Information
for the Descriptive Geometry subject
participant in the BSc course of the University of Miskolc,
Faculty of Mechanical Engineering and Informatics
for full-time mechanical engineering students
Lecturer: Prof. Dr. Zsuzsanna Balajti Óváriné professor
I. The task and purpose of the subject:
The purpose of descriptive geometry is to represent three-dimensional spatial forms using reconstructible mapping, allowing specific spatial geometry problems to be solved directly on the drawing plane.
Among the relationships created between three and two dimensions, a fundamental understanding of Monge's concept and its procedures cannot be replaced for engineering activities. Mastering these concepts provides the foundational freedom needed to effectively select and use visualisation options in modern computer-aided design (CAD) programs.
II. Objectives and Scope of the Course
Monge's representation serves as the foundation for true-to-scale engineering communication. The course covers the following topics:
- Fundamental Spatial Elements: Representation and reconstruction of basic geometric elements in space.
- Spatial Relationships: Mutual positions, containment (fit), parallelism, and intersection of basic geometric elements.
- Auxiliary Projections: Transformation of the plane of projection, target transformations, and their practical applications.
- Perpendicularity: Graphical conditions and representation of perpendicularity in projection planes.
- Revolutions: Rotating a plane into a position parallel to the projection plane.
- Metric Problems: Determining dimensions between spatial elements.
- Polyhedra: Representation of polyhedra, including their intersection with straight lines and the construction of their sections by a plane.
- Curved Elements: Representation of circles, discs, and spheres.
- Surfaces of Revolution: Representation of cylinders and cones of revolution, including their intersections with straight lines and planes.
- Conic Sections: Properties of conic sections and their affine and central collineation relationships.
- Intersections of Surfaces: Constructing intersections between spheres, cones, and cylinders.
- Helices and Helicoid Surfaces: Graphical properties of helices and helicoid surfaces.
III. Course Completion and Assessment
The Descriptive Geometry course is taught in the autumn semester through 2 hours of lectures and 2 hours of practical classes per week. The semester concludes with the granting of a signature (qualifying to take the final exam) and a colloquium (final examination).
III.1. Conditions for Obtaining the Semester Signature
- Lecture Attendance: Regular and diligent attendance at lectures is mandatory. Missing six or more lectures (12+ hours) will result in an automatic refusal of the signature.
- Practical Class Attendance: Regular and diligent participation in practical classes is mandatory. Missing five or more practical classes (10+ hours) will result in an automatic refusal of the signature.
- Drawing Assignments: Students must complete 5 separate drawing assignments to at least a minimum passing level. These must be submitted by the deadlines specified in the Course Schedule. Late submissions require permission from the Dean.
- Midterm Tests: Students must pass 2 written construction tests, earning at least a passing grade (sufficient) on each, or pass the corresponding retake tests. A prerequisite for sitting these tests is the timely submission of all previously issued drawing assignments.
The course instructor (practical class leader) evaluates overall performance before granting the semester signature.
III.1.1. Submission and Evaluation of Drawing Assignments
According to the "Drawing Assignment Specifications," all tasks must be constructed independently by hand on framed, A4-sized technical drawing paper. Submission deadlines are listed in the Course Schedule.
- Defence: Upon submission, students must be able to explain and justify the geometric content and steps of their solution.
- Passing Criteria: To achieve a passing grade (sufficient), the solution must contain no fundamental conceptual errors, and the drawing must meet professional aesthetic standards.
III.1.2. Midterm Construction Tests and Evaluation
During the semester, students will write two 45-minute in-class tests. Test dates are specified in the Course Schedule. Students are only eligible to write these tests if they have submitted all prior drawing assignments by the deadline.
- Grading Scale: Achieving 50% of the maximum points is required for a passing grade (sufficient). Higher grades are scaled approximately linearly.
- Academic Integrity: The use of unauthorised tools, materials, or external assistance during any assessment will automatically result in a failing grade (insufficient).
III.1.3. Rules regarding the retaking of tests and drawing assignments
- Mid-Semester Retakes: Students who do not pass the 1st or 2nd midterm test may take a one-time remedial retake test during the study period, provided they have submitted Drawing Assignments I and II. To obtain the course signature, a passing grade (at least "sufficient") must be achieved on the retake exam for each of the two midterm tests separately.
- Late Signature Replacements: Students who fail to obtain the signature during the regular study period must submit any missing drawing assignments by 12:00 PM on the working day preceding the "Signature Substitute Exam." They must also pass a special written test administered by the department at a time announced in the NEPTUN system.
III.2. Examination Method and Grading
The scope of the final examination covers all knowledge presented in the lectures and practical sessions during the semester, as well as problems that can be derived from and solved using all that information.
