Virtual Reality in Stereometry Training

Abstract

A new stereoscopic system is presented in this chapter for training in stereometry. Virtual reality systems are two types: immersion and non-immersion. An example of a environments with immersion is the desktop computer in which the virtual world is displayed on a special stereoscopic monitor through 3D glasses. The HMD (Head Mounted Display) virtual helmet is an example of an immersion system. The new stereoscopic system can also implement both types of 3D technology. At the same time, the software can generate objects in the .obj file extension to print to a 3D printer or add to a virtual reality device. The 3D technology is one of the fastest-growing. Over the last decade, it has found application in almost all spheres of society, including education. In learning stereometry, a problem may arise in drawing the geometric figures, unlike planimetry, where the drawing 100% coincides with its original. The problem can be solved with the help of specialized 3D software.

1. Introduction

This chapter focuses on the application of 3D technology in teaching the discipline of stereometry. It is known that stereometry is a branch of Euclidean geometry, which mainly studies geometric figures in three-dimensional space. This course examines the properties, and determines the volumes and faces of the surfaces of geometric figures, such as cylinders, cones, truncated cone, sphere, prism and others. According to teachers and experts in mathematics, when studying this discipline, it is of particular importance for students to have a spatial imagination that allows the understanding of the studied geometric figures in three-dimensional space, i.e., in three dimensions: x, y, and z. What distinguishes 3D (volumetric) from 2D (flat) representation is that more information is acquired about geometric figures in three-dimensional space. Spatial thinking is an activity with the help of which it is possible to solve several practical and theoretical problems related to learning. With the use of 3D technology, the student can become part of the learning process, allowing him to actively participate in the study of geometric figures and discuss and discover new dependencies related to the studied material. The evolution of technology and the need to modernize teaching has led to the creation of specialized software applications to assist students in learning the discipline of stereometry. With the help of 3D simulations and information technologies, such as virtual reality (Virtual reality-VR) and augmented reality (Augmented reality-AR), students’ cognitive activities are developed, as well as their interest in geometry (stereometry) increases [1, 2]. 3D modeling software applications, including VR and AR, help visualize complex and abstract shapes that are difficult for students who lack spatial imagination to understand. In the teaching process of mathematics (stereometry), the main goals of the teacher are: To demonstrate the possibilities of modern technologies by using 3D specialized systems.

• To familiarize students with how to work with 3D systems in modeling (generating) and processing geometric objects.

• To show the possibilities of VR/AR in solving a given geometric problem.

In the learning process, a problem may arise when drawing a stereometric figure, unlike planimetry, where the figures are 2D, and usually the drawing almost 100% matches its original. In the three-dimensional space, the figure transferred to a blackboard or white sheet does not match its real image, from where the problem that the teacher faces arises, namely that it is impossible to recreate the spatial figure completely. This problem can be solved using specialized 3D software to generate the geometric object (prism, pyramid, cube, etc.) and their corresponding sections and projections. Software with a suitable user interface can help the student “immerse” into virtual reality and absorb new knowledge of the studied discipline more easily.

The purpose is to perform an analysis of virtual reality systems and to present a new stereo system for stereometry training. The new system can be implemented by devices for VR with and without immersion.

2. Virtual reality

Virtual reality is a part of computer graphics. Today, it is possible for an ordinary user to become a part of VR through a suitable software product. The beginning of this field was given through computer games, which allow a person to experience things that are impossible to experience in real life [3]. Appropriate stereometry software will enable the student to enter this new research environment and interact with it instead of just looking at pictures on a monitor. The technology that enables such an experience is precisely virtual reality. In the current decade, it has become the most popular. VR primarily offers visual experiences on a computer screen or stereoscopic display. At the same time, VR can include additional multimedia, such as audio through headphones or a speaker and text. Interactivity in this virtual reality can be accomplished through the use of a keyboard, mouse, joystick or globe glove. The most used terms in the computer community are virtual reality, virtual environment, virtual world, artificial reality, etc. [3]. These real-time interactive graphics offers the user immersion in a three-dimensional world. Virtual reality systems can be grouped according to the level of immersion they offer the user. Mainly on immersion and non-immersion systems.

