About the application of the R-function for geometric modeling of 3D objects of complex shapes in a virtual educational environment

Nuraliev Fakhriddin Murodullaevich ORCID: 0000-0002-0574-9278 100084, Uzbekistan, Mirzo Ulugbek oblast', g. Tashkent, ul. Amira Temura, 108 nuraliev2001@mail.ru Morozov Mikhail Nikolaevich ORCID: 0000-0002-8591-3700 PhD in Technical Science 424006, Russia, Respubliki Marii El oblast', g. Ioshkar-Ola, ul. Pl. Lenina, dom 3 mikhail.n.morozov@gmail.com Giyosov Ulugbek Eshpulatovich ORCID: 0000-0002-9419-5392 100084, Uzbekistan, Samarkand oblast', g. Tashkent, ul. Amir Temur, 108, of. Tashkentskogo universiteta informatsionnykh tekhnologii imeni Mukhammada al-Khorazmii bek99989@gmail.com

Yorkulov Jonibek 100084, Uzbekistan, respublika Uzbekiston, g. Tashkent, ul. Amira Temura, 108 pythondeveloper0727@gmail.com

I. Introduction

Currently, within the framework of the virtual environment platform, it is necessary to develop algorithms for influencing 3D objects without loss of quality and to simplify the number of polygons. In this article, the learning process is considered as a complex (psychological, physiological and pedagogical) object with an emphasis on building virtual computer models.

The main goal is to create three-dimensional models of the interior, exterior and characters for the virtual three-dimensional educational environment of the university. In addition, the creation of geometric models, technological schemes of equipment, technological equipment, molding, processing and measuring tools based on 3D modeling, including parametric modeling for standard products, automated production of sets of design and technological documentation for 3D models. various products based on database technology in a single information space ^[1-2].

Polygonal modeling and spline modeling are the two most commonly used approaches when creating 3D objects. Both options allow you to create high-quality three-dimensional models. All geometric characteristics of the proposed trigonometric curves of B-splines are similar to classical B-splines, but the ability to adjust the shape is an additional quality that is not characteristic of classical curves of B-splines. The properties of these bases are described earlier and are similar to the classical basis of B-splines. In addition, a uniform and inhomogeneous rational basis of B-splines is also presented. The continuities C ³ and C ⁵ for the trigonometric basis of B-splines and the continuity C ³ for the rational basis are obtained. In order to legitimize our proposed scheme for both basic and periodic curves are constructed. 2D and 3D models are also constructed using the proposed curves ^[2].

Polygonal modeling is most likely the most commonly used form of three-dimensional modeling, which is often found in the animation, film and games industries. Currently, the use of polygonal models is effective in constructing models of smooth surfaces. When constructing internal and external objects, spline functions are considered effective. The use of each method implies compliance with the principle of topology. Usually two types of topology in different variations play an important role. In general, when designing simple parts of simple shapes, it is also advisable to use a triangular topology. Based on the above, scientists have come to the conclusion that the use of the mathematical apparatus of R-functions can significantly simplify the process of describing topological models of geometric areas of almost any complexity. ^[3,4,5]

The R-function method is popular as CSG (Constructive Solid Geometry). All primitives (cube, sphere, cylinder, etc.) are defined by formulas. You can construct any objects as a composition of several primitives by combining (the logical function of the set: and, or, no) the corresponding formulas are as shown below. There is free software for creating 3D CAD objects using CSG. Polygonal models (sometimes called surface models) are effective in 3D CG. To create polygonal models, designers must use special 3D software such as Blender. There are several file formats for polygonal models: Wavefront * .obj, Stanford * .ply, etc. To create educational materials on XR (VR/AR/ MR), successfully used Three.js . Three.js is a JavaScript library for 3D graphics for 3D web content. The laboratory presented by us uses Three.js for creating educational materials on 3D-CG and VR.

The proposed approach to using the R-function for geometric modeling of flat sections, unlike other existing methods and algorithms, makes it possible to describe a geometric model of flat sections of arbitrary shape with their subsequent triangulation quite simply and efficiently. An approach to obtaining discrete models corresponding to implicit analytical models based on the theory of R-functions is developed. An approach to the construction of inhomogeneous discrete models based on the developed universal template is proposed ^[6,7,8].