- Prerequisites: Obtaining the semester signature is a required prerequisite for taking the exam. Exam registration must be completed via the NEPTUN system by 12:00 PM on the day before the exam.
- Format: The exam consists of a mandatory 90-minute written part and an optional oral part.
- Passing the Exam: A passing grade requires a minimum score of 50% on the written exam. Other grades are distributed linearly.
- A student who achieves a result of less than 25% on the written exam is disqualified from taking the optional oral exam to improve the grade.
III.2.1. Final Grade Calculation
A student's mid-semester performance grade (E) is the average of their drawing assignment grades (R) and their two midterm test grades (Z1, Z2) as follows
The grade (E) received for the student's mid-semester work, and the grade (V) at the end of the writing exam form the grade for the exam as follows
III.2.2. Strict rule
The use of any unauthorised device or the acceptance of assistance during an assessment automatically results in an immediate failing grade.
IV. Literature
Required:
- Dr. Balajti, Zsuzsanna: Descriptive Geometry Excercises, Miskolc, 2025.
- Petar Mladinic, Nikol Radovic: Descriptive Geometry, Perspective Monge’s procedure axonometry, Zagreb, 2019.
- Lajos Sándor: Sztereoszkópikus galéria
- V. O. Gordon; M. A. Sementsov-Ogievskii: A Course in Descriptive Geometry, 1980, Moscow
Recommended:
- A. T. Chahly: Descriptive Geometry, 1968, Moscow
- Kh. A. Arustamov: Problems in Descriptive Geometry, 1972, Moscow
COURSE SCHEDULE
| Week | Date | Lecture Topic | Practical Class & Independent Study | Deadlines & Tasks |
| 1. | IX. 7 – 11 | Representation of truncated polyhedra. | Construction exercises | I.1. Drawing assignment |
| 2. | IX. 14 – 18 | Monge's representation as the basis of true-to-scale engineering communication. Representation and reconstruction of basic spatial elements (point, line, plane). Special straight lines of a plane. Containments of basic elements. | In-class construction exercises | |
| 3. | IX. 21 – 25 | Parallelism. Intersections. Creating a new projection plane (Auxiliary projection). | In-classconstruction exercises | I.2. Drawing assignment |
| 4. | IX. 28– X. 2 | Transforming straight lines and planes into special positions. Applications of transformations. Representation and construction of polyhedra. | In-class construction exercises | I.2. Drawing assignment |
| 5. | X. 5 – 9 | Applications of auxiliary projections: intersecting of the pyramids and prisms with straight lines and planes. Determining of the distances and angles between spatial elements. | In-class construction exercises | Submitting of Drawing assignments I. |
| 6. | X. 12 – 16 | Perpendicularity of spatial elements. Rotating a plane into a projection plane. Application: angles between straight lines and planes. | In-class construction exercises | Mid-semester Test I. |
| 7. | X. 19 – 22 | Representation of a circular disc. | In-class construction exercises | II.1. Drawing assignment |
| X. 23 –XI 1 | Academic recess | Home assignment construction tasks | - | |
| 8. | XI. 2 – 6 | Representation of a circular disc and a sphere. Surface points, normals, and tangent planes of a sphere. | In-class construction exercises | II.2. Drawing assignment |
| 9. | XI. 9 - 13 | Representation of cylinders and cones of revolution. Surface points, normals, tangent planes, and line intersections. Planar sections of cylinders of revolution. | In-class construction exercises | II.2. Drawing assignment |
| 10. | XI. 16 – 20 | Planar sections of a cone of revolution. Conic sections. | In-class construction exercises | II.3. Drawing assignment |
| 11. | XI. 23 – 27 | Intersections of cylinders and cones of revolution with intersecting axes (Method of auxiliary spheres). | In-class construction exercises | Submitting of Drawing assignments II. |
| 12. | XI. 30 – XII. 4 | Intersections of cylinders and cones of revolution with non-intersecting axes (Method of a serious of auxiliary planes). | In-class construction exercises | Correction / Late submission of Drawing Assignments |
| 13. | XII. 7 – 11 | Helices and helicoid surfaces. | Construction exercises | Mid-semester Test II. |
| 14. | XII. 14 - 18 | Course summary and exam preparation. | Construction exercises | Retake mid-semester tests |
For independent study, in addition to the problems solved during classroom practices, it is recommended to construct further exercises from the collection titled DESCRIPTIVE GEOMETRY EXERCISES 2025, available at the following link:
https://geometria.uni-miskolc.hu/files/35743/DESCRIPTIVE%20GEOMETRY%20EXERCISES.pdf
DRAWING ASSIGNMENTS
General Instructions:
Solve each assigned task based on the individual variation number obtained from the tables after each task. The variation number corresponds to the sequential number in the official NEPTUN course list. If the list number exceeds the total number of available variations, the variation number will be the remainder of the list number divided by the total number of variations. If the remainder is 0 (i.e., it is evenly divisible), solve the last variation.
