FROM THE EXPERIENCE OF COMPILING METHODOLOGICAL MANUALS

ON DESCRIPTIVE GEOMETRY

L. S. Skriptsov, V. A. Voskresensky

(Presented by the scientific seminar of the departments of descriptive geometry of Tomsk higher educational institutions)

For better mastery of the course in descriptive geometry, students are required independently to complete a number of large homework assignments.

Because of the lack of time, the instructor conducting practical classes in descriptive geometry is unable to dwell in detail on the sequence for solving and drafting the assigned tasks. In connection with this, there arises an acute need to develop special methodological instructions for one assignment or another, so that a student, having acquainted himself with such a manual, can complete the given practical task.

Questions of teaching methodology in higher education have, up to the present time, been studied insufficiently, and little has been said about them in various collections of scientific-methodological conferences. It is true that in the collection of such a conference held in 1965 in Tashkent there are articles by I. I. Kotov [1], N. F. Chetverukhin [2], V. O. Gordon [3], G. F. Bykova [4], and others. However, we believe that methodological developments still contain many shortcomings that should gradually be eliminated, making it possible to develop a definite direction in the creation of up-to-date methodologies, especially those connected with independent work by students in various systems of instruction. Let us note that for correspondence students we have a very extensive methodological literature that helps them complete control assignments. But as for the full-time form of training specialists, here each department acts at its own discretion.

In this article the authors do not intend to impose on everyone their own method for carrying out homework assignments, but wish to share their experience in systematizing certain rules that students need in the process of their self-study.

At the Department of Descriptive Geometry of the Tomsk Polytechnic Institute, all full-time students must complete 3 large homework assignments, which are mentioned in the article published in the present collection [5].

The most labor-intensive assignment is the third one, and we wish to show the methodology for carrying it out (as an exchange of experience).

In each assignment, in our opinion, the following questions should be covered: 1. The purpose of the work; 2. Its content; 3. Explanation of the specific assignment; 4. The sequence of its execution.

Let us consider the questions raised in more detail.

1. Purpose of the Work

Descriptive geometry provides rules for constructing images of bodies (projections) on two or more planes, subsequently used in engineering drawing to construct images of parts of machines and mechanisms. These parts usually consist of a combination of various elementary geometric surfaces, such as prisms, pyramids, cylinders, cones, spheres, etc. The intersections of the surfaces of such bodies produce lines of complex form, which must be shown on the drawing in several projections. The present work teaches the construction of such lines of intersection according to the laws of descriptive geometry and is the concluding independent homework assignment, in which students must demonstrate their ability to apply the theory of descriptive geometry to the solution of practical problems.

In the same work, students become acquainted with the construction of axonometric projections, which are widely used in engineering drawing to clarify the appearance of complex structures that are difficult to understand from an orthographic drawing. Instead of axonometric images, spatial models of structures are sometimes made from paper or cardboard, and for this purpose developments of the surfaces of all elements of the given structure are first prepared. At the instructor’s choice, students carry out either the axonometric projection or the developments of surfaces.

2. Content of the Work

Each student is given an individual assignment in the form of two projections of a complex combination of two or more simple bodies representing part of a machine or mechanism.

The student is required to complete the following tasks:

1. Construct three projections of the proposed part in pencil on format No. 22, observing the GOST standard and the relative thicknesses of visible, invisible, and other auxiliary lines.

2. Construct the lines of intersection of the surfaces of the bodies included in the given part, in three projections, taking visibility into account.

3. Construct a spatial image of the given structure (rectangular isometry or dimetry, as specified by the instructors).

4. Construct the development of the surfaces of the given part and, from them, make a model of the proposed structure. (The choice of the assignment under items 3 or 4 is made by the instructor.)

3. Explanation of a Specific Assignment

For a clearer understanding of the procedure for carrying out the work, an explanation is given for one of them according to the attached drawing. Two projections are given (front and horizontal), with dimensions, of an engineering structure representing a unit of a ventilation installation, consisting of part of the main cylindrical iron pipe I, two cones II and III described about one and the same spherical surface of diameter Ø 96 mm, part of the supply pipe IV, and a branch elbow made up of three cylindrical sections V, VI, VII.

It is required to construct: a) the lines of mutual intersection of the surfaces of all elements of the structure on three projection planes; b) an isometric or dimetric image of the entire structure.

4. Sequence of Performing the Work

1. On a sheet of format No. 22, an inner border is drawn at a distance of 5 mm from the edges of the sheet and 20 mm from the left.

2. In the lower right corner of the border, a title block is placed for the inscriptions of the size indicated on the drawing.