3D models are created using the Virtual Reality Modeling Language (VRML) in the web interface. A geometric shape node based on common web interface functions has been defined for VRML. Integration between the models defined by functions and VRML is proposed to be implemented using the VRML browser plug-in, where a custom node can be called a regular VRML node together with other traditional VRML nodes. Currently, the Blaxxun Contact3D VRML browser has been extended to support this integration. Other VRML browsers are expected to expand ^[9-13].

Currently, the most popular virtual environments in the world are operating, specializing in various fields. For example: vAcademia, Second Life, Virbela, IMVU, Сlassvr.com , Sansar.com .

Three-dimensional distributed multi-user virtual reality systems are especially relevant today. They allow organizing meetings, seminars, conferences, symposiums (see Fig. 1-2) and are used by international corporations.

II. PROBLEM STATEMENT.

It includes the creation of design and geometric models of three-dimensional objects for national virtual university environments, in particular exteriors, interiors, characters (avatars), as well as the development of their computer algorithms. The virtual map shows the design scheme of 3D models of each educational institution, which allows you to move to another educational institution from one place to another in time and space using a teleportation object.

III. METHOD AND FUNCTIONAL MATERIAL PRODUCTION IN THE BASIC THEORY OF R-FUNCTION

Computer geometric modeling for determining and approving the use of an elementary function of three variables of an arbitrary constructive solid as f(x,y,z) and its surface as a null set f(x,y,z)=0 (the so-called implicit surface) was independently expressed by Rvachev ^{[8, 9, 10]} and Rissi ^[11]. Both authors introduced analytical expressions to represent set-theoretic operations. Ricci proposed using ^C1 discontinuous min/max operations for precise descriptions, as well as approximate descriptions to obtain smooth mixing properties of the obtained surfaces. Rvachev's work ^proposed a much more general approach called the theory of R-functions and introduced continuous functions to accurately describe set-theoretic operations. We give the details in the corresponding section. We can say that a three-dimensional field is one of the methods that allow you to write analytical equations of geometry V.L. Rvachev's RFM method. Here are the basic concepts of the RFM method. The R-function is a numeric function with a real variable whose sign is the interval between the axes of numbers until they are completedand   it is determined by the argument labels in the corresponding sections.^[12]

Its arguments are such that it is followedby the logical function sign(z)=F(sign(x),sign(y)), the numeric function z=z(x,y) is called the R-function. Each R-function corresponds to one function of the driven logic. The set of R-functions is closed in the sense that the R-functions overlap. If the set of all overlapping elements H has a nonempty intersection with each branch of the set of R-functions, then the system of R-functions H is called sufficiently complete. ^[11].

The most commonly used complete system with an R-function (-1<alpha<=1 ) R _alpha 

 R ₀ in there are systems:

  ,

in R ₁ there are systems:

In the final state of the R-functions, conjunctions and disjunctions correspond to:

Using the R-function, it is possible to construct an implicit form of boundary equations of regions constructed from some equations of simple regions.^[6]

R-functions can be regarded as an infinitely valuable logical tool. R-functions are used in solving a wide range of problems in mathematics, physics, multidimensional digital signal and image processing, computer graphics and other fields.

IV. THE RESULT OF THE CALCULATED EXPERIMENT

Methods of constructing field geometry equations (i.e. normalized equations) provide a good technological basis for automating the process of organizing these equations. In fact, only the process of constructing predicate equations should be automated, the transition from these equations to simple elementary equations of the geometry of the field is carried out by replacing the symbols of the logical function with the corresponding symbols of the R-function, the symbols of the field are not equal to their corresponding other parts.

Polygon reduction algorithms are not the only way to create a model with fewer faces. Artists will always be able to present a model better using fewer polygons than any reduction algorithm. Polygonal simplification methods offer one solution for developers working with complex models. These methods simplify the polygonal geometry of small, remote, or otherwise insignificant parts of the model, aiming to reduce the cost of visualization without significant loss of the visual content of the scene. This is both a very modern and a very old idea in computer graphics. Back in 1976, James Clark described the advantages of presenting objects in a scene with multiple resolutions, in particular, flight simulators have long used manually created aircraft models with different resolutions to guarantee a constant frame rate. Recently, many studies have been conducted aimed at the automatic creation of such models. When planning to use polygonal simplification to speed up your 3D application, this article will allow you to make a choice among the many published algorithms ^[8].

Figure 3. Managing the complexity of the model by changing the level of detail used to visualize small or remote objects. Polygonal simplification allows you to create multiple levels of detail similar to these.