1. Non-immersive systems (desktop VR): Here, the immersion is relatively inferior than in other systems. It does not require special devices. Its other name is Window on World (WoW) [3]. The user can view the virtual environment through one or more computer screens. At the same time, he is allowed to interact but is not immersed in it. These systems are relatively cheap and therefore are the most used. The effect they offer is satisfactory.

2. Semi-immersive (Fish Tunk VR)—this is an improved version of Desktop VR. Like HMDs, these systems support head tracking, which improves the sensation thanks to the motion parallax effect. They require the use of a monitor. But they do not use the sensors of the immersion systems.

3. Immersive systems—This is the best version of VR systems. These systems allow users to immerse themselves in the virtual environment fully. Such systems are HMD and CAVE. Unfortunately, however, they require a lot of equipment and technique and are therefore not widely used [3].

Main characteristics of immersion systems:

• 3D space navigation interface is provided, such as view, tour, interior view.

• Enables stereoscopic viewing, i.e. improves the perception of the sense of depth in space

• Realistic interaction is offered that allows manipulation

3. VR systems

VR can be enhanced by hearing technology. VR devices can be divided into three categories [4]:

• Desktop PC based

• VR all in one

• Mobile based

One of the most used virtual reality systems that offer the most excellent immersion effect is the virtual helmet. Compared to other systems with an immersive effect, they are relatively cheap; the CAVE system can be given as an example of an expensive one. The HMD virtual helmet provides a truly immersive VR experience through two head-mounted displays, while desktop-based helmets require the helmet to be connected to a powerful gaming PC. Each eye has its own display. Unfortunately, however, this category of virtual reality device is not widely used because it is quite expensive. An all-in-one is a mid-range option; these devices do not require a computer connection. The low-level mobile VR product relies on VR Head Set. The main advantage is that it has a relatively low price and is convenient to work with. Today, many schools in Europe and worldwide are already improving the educational process through innovative modern methods based on new technologies. VR is one of the most advanced technologies, and it is normal for it to be used in primary and secondary schools today. It allows students to grasp complex concepts and definitions more easily. The main goal is to support student-centered teaching. There are documented results from using VR in secondary school mathematics learning, with increased positive attitudes and interest in mathematics reported after using VR [5]. Another study concluded that students and teachers are open to a form of learning through VR and AR (Augmented Reality) [5]. In this way, students are helped to understand better geometric shapes that are difficult to represent. Virtual reality enables the user to use and interact with the virtual world similar to the real one [6]. VR finds application in many fields, such as flight simulation, motion simulation in physics and chemistry, etc. The output channels of virtual reality correspond to the human senses: vision and hearing. Vision is the most basic sense used. Stereoscopic vision is a primary human mechanism for depth perception. The techniques for stereoscopic vision are HMD, shutter, passive, and others. The main steps in implementing Virtual and Augmented Reality are.

1. Recall of facts and concepts

2. Understanding the new

3. Application of learning

4. Analysis: the relationship between concepts

5. Justification

6. Creativity

Let us summarize that the 3D scene can be observed through a virtual reality viewer, such as an HMD or a normal desktop monitor. The user can move around the virtual environment using a mouse and keyboard. In this scene, 3D geometric figures are visible, allowing us to understand mathematical functions that are generally difficult to draw.

