PSYCHOLOGY AND EDUCATION (2021) 58(2): 5710-5718 ISSN: 00333077 Axonometry New Practical Graphical Methods For Determining System Parameters Malikov Kozim Gafurovich Senior Lecturer, Department of Engineering Graphics and Teaching Methods, Tashkent State Pedagogical University named after Nizami Abstract Some practical graphical method of determining the coefficients of variation and angles between them, which are the main parameters of the axonometric system, first describes the development of a model for generating axonometric projections and using it to determine the basic parameters of axonometry graphically. Key words: Axonometry, isometry, projection, transformation, parallel, plane, perpendicular, horizontal, bisector plane. Article Received: 18 October 2020, Revised: 3 November 2020, Accepted: 24 December 2020 Introduction. shortened, the angular values between them do not The practical graphical method of determining the change, and the triangle of traces is an equilateral coefficients of variation of the axes of the triangle, as noted above, described in the Q plane axonometric system and the angles between them in their true magnitude. has not been raised as a problem until now [1]. Therefore, all elements of isometry, i.e. Because the results obtained were graphic works axonometric axes, are reduced to a theoretically performed manually, they were approximately 0.5- determined coefficient of variation of 81.6496 mm 1 mm accurate. Therefore, graphical methods were and the angular values between them are described of great importance in their time, although they as 1200. The triangle of traces is also depicted in could not compete with analytical methods for real terms with sides of 141.4214 mm. high-precision results. Figure 2 shows a dimetric representation of the In one of his collections, Monge compared his Cartesian coordinate system when the axonometric creative research and applied stereometry to Y axis is at an angle of 61 52'28 '' to the plane of planametry, telling his colleague Oliver: I would the axonometric projection plane Q, and its tip suggest that the solution of these examples be done touches Q [2]. by means of graphical constructions. Students In this case, the trace triangle of the Cartesian would be amazed to see the accuracy and ease of system is depicted at an angle of 26 46'42 '' to the the second column's solutions and proofs. ” plane Q. For now, in the development of science and The analysis of the elements of dimetry formed by technology, we will consider the graphical the applied graphical method showed that they are definition of the above problem, as the creation of equivalent to the parameters determined by the graphics using information and computer theoretical analytical method. The coefficients of technology, for example, in Auto CAD provides variation were 0.9428 along the X and Z axes and very high accuracy (0.00000001-one hundred 0.4714 along the Y axis, and the angles between million). the axes were ∠XOZ = ∠YOZ = 131 25'36 '' and In Fig. 1, the plane of axonometric projections Q ∠XOY = 97 10'54 ''. And the trace triangle and the coordinates in space are 100 mm long and showed that the length of the base of an equilateral intersect at a homogeneous angle 35 15'52 '', the triangle is 141.4214 mm and the length of the sides Cartesian coordinate plane system intersects to of an equilateral triangle is 131.6197 mm. form a triangle of Gaussian traces. The resulting Figure 3 shows a triangle of traces formed by the axonometric projection is isometry, in which the intersection of the plane of axonometric axonometric axes from one point-coordinates are projections Q with the Cartesian system in space. 5710 www.psychologyandeducation.net PSYCHOLOGY AND EDUCATION (2021) 58(2): 5710-5718 ISSN: 00333077 It was observed that all the parameters of the the Y axis was visible, and the equilateral sides of dimetry did not change and the upper right part of the triangle of traces formed were 106.9341 mm. Figure 1 Figure 2 Figure 3 Thus, the Auto CAD program, which is a modern graphics program, has sufficient right-angled projections of objects. Such attempts capabilities to determine the coefficients of have been made by many scientists and variation of the axes of the axonometric system researchers. Of particular importance is the and the angles between them in a practical construction of standard axonometrics in the graphical way. method of additional right-angled projection, The main part developed by Kolotov. In determining the standard axonometric The first essence of this method is that the parameters using the auxiliary right-angled projections of