I. DRAWING ASSIGNMENT
I.1. Representation of a truncated polyhedron
Design and sketch a truncated shape created from a cube using axonometric projection. Mark its vertices in alphabetical order with the letters A, B, C, etc.
Based on your variation number, construct three principal views of the truncated shape. The distances of point A from the projection planes are given in the table below:
| Variant number | 1 | 2 | 3 | 4 | 5 | 6 |
| K1 | 30mm | 30mm | 20mm | 20mm | 10mm | 10 mm |
| K2 | 10mm | 20mm | 10mm | 30mm | 20mm | 30 mm |
| K3 | 20mm | 10mm | 30mm | 10mm | 30mm | 20 mm |
I.2. Representation of the prism
Given a straight-line m and a point A not lying on it.
Using auxiliary projection planes, construct the projections of a right prism. The base of the prism is either a square (1) or a regular triangle (2). The axis of the prism lies on line m, and A is one of the vertices of the base polygon. Furthermore, the height of the prism (mh) is one and a half times the side length of the base polygon. Show visibility (hidden/visible lines drawn dashed/continuous).
(Start the construction by introducing a new projection plane connected perpendicularly to the first projection plane K1.)
II. DRAWING ASSIGNMENT
II.1. Representation of the disk
Given a straight line t in a frontal straight-line position and a point P not lying on it. Represent a circular disc with axis t, where P is a point on the circumference.
Construct the first projection (ellipse) of the circular disc, determining:
- the major axis AB and the minor axis of CD,
- tangents and osculating circles at the endpoints of the axes,
- the tangent line e at point P.
Draw the first projection ellipse of the disk, then indicate the visibility of the circular disk and its axis of rotation.
II.2. Intersections of a cone with a plane
Intersect a right circular cone resting on the projection plane K1 with a second projecting plane V2 (perpendicular to K2) according to your variation number:
| Variable number | 1 | 2 | 3 |
| Section | e | p | h |
Construct the following for the first projection:
- The ellipse section curve (e), specifying the major axis AB and minor axis CD,
- The parabola section curve (p), specifying the vertex T, the axis line t, and the directrix line d,
- The hyperbola section curve (h), specifying the real axis AB, imaginary axis CD, and the asymptotes u and v.
For the resulting first projection curve, determine:
- The focal point F (or focal points F1 and F2),
- A point P in a general position, and the tangent line e passing through it.
Draw the first image of the section curve using the hyper-osculating circle(s). Represent the part of the cone between the base plane and the cutting plane according to visibility.
II.3. Intersection between the rotating cone and cylinder
Construct the intersection curve of a right circular cone (with a first projector axis perpendicular to K1) and a right circular cylinder (with a second projector axis perpendicular to K2) such that the intersection curve has a self-intersection point (double point).
Determine the following on the intersection curve:
- The self-intersection point,
- The points lying on the contours of the cylinder and the cone, along with their tangents,
- The lowest points,
- A few points in a general position, constructing the tangent line for at least one of them.
Represent the part of the conical solid that lies outside the cylindrical solid, clearly indicating visibility.
Notes
According to the schedule, the drawings must be executed in pencil on a single sheet of A4-size (210 mm × 297 mm) drawing paper. Subject to individual approval, drawings may also be done in ink on tracing paper. Technical drawing standards for lines and lettering (MSZ EN ISO 128-20) must be applied.
For line work, use line groups with thicknesses of 0.13 mm or 0.18 mm (thin), 0.35 mm (thick), and 0.70 mm (extra thick). A font size of h = 3.5 mm must be used for lettering the figures, and h = 7.0 mm for the title frame.
When drawing in pencil, three distinct line thicknesses must be used as specified above. The lines of correspondence between views must be clearly visible dashed lines. Consistent graphic symbols must be used for identical content.
- Thin lines (0.3 mm, H or HB pencil): construction lines, projecting lines.
- Medium lines (0.5 mm, HB pencil): given elements and hidden (invisible) parts of the final result using dashed lines.
- Thick lines (0.7 mm, B or 2B pencil): final results and the sheet frame using solid lines.
The assignment title, designer's name, student group number, signature, academic year, date, task number, and variation number must be indicated in the title block in 7 mm font, as shown in the attached template.
Further details will be provided by the lab instructor.