4. Stereo technologies

Anaglyph technology is achieved by two identical, differently filtered images, respectively for each of the eyes [7]. The two images are placed on the screen with a slight offset from each other. No specific equipment is required here, just a pair of anaglyph glasses. This technology is most commonly seen in magazines and in the past when showing movies in movie theaters. 3D polarizing technology uses two images: for the left, and for right eye, they are superimposed on the screen before they pass through polarizing filters. Polarization is 45 for one eye and 135 for the other. There are two filters here, one for each eye [7]. The necessary equipment is two projectors aimed at the same screen; a polarizing filter is placed in front of each projector, allowing light to pass through. Shutter3D technology—with it, the frame alternates sequentially, first for one eye and then for the other eye; the frequency is 120 Hz by 60 Hz for each eye [7]. For this purpose, active shutter glasses are used, which darkens the glass of the eye, which should not receive information. 3D Display technology improves the perception of the 3D scene, from which the applications become more qualitative. Different 3D technologies are suitable for different applications, so it is important to know them thoroughly [4, 8]. Generally, these technologies are classified into two categories: stereoscopic and real 3D. The stereoscopic technique is based on binocular parallax by presenting separate images [4] to each eye. Motion parallax can also be simulated by adding a head tracking system. Unfortunately, not everyone can perceive 3D information through 3D Display technology, as (2–3% of the population are stereo blind) [4]. In turn, stereoscopic displays are divided into two subclasses: stereoscopic and autostereoscopic. With autostereoscopic displays, the user does not need to have special 3D glasses. Stereoscopic displays are based on blocking each eye from seeing the image corresponding to the other eye. This is achieved through glasses using various Display technologies. They are mainly used for simulation and training. Many of these 3D scenarios are impossible or too expensive to simulate. Because most of them are too risky to simulate or too expensive. To be as realistic as possible, it is necessary to use immersive 3D systems. Low-cost 3D stereo displays are used in middle school student learning. The aim is to increase the level of understanding on the part of the students.

5. Modern mathematics approaches for students

It is known that the main concepts related to the assimilation of knowledge in the discipline of stereometry are: axioms, definitions, theorems and their proofs, and, most importantly, the solution of tasks [9]. The role of the functions is as follows:

• To prepare.

• To acquire new knowledge.

• For control and evaluation.

Today, with the development of information technology, it can be said that the learning process is a function of objects: student, mathematical knowledge, teacher, and “virtual reality”, with the teacher managing the learning process with the help of a computer. It is expected that with the developed StereoMV stereoscopic system, the efficiency of the learning process will increase. The material and applied tasks included in the system will be ordered by difficulty and complexity, starting from easier ones and ending with more complex tasks. Lessons can be solved both individually, and students can be divided into separate groups [9]. The course on the discipline of stereometry is a continuation of the course on planimetry. Very often, in the study of stereometry it is necessary to use various definitions and theorems of planimetry to clarify stereometric concepts, i.e., the solution of some stereometric problems is reduced to the solution of planimetry problems. The developed new stereo system makes it possible to build a spatial model of the studied object. At the same time, in the traditional way of learning, the drawings are mainly presented in a plane, which can make it difficult for the students and lead to confusion. The basic knowledge related to the study of stereometry in the middle course uses knowledge obtained already in the second school grade (8-year old students). This knowledge is related to planimetry and algebra, and students get to know them before the specific study of spatial objects. In the second grade, students study the types of triangles, measure the lengths of the sides of a triangle, square, and rectangle with a ruler, as well as study units of measurement (cm, dm, m). The teaching material of the third grade is related to the introduction of basic geometric knowledge, as students learn to use drawing tools (ruler and protractor), study-specific definitions, and the concept of an angle (straight, acute, and obtuse). Students learn to use the square grid, which is a propaedeutic of the Cartesian coordinate system introduced in the upper course. By measuring, they learn to distinguish which shape is a square and which is a rectangle. The fourth-grade learning material is all about learning and drawing a square and a rectangle on the square grid. The concept of the area of a geometric figure and the units of area measurement are introduced. After the fourth grade, the area of the studied figures from geometry is expanded, the properties of the figures are examined, and conclusions are formulated based on experience and observation. A habit of deductive thinking is formulated. The new geometric concepts of students at this age are: straight parallelepiped, cube, vertex, edge, wall, etc. The formula for volume and units of volume measurement are derived experimentally. One of the essential concepts in planimetry, which is the basis of stereometry in the study of a regular pyramid, is similar triangles and similar polygons. The system provides an opportunity to form initial knowledge of stereometry, implementing the material laid out in the textbook for the fifth grade. In connection with the teaching of geometry in IX–X grades, the problem of improving the methodology to develop spatial thinking takes on great importance. The development of this type of thinking in students is carried out through a unique system of tasks that meet the regularity of forming logical thinking. In the tenth-grade geometry course, students use cabinet projection and ways to depict polyhedra such as prisms and pyramids. Among the stereometric tasks of particular importance are the tasks of constructing a section of a polyhedron with a plane. In reality, these sections represent a polygon, which students are familiar with from the sixth-grade course and it is an essential element of the stereo system. The necessary knowledge related to areas are regular/irregular polygon and polyhedra. In the developed module of the system, there is a partial presentation of the material related to the above course on introductory sections of geometric bodies. In studying the topics associated with the cylinder and cone, the questions of depicting a circle and regular inscribed and circumscribed polygons are considered. The necessary propaedeutics for an upper course in stereometry are carried out through the course from I–III and IV–V classes, respectively. Up to this point, the measurement of sections in the planimetry course has been approached inductively. In this way, one goes outside the confines of the classroom by measuring objects with a meter. It is shown that there exists a mutual, unambiguous and reversible correspondence between the objects and the set of positive numbers [9]. The upper course students have the knowledge obtained from the elementary and intermediate courses about the volumes of geometric figures. and this knowledge should be expanded, ie. raised to a higher level. In Bulgarian schools, similar triangles and the signs of their similarity are first examined, then similar polygons are introduced. As it is known, two triangles are similar when there is a correspondence between their vertices, i.e., their angles are equal, and their sides are proportional. The Similar Polygon Theorem states, “If two polygons are divided by their diagonals into similar triangles, then they are also similar.” Here the theorem of dependence between the perimeters and their corresponding sides is introduced. The formula for the volume of a pyramid can be obtained by the method of indivisibles of the Italian mathematician Cavalieri [9]. The pyramid’s height is divided into a number of proportional parts, after which sections parallel to the base are drawn. There is a dependence between the area of the base and its parallel section. A polyhedron’s section with a plane, is a polygon with sides intersecting the walls of the polyhedron with a plane.

