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Pseudo-holographic Displays as Teaching Tools in Mathematics
Monica Ciobanu
Department of Engineering and Computer Science
Vasile Goldis Western University of Arad,
Bulevardul Revoluției 94, Arad, Romania
Phone: 0257 211 884
cmmciobanu@gmail.com
Antoanela Naaji
Department of Engineering and Computer Science
Vasile Goldis Western University of Arad,
Bulevardul Revoluției 94, Arad, Romania
Phone: 0257 211 884
anaaji@uvvg.ro
Ioan Dascal
SC Antaco Romania SRL, Arad, Romania
Bd. Iuliu Maniu, Nr. 31-33, Sc. B Et. 2 Ap. 10, Arad, Jud. Arad
nelu_dascal1@yahoo.com
Ioan Virag
Automation and Computer Science Faculty
Politehnica University of Timisoara,
Piața Victoriei 2, Timișoara 300006, Romania
Phone: 0256 403 000
ioan.virag@aut.upt.ro
Abstract
Due to the fact that the abstract mathematical notions are often hard to grasp by students,
without linking them to a specific physical representation or correlate them with regular facts, our
team came up with the idea to create and implement a set of interactive teaching tool packages
(ITTPs) to help them overcome such issues. The ITTPs consist of theoretical modules and
applications in the framework of a Hungarian - Romanian cross-border project, developed with the
aid of existing open source software and technologies. The focus of the Romanian team was on
basic elements of linear algebra and analytic geometry, showcased using an auto-stereoscopic
display for the purpose of image visualization. This paper presents an overview of similar
technologies and the applications developed for representing pseudo-holographic images of
geometric objects, namely vectors, straight lines, planes and conics. One of the developed packages
was written using web technologies in order to make the applications available on mobile devices,
as well as to eliminate the need to install any additional software.
Keywords: Autostereoscopic display, pseudo-holographic images, teaching tools
1. Introduction
The trends in modern education lead to the development of online courses due to their benefits
especially in terms of flexibility, knowledge achievement and costs. Getting education by integrating
ICT in the learning process motivates learners and offers skills that sometimes lack resources in
traditional education. Also, the online environment is changing continuously representing a great
opportunity for learning, particularly for individuals who want to advance their career.
In the last years there has been an increasing development of the massive open online
courses (MOOC) (Provision Company webpage, 2016), materialized in the form of multiple
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platforms, offered to large numbers of students, using technologically enhanced resources and
activities, such as virtual laboratories, simulations and cognitive tutors.
Although many people still consider traditional education better in achieving knowledge,
online learning proves to be a great alternative for the study many fields and to boost the level of
self-motivation. In this way, students have the chance to study at their own pace and in their own
time.
In this context, considering the potential and the benefits of e-learning platforms, we created
and implemented some courses for mathematical disciplines, adapted to online education
(Ammeraal, Zhang, 2007).
The courses were developed as interactive teaching tools packages (ITTPs), containing both
theoretical modules and applications (Andrica, Topan, 2004; Atanasiu, Munteanu, Postolache,
2003). The topics related to ITTPs are: mathematical analysis, probability, numerical calculus,
computational algebra, linear algebra and analytical geometry. The corresponding applications were
made using Maxima or were created using Java (for Android), Java 3D and JavaScript. All of the
developed applications are open-source.
These results were obtained in the framework of a Hungarian - Romanian cross-border
project (see Acknowledgement).
The aim of our project was to facilitate the understanding of mathematical abstract notions
by combining theory with interactive applications. The ITTPs provide students with a new tool in
order to make the content more accessible and help them understand and learn mathematical notions
(Ciobanu, Naaji, Virag, Dascal, 2016).
For some applications, an autostereoscopic display was used for visualizing 3D objects as
pseudo-holographic images, as well as the outcome of parameter changes inside mathematical
formulas. For each application we developed a version which can be visualized on a classical 2D
display and another one adapted for the 3D pseudo-holographic device (Ciobanu, Ploscar, Dascal,
Virag, Naaji, 2015). This new approach (of using a 3D autostereoscopic display for education)
differentiates our tools from other existing contributions.
