# Information

about the Descriptive Geometry subject

## to participant in the BSc course of the University of Miskolc,

Faculty of Mechanical Engineering and Informatics

for full-time mechanical engineering students

## Lecturer: Dr. habil. Zsuzsanna Balajti Óváriné associate professor

## I. The task and purpose of the descriptive geometry

The aim of descriptive geometry is to represent the shapes of three-dimensional space in a reconstructable mapping and to solve specific spatial geometry tasks in the plane of the construction drawing

Among the relationships that can be established between three and two dimensions, knowledge of the concepts and procedures of Monge's approach is irreplaceable during engineering activities. Knowing these guarantees you the freedom to choose visualisation options with computer programs.

## II. The thematic of the subject

Monge's representation is the basis of true-to-scale engineering communication. Representation and reconstruction of the basic elements of the space. The mutual position, fit, parallelism and intersection of the basic geometric elements. Transformation of the plane of projection, target transformations and their applications. The condition of perpendicularity on the projections. Rotating a plane into a position parallel to the plane of projection. Dimensions between space elements. Representation of polyhedral, cutting with a straight line, cutting with a plane. Representation of circle and disc. Representing a sphere, a cylinder of revolution and a cone of revolution, intersecting it with a straight line and cutting it with a plane. Conic slices and their affine and central collineation relations. Editing sphere, cone and cylinder interactions. Helix, helicoid surfaces.

## III. The way to finish the subject

The Descriptive Geometry subject is taught in the framework of 2 hours of lectures and 2 hours of practice in the autumn academic semester. The semester finishes with a **signature** and **colloquium** for mechanical engineering students.

### III.1. Conditions for obtaining a signature

- Regular attendance at the lectures with due diligence, failure to attend six lectures (12 hours) or more will result in the final refusal of the signature.
- Regular participation in practical classes with due diligence, missing five practical classes (10 or more hours) will result in the final refusal of the end-of-semester signature.
- Complete 5 construction drawing tasks separately at a minimum grade and submit them by the deadline specified in the Schedule (the faculty dean's permission is required for submission after the deadline).
- 2 written knowledge surveys composed of constructing tasks, each with at least a sufficient grade, or completing them in correctional written knowledge surveys. The prerequisite for the possibility of written knowledge survey is the submission of the construction drawing tasks issued by the deadline.

To acknowledge the semester and obtain a signature, the performance is evaluated by the practice leader.

#### III.1.1. Submission and assessment of the construction drawing tasks to be submitted during the semester

According to the "Construction Drawing Task specification", the construction drawing tasks must be *independently constructed* on an A4-sized technical drawing sheet framed by your own hand.

The submission deadlines are included in the "Schedule".

*When submitting a constructing drawing, the content of the task must be interpreted!*

In order to achieve a sufficient grade, there must be no fundamental content errors in the solution of the task and the constructing drawing must also be aesthetically acceptable.

#### III.1.2. The qualification, date and evaluation of constructing exercises as completed during the semester

During the semester, the student must write two closed-door papers in 45 minutes each. The dates of the closed assignments are included in the "Schedule".

The mid-semester construction exercises can be written if the previously issued constructing drawing assignments have been submitted by the prescribed deadline.

50% of the achievable performance is required for a sufficient grade for construction exercises; the other grades are approximately linear.

The use of unauthorized tools and assistance during the assessment will automatically result in an insufficient grade.

#### III.1.3. The conditions of the correction of the written knowledge surveys and additional submission of construction tasks

Those students who did not complete the 1st and 2nd constructing exercises satisfactorily may write a one-time remedial constructing exercise during the study period if the I. and II. constructing tasks are submitted by them*. *

In order to obtain the signature, the results of the corrective constructing tasks must also be at least sufficient for each piece. * *

Those students who did not obtain a signature during the study period must submit the missing constructing drawing tasks by 12:00 on the working day preceding the "Signature Substitute Exam" and/or write a paper in private with at least sufficient results at the time announced by the department in the NEPTUN system.

### III.2. Method and grade of the final exam

At the end of the semester, there will be an exam on the subject. The topics of the final exam is all topic covered in the lectures and practices.

The condition for the student may take the exam is the obtained Signature. Registration for the final exam can be done through the NEPTUN system until 12:00 the day before the final exam.

The exam consists of a mandatory 90-minute written part and optional oral part. ** **

*A sufficient grade requires 50% of the maximum score that can be obtained on the written final exam; the other grades are approximately linear. *A student who does not obtain even 25% of the available score in the written part of the exam may not go to correct with the oral part.