6. Advantages of 3D software systems

In the theory of mathematical education, which deals with spatial and geometric imagination development, several experts and researchers are united in their opinion that this type of imagination is poorly developed in some students [10]. To help these students in the process of learning mathematics, in particular in the discipline of stereometry, there is a need to create software applications. These applications should enable teachers and students to visualize studied geometric figures, explore geometric relationships and concepts, and make and test assumptions in a dynamic learning environment by manipulating the studied objects, such as: constructing, dragging, rotating, and others, to train to be more effective, more accessible and enjoyable. The classical method of teaching with chalk, ruler, and compass ensures the understanding of geometric drawing in stereometry, the construction being done in the imagination but in a flat structure. To achieve better results in training in this discipline, some software applications have been created in recent years, such as CABRI 3D, GEOGEBRA, DALEST and others, the use of which opens up new opportunities for the educational process.

7. Methods and results

The stereoscopic system generates the following two types of objects studied in the discipline of stereometry:

• 3D edged geometric figures: polygons (regular), cube, parallelepiped, prism and pyramid;

• Rotational geometric bodies: cylinder, cone, sphere and torus.

There are two main techniques involved in generating a regular polygon in the stereoscopic system; the first is the traditional one that converts a 2D polygon into a 3D object. This is done by extruding a two-dimensional graphic into a three dimensional one. The extrusion technique creates a three-dimensional object from a two-dimensional that moves along a set trajectory. The trajectory can be a rotational or translational motion or a trajectory defined by an arbitrary curve. In the research on the generation of a quadrangular pyramid, a new boundary method is used, which is based on Cavalieri’s Indivisible method and Newton’s boundary method [11, 12]. The idea is as follows—the geometric figure must be divided into sections parallel to a given ruler. This approach can be applied to any regular polygon. Next, determine the relations of the parallel sections. In the case of four vertices, i.e., a square (regular polygon), the ratio of the segments is 1:1. Polygon vertices can grow without limit. Regular polygons generated in this way differ from traditional programming polygons because they do not need to be converted from a 2D to a 3D object by extruding. They are “mathematically more accurate” and more flexible than traditional ones. They have many advantages, the biggest one being that virtual reality devices can visualize them. At the same time, they can be exported to files with the extension .obj. It follows that the polygon is placed in the center of the coordinate axis and its vertices are calculated by the length of its side by the parameter a. To parametrically define a regular quadrangular pyramid, the values for each vertex of the three-dimensional space should be defined. The parameter “a” is the length of the side of the polygon that is the base of the pyramid. The parameter h is the height of the pyramid. When the value of vertex No. 5 is a number other than zero on the abscissa or ordinate, the final result will be a tilted pyramid. The system also enables the generation of a truncated pyramid. The methods for 3D geometric modeling and their schematic representation is done in two ways: edge and boundary. The edge representation (Figure 1) requires the following information:

• The object’s shape is represented by its edges (E1, E2,…).