Figure 4. Marking up 3D objects using the treecube method.

A filling algorithm for matching each node to a cell of a suitable size according to its weight value. as a 3D file system extension from a tree map.

        The Slice and Dice algorithm

node level: 3i - on the z axis

node level: 3i + 1 - along the abscissa axis

node level: 3i + 2 - on the y axis

strictly following the order in the average aspect ratio of the layout: tend to be large.

    Ordered treecube algorithm

1. Select a specific node called "pivot point" from the list.

2. Distribution of the remaining nodes: L1, L2-1, L2-2 and L3 (placed in "V1", "V2-1", "V2-2" and "V3" respectively)

3. Recursive layout ? L1 ? V1, L2-1 ? V2-1, L2-2 ? V2-2, L3 ? V3

while maintaining the order of approximately the average aspect ratio: tend to unity. ^[8]

 During this work, i.e. the creation of a national virtual university platform, geometric modeling of the design of exteriors, interiors and characters in the field of information technology based on 3D technologies is provided. We know that visualization uses geometric splines and polygonal mesh construction methods. The project offers new methods of geometric modeling of 3D objects: the F-Rep function for the representation of multidimensional geometric shapes, the method of constructive logical-algebraic R-function (RFM). This method allows you to depict 3D objects of high complexity.

The total preference score (S) of option I is the sum of the average scores of all desktop applications for creating 3D models for each criterion.

  W _j= value for criterion j

S _ij= score for option i according to criterion j

n = number of criteria to be considered

         Si_{3D Max}=(100 * 0,2) + (25* 0,4) + (100 * 0,3) + (100*0,1) = 70.1

Si_Blender =(50 * 0,2) + (75* 0,4) + (75 * 0,3) + (50*0,1) = 67.5

 Si_Maya=(0* 0,2) + (100* 0,4) + (0* 0,3) + (0*0,1) = 40

 {Table1} Calculate the total preference scores

     Based on the methodology, materials and feedback from experimental groups, traditional education lessons are now taught by professors, and by the end of the lesson 20 out of 30 students master the subject. It was found that 27 out of 30 students would master if they were taught by professors through a virtual three-dimensional learning environment. As a result, it will increase the skill level by 40%.

Now we will consider the sequence of the design process of a given historical object on the territory of Uzbekistan using the R-function method.

(1) here in this sequence the cube is given in the coordinate system x,y,z. In this case, the origin of the coordinate point will be (0,0,0). In order to provide illumination of the internal structure of the object, another cube is drawn inside the cube. This process can be represented using formula (2).

    (2) in the formula, we take the difference of the second cube inside the first one with accuracy n and subtract the values of each side a, b, c. At the same time, we enter a designation equal to a=b=C. n is the thickness of the wall.

  (3) This formula (1) also represents the opening of a door inside a cube. Here we introduce the expression m=a-n.

  (4) Write down the following logical expression and form a general view of the cube.

    (5) we draw a ball inside the cube. (6) we form a ball from the created ball at the x coordinate by raising a by one. (7) according to this formula, a general view is created based on the combination of the ball and the general dome. ^[9]

As a conclusion, it is worth noting that the creation of 3D objects using modern design programs based on the R-function takes less time and RAM.

a                                                             b                                                      c

Fig. 5. View of the historical object created using the R-function, a – view of the 3D model on the left with texture, b - color image in RGB mode, c – full–face view of the 3D model taking into account the provision of texture

 conclusion

 In virtual reality environments, each object is represented by a three-dimensional model. The main task is to create custom models that control them. A three-dimensional model of a character is represented by a depth map, dots, a polygonal model, a parametric model describing anthropometric, full-face and profile features of a human face. When designing an environment in the virtual world, the interests of the avatar come first. When testing and implementing the proposed virtual 3D environment in the activity, geometric models and 3D objects for the virtual environment were created, in particular, the design of the exterior, interior and characters, and their computer algorithms were developed using the example of creating a historical object using the R-function, as well as algorithms for reducing the volume of polygons and simplifying the number of polygons that function without loss of quality of 3D objects in a virtual software environment.

National historical sites are used to test the presented concept in real conditions. This article substantiates the use of three-dimensional objects for the virtual reality environment by modeling them and importing the created models into the virtual environment. Analysis of algorithms and methods for creating 3D models using the R-function has shown the effectiveness of developing models created in modern programs.