3. Three projections of the proposed structure are constructed; moreover, the third projection is made with the aid of the permanent straight line of the drawing, as shown in Fig. 1 for points 1, 4, 5. The scale of the drawing is chosen according to the instructor’s directions. The projections are placed closer to the left edge of the border and should not occupy more than 2/3 of the area of the entire sheet. The inscriptions in the title block are executed in type No. 5, in lowercase letters, as shown on the drawing.

4. The projections on \(H\) and \(W\) of the lines of intersection of links V, VI, VII of the left branch are constructed as ellipses according to the dimensions of the minor and major axes, by any rules set forth in textbooks on drafting.

5. The frontal projection of the line of intersection of the conical surfaces II and III, circumscribed about one sphere, is constructed. In this case one should recall Monge’s theorem, which states: “If two surfaces of the second order (second degree) are inscribed in or circumscribed about a third surface, also of the second order, then they intersect each other in two plane curves of the second order.” In our case the conical surfaces and the sphere are surfaces of the second order; consequently, the cones will intersect in two ellipses. Since the axes of the cones are at the same time parallel to the frontal plane of projection, the ellipses will be projected onto it as straight lines, as is known from the theory of descriptive geometry.

For constructing one of the ellipses, two end points of its projection are marked on the plane—one (2) at the intersection of the left contour generators of the cones and the other (3) at the right generators, which are then joined by a straight line, giving the required projection of one of the ellipses.

To construct the projection of the second ellipse (a straight line), point 1 of the intersection of the right contour generator of cone III and the left generator of cone II is marked. It will be the beginning of the projection of the second ellipse.

The second end point of the projection of the ellipse is found at the intersection of the extended left contour generator of cone II with the extended right contour generator of cone III. However, this point cannot be found within the limits of the drawing. To construct the line passing through the inaccessible point, one may use the method of similar triangles with mutually parallel sides. Arbitrary points \(A\) and \(B\) are chosen on the contour generators of cones II and III and are joined to the known given point \(C\) (1), forming triangle \(ABC\).

An arbitrary point \(A_1\) is chosen on the extension of the right generator of cone III, and through it lines are drawn parallel to the sides of triangle \(ABC\), namely \(A_1B_1 \parallel AB\), \(A_1C_1 \parallel AC\), and \(B_1C_1 \parallel BC\). Through the vertices \(C\) and \(C_1\) a line is drawn, which will be the required projection of the second ellipse; however, it must be ended at points 4 and \(4_1\), where it meets the projection of the first ellipse.

1. The horizontal projections of points 1, 2, 4 are constructed; moreover, to find point 4, first a horizontal secant plane \(Q_4\) is drawn through points 4 and \(4_1\), giving in its section with cone III a circle, which is projected onto \(H\) in true size, and point 4 is then marked on it. Additional points are found in the same way.

Fig. 1.

Visible drawing labels include: a), b), c); scale M 1:2.

Title-block labels in the drawing:

• Completed by
• Group No.
• Date
• Accepted by
• Grade
• Work No.
• Record-book No.

ellipses 5, 5₁, 6, 6₁ and 7, 7₁, then are joined by a smooth curve and, finally, transferred to the profile plane of projections.

1. From the vertex of cone \(S_2\), the horizontal projections of the contour generators of cone II are drawn; they must be tangent to the horizontal projection of the first ellipse, to the ellipses of intersection of link V with cone II and with the sphere \(\varnothing 96\) mm. The exact tangency of these curves is a check on the correctness of the constructions performed.

2. On the profile projection, from the vertices \(S_1\) and \(S_2\) of the cones, the contour generators of the cones are drawn, tangent to the spherical surface \(\varnothing 96\) mm, and also to the projections of the constructed ellipses.

3. Construction of the axonometric image shown in Fig. 1, б) begins with the selection of the isometric axes as projections of certain coordinate axes on the complex drawing. For simplicity in further constructions, it is recommended to make the coordinate axes coincide with the axes of symmetry of the given structure, taking point \(O_3\)—the center of the base of cylinder 1—as the center. Thus, the coordinate axes will be \(O_3X_3\), \(O_3Y_3\), \(O_3Z_3\), which are projected onto the isometric axes indicated above.

4. The construction of the isometry is carried out at the same scale at which the complex drawing was made (in our case \(M1:2\)). The coordinate dimensions of points are transferred to the isometric axes without distortion, i.e., the distortion coefficients are taken equal to \(p=q=r=1\).

If the free space on the sheet does not allow the axonometry to be placed, then the scale of its execution should be taken smaller, as recommended by GOST.