  1. Vertices metric information (A, B, C,…).

  2. Partial topological information (X1, Y1, Z1,…).

• The boundary representation of a regular quadrilateral pyramid presented in (Figure 2) requires:

  1. List of all its walls (W1, W2,…)—W6 is the section of the pyramid

  2. The walls are described by the edges (E1, E2,…).

  3. Edges are represented by vertices (A, B, C,…).

  4. Vertices are represented by three-dimensional coordinates (X1, Y1, Z1,…).

Figure 1.

Edge representation of the base of the pyramid.

Figure 2.

Limit representation of a pyramid.

For the parametric setting of a regular quadrangular pyramid (Table 1), values for the coordinates of each vertex of the three-dimensional space must be set. The pyramid’s height is set with parameter h, and the length of the base with parameter a. When vertex number five values for the X or Y coordinates is a non-zero number, the result will be a tilted pyramid. The first four vertices form the base of the pyramid, and the fifth is the top of the pyramid. The peaks are crawling in the same direction in the counterclockwise case, starting from the IV quadrant (Table 2).

No. vertex	X	Y	Z	No. face	vertex
1	a/2	−a/2	0	1	1,2,3,4
2	a/2	a/2	0	2	1,2,5
3	−a/2	a/2	0	3	2,3,5
4	−a/2	−a/2	0	4	3,4,5
5	0	0	h	5	4,1,5

Table 1.

The parametric values of a pyramid.

No. vertex	X	Y	Z	No. face	vertex
1	1.5	−1.5	0	1	1,2,3,4
2	1.5	1.5	0	2	1,2,5
3	−1.5	1.5	0	3	2,3,5
4	−1.5	−1.5	0	4	3,4,5
5	0	0	4	5	4,1,5

Table 2.

Precise pyramid values.

8. Regular/tilted/truncated pyramid

The algorithm for building a regular/ tilted /truncated pyramid (Figures 3 and 4) is as follows:

1. From the StereoMV object panel, select the pyramid shape. The base type is chosen, which is a regular polygon with 3–6 vertices.

2. The user sets the length of the main edge.

3. The system builds a polyhedra along the third axis by setting a height value.

4. The value entered by the user for the abscissa/z coordinate is checked. If the value entered is a non-zero number, the final result will be a tilted pyramid; otherwise, a straight pyramid is generated.

5. The system also offers an additional option: generating a truncated pyramid. When selecting this option, the user must enter values for both heights.

Figure 3.

Straight a regular pyramid.

Figure 4.

Truncated pyramid.

The traditional way to generate a regular polygon is through trigonometry. It is characterized by the number of vertices and radius—its length. To generate a cone, the traditional method (Figures 5 and 6) is used. The following are set:

1. The base—a polygon with 100 vertices

2. The radius of the base is set.

3. The center of the circle rises along the third axis by extrude.

4. If the number on the abscissa or z coordinate is a non-zero number, the result will be an inclined cone. Figure 5 shows the image of a straight cone and its dimensions. If the values for x and z are 0, then this Cone is straight. Figure 6 shows the figure of a cone that is tilted to the left, and in its dimensions, as the value for x is −6 and z = 0. When the value of x is +6 and z = 0, the Cone will be inclined to the right.

Figure 5.

A straight cone.

Figure 6.

Inclined cone.