geometric figures given in an projection method, we obtained the Moon axis at arbitrary situation are intended to construct and an angle of 35 15'52 'and 61 52'28' 'to the bring to a point their true appearance or length or axonometric plane, respectively, in the graphical size by depicting them in a new projection determination of the basic parameters of the direction at right angles to an imaginary plane standard isometry and dimetry. parallel or perpendicular to them in thought space. We consider that the above standard axonometric parameters can also be determined directly using Figure 4 In this case, the new projection Also, the second essence of this method direction s is obtained perpendicular to the is the indirect representation - transformation of imaginary plane R bounded by the dashed lines: all points of the thought plane R and the figures s ⊥ R. lying in it to the V plane. In this case, the plane R is located on the plane V as R1 - it merges. 5711 www.psychologyandeducation.net PSYCHOLOGY AND EDUCATION (2021) 58(2): 5710-5718 ISSN: 00333077 To understand the content and essence frontal position. The frontal projections of the of this method, let’s take an AV cut in an intersection ends intersect with the connecting arbitrary situation and take a creative approach lines passing through A1 'and V1' and the arcs of to determining its true length, Figure 4. Some radius RXA 'and RXV' passing through the A and sources have shown that this problem can be B ends of the section in space to form the frontal solved using the auxiliary right-angled projections A1 "and V1". In the plane V, the projection method. Since this method is rare in projections of these arcs are straight lines A "A1" most publications published in recent years, we and V "V1" parallel to s ''. will consider it in a clear picture and drawing 2. Since the plane R is a horizontal [3,3,4]. projection, for indirect projection-transformation of the AV section is carried out with R and V and This problem is solved by the following using the bisector plane (Qbis) of the planes R1, algorithm based on the content and essence of which are expected to be placed on V as follows: this method: 2.1. The trace of the bisector plane is 1. According to the property of parallel drawn by dividing QbisN by R and the angle projections, if the AV cross-section is projected between the horizontal traces of the planes R1 and onto a plane parallel to it, it is described in this V, which are expected to be located. In the plane at its true length. Based on this, the new drawing, the trace of this plane is the line QbisN, projection direction s (s ', s ") perpendicular to it is the bisector line dividing the angle between Rn determined by passing the plane R in the and Vn and the horizontal R1n of R1, which is projection position through the AV section (or expected to lie on top of it (Vn≡R1n). parallel to it). For example, if the R plane passes 3. To place all points of the plane R through the AV section, A'V' ≡ RN and s' ⊥ RN; s and itself in the plane V as R1, a new projection "⊥ will be RV, Fig. 5. The thought R forms a two- light direction perpendicular to the plane R is sided angle with the edge RV, with the plane V projected along the s (s', s' ') to the bisector plane and the position itself being V1 to R1. of its points, and then to the plane V in the If this R plane is joined to the V plane direction of the returning light. using the rotation method, then all the lying This can be seen in the example of point A geometric figures, especially the AV cross-section in Figure 5. However, in the method of rotation of and the bounded scale itself, will be depicted in the point A, it was noted in detail above that the real V1 plane. To do this, the plane R is rotated position A1 "is determined by rotating it around around its frontal trace (as the axis of rotation) and Rv and joining it with V. A new projection of merged with the plane V. As a result, the actual point A in the direction of light s (s', s' ') at an length of the AV section lying in it is described in angle d to the bisector plane, then The projection A1'V1 ': A1'V1' ≡ AV. Because its ends A1 'and returned from point D 'is projected onto the plane V1' are placed on V using arcs of radius RXA 'and V using light directions. RXV'. Then the AV cross section comes to the 5712 www.psychologyandeducation.net PSYCHOLOGY AND EDUCATION (2021) 58(2): 5710-5718 ISSN: 00333077 Figure 5 In this case, the direction of the projection the return light from it at an equal angle: ⊥ A1 light returning to D'A1 'is parallel to the main "V1"; projection direction in the drawing, since the plane 3.