Course detail
Constructive and Computer Geometry
FSI-1KG Acad. year: 2021/2022 Winter semester
Principles and basic concepts of three-dimensional descriptive geometry. Perspective transformation. Orthographic projection. Curves and surfaces. Intersection of plane and surface. Piercing points. Torus, quadrics. Helix, helicoid. Ruled surfaces.
Descriptive geometry is supported by a computer.
Supervisor
Department
Learning outcomes of the course unit
Students will acquire the basic knowledge of three-dimensional descriptive geometry necessary to solve real life situations in various areas of engineering.
Prerequisites
The students have to be familiar with the fundamentals of geometry and mathematics at the secondary school level.
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 REQUIREMENTS: Draw up 5 semestral works (each at most 2 points), there is one written test (the condition is to obtain at least 5 points of maximum 10 points). The written test will be in the 9th week of the winter term approximately.
FORM OF EXAMINATIONS: The exam has an obligatory written and oral part. In a 90-minute written part, students have to solve 3 problems (at most 60 points). The student can obtain at most 20 points for oral part.
RULES FOR CLASSIFICATION:
1. Results from seminars (at most 20 points)
2. Results from the written examination (at most 60 points)
3. Results from the oral part (at most 20 points)
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
Language of instruction
Czech
Aims
The course aims to acquaint the students with the theoretical basics of descriptive geometry. It will provide them with a computer aided training in basic parts of geometry.
Specification of controlled education, way of implementation and compensation for absences
Attendance at seminars is required. The way of compensation for an absence is fully at the discretion of the teacher.
The study programmes with the given course
Programme B-MAI-P: Mathematical Engineering, Bachelor's
branch ---: no specialisation, 5 credits, compulsory
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
Syllabus
1. Extension of the Euclidean space. Mapping between two planes. Collineation and affinity.
2. Methods for mapping three-dimensional objects onto the plane – central and parallel projections. Introduction into the Monge's method of projection (the two picture protocol) – the orthogonal projection onto two orthogonal planes.
3. Monge's method: points and lines that belong to a plane, principal lines, 1st and 2nd steepest lines.
4. Monge's method: rotation of a plane, circle that lies in a plane. 3rd projection plane (profile projection plane).
5. Rectangle and oblique parallel projection, Pohlke's theorem. Axonometry.
6. Axonometry: points, lines, planes, principal lines.
7. Axonometry: Eckhard's method. Elementary solids and surfaces.
8. Elementary surfaces and solids in Monge's method and axonometry. Intersection with stright line and with plane.
9. Curves: Bézier, Coons, Ferguson curves. Kinematic geometry in the plane. Rectification of the arc.
10. Helix: helical movement, points and tangent lines in Monge's method and axonometry.
11. Surfaces of revolution: quadrics and torus. Right circular conical surface and its planar sections. Hyperboloid as a ruled helical surface.
12. Helical surfaces: helical movement of the curve, ruled (opened, closed, orthogonal, oblique) and cyclical surfaces.
13. Developable surfaces: cylinder and right circular cone with curve of cut.
Exercise
14 hours, optionally
Teacher / Lecturer
Syllabus
1. Conics: definitions of ellipse, parabola, hyperbola. Points, tangents and points of tangency of the conics, hyperosculating circles.
2. Collineation and affinity. Conics: affine image of a circle.
3. Conics: construction of ellipse by trammel method, Rytz's axis construction of ellipse. Monge's method: points, lines, planes.
4. Monge's method: basic geometrical relationships – the relative positions of points, lines and planes, angles, distances.
5. Monge's method: circle that lies in the plane, basic solids.
6. Axonometry: points, lines, planes. Square and circle in projection planes. Circles and squares in the horizontal, frontal or profile planes.
7. Axonometry: basic geometrical relationships – the relative positions of points, lines and planes. Projection of basic solids.
8. Axonometry: Eckhard's method. Monge's method and axonometry: intersection of the stright line with a basic solid.
9. Written test. Monge's method and axonometry: intersections of the polyhedron or cone with a plane.
10. Kinematic geometry: points and tangents of cycloid, evolvent, epicycloid, etc.
11. Helix: points, tangent lines. Helix in Monge's method and in axonometry.
12. Surfaces of revolution: intersections of the quadric surfaces with a plane. Helical surfaces: ruled surfaces.
13: Helical surfaces: cyclical surfaces. Developable surfaces: cylinder and right circular cone with curve of cut.
Computer-assisted exercise
12 hours, compulsory
Teacher / Lecturer
Syllabus
1. Conics.2. Computer: Rhinoceros 3D: Line, Ortholine,Circle, Ellipse etc.
3. Mongean system of descriptive geometry.
4. Computer: Rhinoceros 3D: Polygon, Plane etc. Mapping between planes. Mapping between a circle and a ellipse.
5. Mapping of circle.
6. Computer: Rhinoceros 3D: Line, Plane, Circle, Polygon in 3D. A line perpendicular to a plane surface, a plane surface perpendicular to a line, true length projection of line AB, distance from a point to a line etc.
7. Basics of an axonometric projection.
8. Computer: Rhinoceros 3D: Elementary solids and surfaces – Intersect, Subtract, Slice.
9. Slice and intersection of geometric solids and surfaces.
10. Computer: Kinematic geometry, Rhinoceros 3D: Helix.
11. Torus, cylinder,cone etc. Helix, projection of helix, helicoids.
12. Computer: Rhinoceros 3D: Helix, helicoid. Rotation surfaces.
13. Computer: Ruled surfaces. Deployable surfaces.