conditional on the following: Active attendance at seminars and
three seminar works per 10 points. Each work contains two parts: graph (max 5 points)
and Rhinoceros model (max 5 points). Course credit: minimal one point in each part of
each work and 15 total point.
Examination: written part consists of three drawing (20 + 20 points) and one calculation
(20 bodů). The last 10 points is possible to obtain in oral part of examination.
excellent (100 - 90 points),
very good(89 - 80 points),
good (79 - 70 points),
satisfactory (69 - 60 points),
sufficient(59 - 50 points),
failed (49 - 0 points).
Language of instruction
The gist of the subject is to introduce students basic knowledge of projective geometry and computer graphics which is used in CAD systems and graphics modelers. Students will learn the classic geometry and computer graphics as well. The base of the subject is in connection of theoretical knowledge with the work in graphics modelers. Students will work with graphics software Rhinoceros.
Specification of controlled education, way of implementation and compensation for absences
If a student does not satisfy the given conditions, the teacher can set an alternative condition. The seminars are compulsory, lectures are optionally but very recommend.
Type of course unit
17 hours, optionally
Teacher / Lecturer
1. Euclidean space, topologic dimension, curve, surface, solid. Projective space, dividing ratio and cross ratio, projection
2. Basic mappings in plane and space, their analytic representation (rotation, translation, axis and central symmetry, homothety), analytic representation of parallel and central projection).
3. Analytic curves, Point function, tangent and normal of curve, curvature. Analytic surfaces, isolines, tangent plane, normal, normal and Gaussian curvature (basic information)
4. Focus and projective attributes of conics, circle - ellipse affinity, Triangle, stripe and Rytz construction.
5. Curve representation in CAD systems, affine point combination, control points. Beziere curves, B-spline curves and surfaces, NURBS curves.
6. Fundamentals of kinematic plane geometry (motion, fixed and moving centrode, circle arc rectification, rolling motion, cycloid and involute curve - synthetic and analytic construction, animation principle, software modeling)
7. Elementary surfaces and solids (prism, pyramid, cylinder, cone, sphere) two-plane (Monge) Monge projection (MP) and orthogonal axonometry (OA), NURBS surfaces, NURBS representation of elementary curves and surfaces.
8. Slices of solids, the intersection of line and solid, intersection of solids - Monge's projection and axonometry solutions
9. Helix, analytic representation, MP and OA projection.
10. Methods of surface generation in graphic system, Basic generating principles. Developable surfaces (cylindric and conic surface, curve tangent surface, transition surfaces). Undevelopable surfaces (conoid, cranc mechanism surface, oblique transition surface) - analytic representation, computer modeling
11. Surfaces of revolution (torus, rotation quadric) - Monge's projection and axonometry, - analytic representation, computer modeling
12. Skrew surfaces, cyclic and linear surfaces, - Monge's projection and axonometry, analytic representation, computer modeling
13. Hausdorff dimension, fractal. Self-similarity and self-afinity, random walk method, midpoint method, L-systems
35 hours, compulsory
Teacher / Lecturer
1. Introduction to the computer graphics - raster and vector image. Image processing, CAD data visualization. Rhinoceros - introduction, view setting, elementary examples and commands.
2. Image and color models. Solids in Rhinoceros (colour, solids operation, rendering)
3. Cross-ratio, Transformations of the plane, Rhinoceros - arrays
4. Curves and surfaces in computer graphics - NURBS. General surfaces - boundary curves, revolution surfaces, sweep and offset surfaces.
5. Conics, projective and focal attributes, affinity
6. Computative problems solution
7. Kinematic geometry in the plane, cycloids, epi- and hypocycloids, evolventa, derivation of the equation of the kinematic curve
8. Basic positional and metric problems in Monge projection
9. Basic positional problems in axonometry
10. Monge and axonometric projection of the elementary solids
11. Methods of surface modelling, basic examlpes (derivation in projective space and corresponding model in Rhinoceros)
12. Planer cuts of the elementary solids, line and elementary surface intersection (Monge and axonometric projection + corresponding model in Rhinoceros)
13. Helix, circular and helical surfaces (Monge and axonometric projection + corresponding model in Rhinoceros, simple analytical problem)
Presence in the seminar is obligatory.