The traditional method of generating a regular polygon is based on trigonometry and is characterized by the number of vertices and radius. To construct a prism and a pyramid through this method, the 3D graphics modeling technique is used—extrusion, which is a big drawback. The new authoring method for generating polygons is characterized by the number of vertices and the side length of a regular polygon. This method gives a more accurate result than the traditional one because it does not need to be extruded. It uses only the ratio of parallel segments, not trigonometry, which differs from the traditional method. The author’s method accurately results because it does not use trigonometry. A disadvantage of the new approach is that it requires a lot of computing power. Also, another limitation is that for now it is limited to 6 vertices. The new method can be applied to any regular polygon with unlimited vertices growth. The proof that the method works is that it allows visualization through a 3D virtual reality library and the ability to export to a .obj file.This paper uses the author’s new boundary method to generate a regular quadrilateral pyramid. Values are set for the 4 vertices and the length of the base, respectively. Next, set a value for the height. While for the geometric body cone, the traditional method is used. Defines the base by a number of vertices, radius, and extrude height.

Before starting the process of 3D printing or adding a 3D model to a virtual/augmented reality device, it is necessary to create a file (.obj) containing the necessary information about the geometric object’s spatial characteristics. This type of data is stored in text format and can be modified using a text editor (eg, notepad). These files contain the following information about the geometry of the object:

1. “v”—vertices of the geometric body;

2. “vn”—normal, which is a parameter for the light source;

3. “vt”—texture coordinate;

4. “f”—wall.

The stereoscopic system makes it possible to represent geometric objects in two ways—solid and wireframe mesh. The wireframe mesh representation will make it possible to understand the geometry of the three-dimensional model. While the solid gives a finished look to the geometric figure. These are its visual characteristics. It is preferable that before exporting the geometry object to a .obj file, it should be generated as a solid body. The Canvas3D class is a component of the AWT interface library and extends a two-dimensional object into a three-dimensional one by including the necessary 3D information. It represents the canvas where the three-dimensional objects are drawn. One of the modes for stereoscopic visualization is Mixed Immediate. The stereoscopic system uses this mode for stereo visualization through active glasses. The advantage of StereoMV is that it uses the Java3D library. And so it can be realized by a virtual reality device, such as:

1. Traditional Desktop

2. 3D active visualization—improved Desktop

3. 3D passive visualization—improved Desktop

4. HMD—an immersive system using a Desktop PC

5. CAVE—immersion system

In Figure 7 presented: traditional, anaglyph, and stereo visualization of a objects. After exporting the geometry object to a .obj file. It can be added to an augmented or virtual reality device. The article uses the 3D viewer program in Mixed Reality mode using animation (Figure 8). In (Figure 9) HMD mode, each eye needs to have its own display. For this purpose, two canvas3D objects for the left and right eye need to be added.

Figure 7.

Traditional, anaglyph, and active visualization.

Figure 8.

Mixed reality 3D viewer.

Figure 9.

Stereo cone—2 Canvas3D objects for HMD, left and right eye.

9. StereoMV

The author’s software StereoMV (Stereo Math Vision), is the result of dissertation work on the topic “Stereoscopic Training System”. It can be used to teach students about stereometry. It is aimed at the middle course of study. In the future, it will also include an upper course. The Java programming language and, more precisely, Java3D was used as a means of implementation. 3D active visualization of a non-immersive system was used at the beginning of the system development. After the thesis’s successful defense, the software’s development continued so that it could be implemented by an immersive system—HMD (virtual helmet). At the moment, the software also includes another direction, which is in the field of medicine. The goal is to create stressful situations through virtual reality that will take part in heart rate analysis by making recordings using a Holter device.

10. Conclusions

The software StereoMV proposed in this chapter can be used in teaching stereometry. It will be able to help develop spatial thinking in students who do not have it. VR systems are tracked, both with and without immersion. The stereo system can work with any VR device thanks to its Java3D library. The use of virtual and augmented reality in teaching mathematics, and in particular stereometry, is not new. There are documented results of the benefits of 3D technology in mathematics education. The conclusion is that VR helps students discover and track mathematical dependencies. At the same time, teachers are also open to this kind of training. The system’s main advantage is that for the generation of 3D objects, a new author’s boundary method is used with high accuracy compared to the traditional one which is using trigonometry. With this method, extruding the 2D polygons into 3D objects is unnecessary. This chapter uses the technique to generate a square (a polygon with four vertices). But the same rules can be applied to any regular polygon. The number of vertices can grow indefinitely. The chapter also presents modern educational approaches to teaching stereometry through the new author system.