The student receives a grade (E) for the mid-semester work, which is the average of the grades (R) for the constructing drawing tasks and the grades for the constructing knowledge surveys (Z1, Z2) as follows

The grade (E) received for all work of student in the mid-semester, and the grade (V) at the end of the writing exam will form the final exam grade as follows

** **

*Using unauthorised tools or help during the assessment will automatically result in an insufficient grade!*

## IV. Literature

#### Required:

- Óváriné Balajti, Zsuzsanna: Practice Worksheets for Mechanical Engineering Students _2025.pdf
- 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
- Geiger János: Ábrázoló geometria, 2015.
- Bancsik Zs., Juhász I., Lajos S.: Ábrázoló geometria szemléletesen, elektronikus könyv, 2007.

**Recommended:**

- A. T. Chahly: Descriptive Geometry, 1968, Moscow
- Geiger János: Ábrázoló geometria feladat gyűjtemény 2012.
- Geiger János: Ábrázoló geometria, Csavarvonal, csavarfelületek
- Petrich Géza: Ábrázoló geometria,Tankönyvkiadó, Budapest, 1973.
- Popa-Müller Izolda: Ábrázoló geometria, Sapientia tankönyvek, Műszaki tudományok, Scientia Kiadó, Kolozsvár.

# SCHEDULE

Week |
Date |
Lecture |
Practice and independent preparation |
Task |

1. |
IX.9.- 13. |
Monge's representation, true-size engineering communication. Representation and reconstruction of the space basic elements (point, line, plane). Special straight lines on the plane. Lying of the basic elements. | Practice Worksheet 1. | I.1. drawing task |

2. |
IX.16.- 20. |
Parallelism. Intersection. Creating a new plane of projection. | Practice Worksheet 2. | |

3. |
IX.23.-27. |
Transforming straight lines and planes into a special position. Applications of transformation. Representation and creation of polyhedral. | Practice Worksheet 3. | I.2. drawing task |

4. |
IX.30.- X.4. |
Applications of the transformation of the projection plane system: intersecting of the pyramid and prism with a straight line, a plane. Determining of the distance and angle of the elements of the space. | Practice Worksheet 4. | |

5. |
X.7.–11. |
Perpendicular space elements. Rotate a plane to a position parallel to the image plane. Application: angle of space elements (straight lines and planes). | Practice Worksheet 5. | Submission of drawing tasks I.1. and I.2. |

6. |
X.14.– 18. |
Representation of a circle and disc. The affine relationship between the circle and the ellipse. | Practice Worksheet 6. | I. Written knowledge survey |

7. |
X.21. Teaching break |
Representation of a sphere, its surface point, normal, tangent plane, intersections with a straight line and plane (independent preparation). | Practice Worksheet 7. | II.1.drawing task |

8. |
X.28.– 31. |
Representation of a rotating cylinder and cone, surface point, normal, tangent plane, and intersecting with a straight line. Intersection of a rotating cylinder with a plane. | Practice Worksheet 8. | II.2.drawing task |

9. |
XI.4.-8. |
The intersection curves of a rotating cone with a plane. Cone section curves on the plane. | Practice Worksheet 9. | II.2.drawing task |

10. |
XI.11.– 15. |
The intersection curve of a sphere, rotating cylinder and cone (plane slicing method). | Practice Worksheet 10. | Submission of drawing tasks II.1. and 2. |

11. |
XI.18.– 22. |
The intersection curve of rotating cylinders and cones of rotation with an intersecting pair of axes (method of auxiliary spheres). | Practice Worksheet 11. | II.3.drawing task |

12. |
XI.25.–29. |
The intersection curve of rotating cylinders and cones of rotation with a non-intersecting pair of axes (method of a serious of auxiliary planes). | Practice Worksheet 12. | II.3.drawing task |

13. |
XII.2.– 6. |
Helix. Helicoid surfaces. | Practice Worksheet 13. | Submission of drawing task II.3. II. Written knowledge survey |

14. |
XII.9.- 13. |
Summary. Preparation for the final exam. | Practice Worksheet 14. | Corrective written knowledge survey |

For independent preparation, it is recommended that additional tasks be solved from the example library János Geiger: Descriptive Geometry Task Book, in Hungarian, University Publishing House 2012).