1. At the center \(O_2\), the isometry of the circumference of cylinder 1 is constructed in the form of an ellipse with major axis equal to \(1.22d\), and minor axis \(0.7d\), where \(d\) is the diameter of the circle, equal to 100 mm. The magnitudes of the axes \(1.22d\) and \(0.7d\) can be found graphically, as shown in Fig. 1, в), where the given circle \(d_0=100\) mm is constructed and two mutually perpendicular diameters are drawn—horizontal and vertical. The ends of the diameters \(C_1\) and \(D_1\) are connected by a straight line, whose length will be equal to the minor axis of the ellipse. To find the magnitude of the major axis of the ellipse, from points \(C_1\) and \(D_1\) arcs of circles of radius \(C_1D_1\) are drawn until they intersect in points \(A_1\) and \(B_1\). The length of line \(AB\) will be equal to the major axis of the ellipse. At point \(O_2\) these axes are constructed, and an ellipse is drawn along them by any method; however, for simplification it is recommended to replace axonometric ellipses with ovals closely approximating them. The technique for constructing such ovals may be found in books [6, 7].

2. Isometric axes are constructed at the center \(O_{3-2}\) for the lower circumference of cylinder 1, and at the centers \(O_{4-2}\), \(O_{5-2}\), \(O_{6-2}\) for the circumferences of the cylindrical elbows. On these axes, ellipses are drawn as projections of the corresponding circumferences by the method indicated above.

3. At the center \(O_2\), the isometry of the spherical surface, to which cones II and III are tangent, with diameter \(=96\) mm, is also constructed in the form of a circle, as in the complex drawing, but with a diameter equal to \(1.22 \times 96\), as follows from the theory of descriptive geometry.

4. The isometric projections of the vertices of cones \(S_{1-2}\) and \(S_{2-}\) are constructed according to the coordinates of these points taken from the complex drawing; from them the contour generators of cones II and III are drawn, tangent to the spherical surface. The generators of cone III pass smoothly into the ellipse constructed at the center \(O_2\).

5. The construction of the line of intersection of cones II and III, as well as of the ellipses of intersection of the surfaces of cylindrical links V, VI, VII, is performed by the points belonging to them, as shown, for example;

To obtain the isometric image of point \(6_2\) (Fig. 1, b), from the center \(O_2\) first lay off the value of the coordinate \(z_6\), taken from the multiview drawing; from the end of this coordinate, parallel to the axis \(O_2X_2\), lay off the coordinate \(x_6\), and from its end—the coordinate \(y_6\), parallel to the axis \(O_2Y_2\). The end \(6_2\) will be the required point. In this way the positions in isometry of several other points of the curves should be found and connected by a smooth curve.

1. The constructed image should be outlined in pencil or in ink, but only the visible contour. In axonometry invisible lines are not shown, so that the visual clarity and legibility of the image of the object are not lost. The thickness of visible lines is chosen as \(0.5\) mm or \(0.75\) mm, depending on the size of the drawing.

2. After outlining, the drawing is lettered. All points are assigned numbers or letters with the corresponding marks on the projections. Auxiliary lines connecting the projections of points must not clutter the drawing or obscure it. Such lines are drawn thinly, in colored ink or pencil.

Conclusion

The methodology we have compiled for carrying out a large assignment that concludes the course in descriptive geometry makes it possible:

1. To accustom students to independent work using the necessary literature.

2. To draw the attention of learners to the interconnection between the multiview drawing and the axonometric image.

3. To save a considerable amount of time for the teaching staff on consultations and explanations during classroom sessions.

LITERATURE

1. I. I. Kotov. Main directions of research work in applied geometry and engineering graphics. Proceedings of the II Conference on Descriptive Geometry and Engineering Graphics of Higher Educational Institutions of the Uzbek SSR. Issue 60. Tashkent. Publishing House of the Academy of Sciences of the Uzbek SSR, 1968.

2. N. F. Chetverukhin. On programmed instruction and its application in the course of descriptive geometry. Proceedings of the II Conference on Descriptive Geometry and Engineering Graphics of Higher Educational Institutions of the Uzbek SSR. Tashkent. Issue 60. Publishing House of the Academy of Sciences of the Uzbek SSR, 1968.

3. V. O. Gordoi. On the content and organization of classes in the basic course of descriptive geometry. Proceedings of the II Conference on Descriptive Geometry and Engineering Graphics of Higher Educational Institutions of the Uzbek SSR. Tashkent. Publishing House of the Academy of Sciences of the Uzbek SSR, 1968.

4. G. F. Bykova. The dynamic drawing as a means of instruction in the course of descriptive geometry. Proceedings of the II Conference on Descriptive Geometry and Engineering Graphics of Higher Educational Institutions of the Uzbek SSR. Tashkent. Publishing House of the Academy of Sciences of the Uzbek SSR, 1968.

5. B. L. Stepanov, V. A. Voskresensky. On the question of students’ independent work in descriptive geometry. The present collection.

6. V. A. Fedorenko, A. I. Shoshin. Handbook of Machine-Building Drafting. Moscow, Gostekhizdat, 1963.

7. I. M. Mogilny. Technical Drawing. Moscow, “Mashinostroenie,” 1966.