# DRAWING TASKS

## I. DRAWING TASK

### I.1. Truncated polyhedron representation

Design and axonometrically sketch a truncated shape created from a cube with edge ** a**=

**40**

*mm*, then mark its vertices in order with the letters

**A**,

**B**,

**C**, ...! According to the variation number, create three ordered views of the truncated shape so that the distances of point

**A**of the shape from the planes of projections are given in the table!

Variant number | 1 | 2 | 3 | 4 | 5 | 6 |

K_{1} |
30mm |
30mm |
20mm |
20mm |
10mm |
10 mm |

K_{2} |
10mm |
20mm |
10mm |
30mm |
20mm |
30 mm |

K_{3} |
20mm |
10mm |
30mm |
10mm |
30mm |
20 mm |

### I.2. Representation of the prism

Given a straight-line **m** and a point **A** that does not lie on it. Using new projection planes, construct the projections of the straight prism based on the square (1) or regular triangle (2), whose line of height is the **m** and **A** is one of the vertices of the base polygon; furthermore, the dimension of the height *mh* is one and a half times the side length of the base polygon! Show the visibility too! (Start constructing by creating a new projection plane connected to the first projection plane!)

## II. DRAWING TASK

### II.1. Representation of the disk

Given a straight-line **t** in the frontal straight-line position and a point **P** that does not lie on it. Represent the circular disc with axis **t**, of which **P** s a point on the circumference! Construct for the first projection ellipse of the circular disc

- the major axis
**AB**and the minor axis of**CD**, - tangents and hyperosculating circles at the axis’s endpoints,
- the tangent straight-line
**e**at point**P**! Sketch the first image ellipse, then indicate the visibility of the circular disc and its axis!

### II.2. Intersections of a cone with a plane

Intersect a rotating cone lying on the projection plane **K**_{1} with a second projection plane **V**_{2} according to variation number!

Variable number | 1 | 2 | 3 |

Section | e |
p |
h |

Determine the

- first projection of the curve of the ellipse section
with the major axis*e***AB**and minor axis**CD**! - first projection of the curve of the parabola section
with the axis point*p***T**, the axis straight line**t**and the directrix straight line**d**! - first projection of the curve of the hyperbola section
with the real axis*h***AB**and imaginary axis**CD**, furthermore the asymptotes**u**and**v**!

Construct

- the focal point
**F**, or focal points**F**_{1}and**F**_{2}, - a point
**P**in general position, lying on its tangent straight-line**e**of the curve of the first projection!

Draw the first projection of the section curve using the *hyperosculating circle(s)*! *Visually depict the part of the cone body between the base plane and the cutting plane!*

### II.3. Intersection between the rotating cone and cylinder

Construct the intersection curve of a rotating cone with an axis in first projector line position and a rotating cylinder with an axis in second projector line position in such a way, that it has a self-intersection point! Determine

- the self-intersecting point,
- the points lying on the contours of the cylinder and the cone with the tangents,
- the lowest points
- a few points in general position with the tangent in one from it of the intersection curve!

*Represent the part of the conical body outside the cylindrical body with an indication of visibility!* ** **

### Notes

According to the Schedule, the drawings of the construction drawing tasks must be made in pencil on one A4 (210mmx297mm) size technical drawing sheet. The standards of lines and letters (MSZ EN ISO 128-20) must be applied. The letters size of *h*=**3.5**mm should be used for lettering the figures, and *h*=**7.0** mm for labelling the frame.

The orderliness between the views should also be promoted through the use of graphic elements, and the same content should be marked consistently by the same graphic elements. Based on an individual decision, it is possible to *draw with ink on pause*.

The standards of lines and letters (MSZ EN ISO 128-20) must be applied. To draw the lines, use the line thickness of **0.13**mm or **0.18**mm thin, **0.35**mm thick and **0.70**mm highlighted line thickness.

In case of the *drawing with a pencil*, three different line thicknesses should be used in accordance with the above-mentioned line thicknesses.

For *thin line thickness*, **0.3**mm is used, and it is recommended to use the H or HB pencil for drawing *constructing lines* and *order straight line*s.

The thickness of **0.5**mm is used *for the middle line thickness*, and it is recommended to use the HB pencil to draw *the given elements* and *the invisible part of the final result with a dashed line*.

*For thick line thickness, ***0.***7**mm is used, and it is recommended to use the pencil marked B or 2B to draw the part of the final result, which is to draw with a continuous line* and* the frame.*

The title of the task, the designer's name, student group number, signature, the academic year and semester, the task number and the variant number should be indicated on the drawing sheets - based on the attached sample - in **7**mm letter size. More detailed information is provided by the practice leader.