2. Technical Background and Related Work
The proper rendering of three-dimensional (3D) content cannot be achieved using traditional
two-dimensional (2D) displays. This lack of the ability to properly display cues related to the depth
of the rendered 3D objects makes it very hard for students to grasp the proper spatial representation
of the discussed subjects.
The cues needed to understand three-dimensional content displayed on a two-dimensional
display are emulated with the help of mathematical algorithms, but the resulting images are still
missing some essential pieces that could be properly represented only by using a 3D display device.
According to (Geng, 2013) there are four major physical (physiological) and five
psychological depth cues the human brain uses to gain a 3D sensation based on the viewed images.
The physical depth cues are:
Accommodation – the measurement of tension in the muscle that changes the focal length of
the eyes’ lenses in order to bring into focus a specific 3D object in the scene, allowing us to
perceive its 3D depth.
Convergence – the difference between the viewing directions of the eyes when looking at
the same spot in a 3D scene.
Motion parallax – nearby objects appear to move faster in a 3D scene than those that are far
away.
Binocular disparity – refers to the fact that our left and right eyes see slightly different
perspectives of the same 3D object. Thus it brings into focus objects at different distances.
Also according to (Geng, 2013) the human visual system can interpret depth by extracting
psychological cues from 2D monocular images. These psychological depth cues are:
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Linear perspective – the appearance that parallel lines converge at a distant vanishing point
on the horizon. This technique was often used by Renaissance painters to depict buildings
and scenes that appeared more realistic than those painted in the Middle Ages.
Occlusion – the ability of the human visual system to perceive an overlapped 3D object as
laying farther away than the object in front of it.
Shading – when objects are casting shadows upon another object in the 3D scene, this visual
cue helps us determine which object is closer to the viewer. It also helps determine the
spatial orientation of the object.
Texture – objects with smooth textures are usually interpreted as being farther away since
we cannot distinguish too many surface details.
Prior knowledge – 3D shapes and sizes of common geometrical structures can help the
viewer infer depth cues due to the fact that the brain already knows how they interact with
the surrounding light sources.
Depending on their construction principle, some of the actual 3D displays are able to
provide all these depth cues, while others provide only some of them.
The current 3D display technologies fall in two major categories: stereoscopic displays
(when special eyewear is used to induce the 3D depth sensation) and autostereoscopic displays
where different techniques provide a sense of 3D depth without the use of any kind of 3D glasses.
Autostereoscopic displays, according to (Geng, 2013), fall into three major construction
types: multiview 3D displays, volumetric 3D displays, and holographic displays, although the latest
are more commonly referred as pseudo-holographic displays since they do not use real holograms.
The multiview 3D displays usually provide a finite number of different perspectives to a
limited number of users by using parallax barriers (Gocheva, Rahnev, 2008; Iltina, Iltins, 2013), or
lenticular lenses (Iordan, 2009; Jeon, Park, Gwag, Lee, Kwon, 2014) that block the view of a
specific set of pixels for the other users. These displays usually provide a lower resolution because,
for a display with N views (perspectives), the actual resolution of each view is 1/N of the initial
display resolution.
Volumetric 3D displays use individual voxels that provide accurate volumetric information
regarding the rendered 3D objects. The major enhancement of these displays is that they provide a
360-degree view of the displayed model, and not only one single perspective for each viewer as the
multiview 3D displays. By their construction principle, the volumetric 3D displays fall into two
major categories: Static Screen Volumetric 3D Displays as in Figure 1 that have no moving parts
and Swept Screen Volumetric 3D Displays as in Figure 2 that usually involves a rotating screen and
relies on the persistence of vision.
Figure 1. 3D LED RGB Cube
source: https://www.youtube.com/watch?v=Pswc-hIvRrM
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Figure 2. Perspecta volumetric display
source: http://www.3dcgi.com/cooltech/displays/ActualityMilitary3D_sm.jpg
Pseudo-holographic displays are capable of displaying individual hogels (holographic
picture elements) (Kim, Yoon, Yoon, Ju, 2015). Their major advantage is that the displayed 3D
image seems to “float away” from the display providing the viewer with an experience close to an
actual hologram. Depending on their construction principle, these devices provide a single or
several different perspectives of the 3D object to multiple users at the same time.
During project development our team used a HoloVision pseudo-holographic display
manufactured by Provision (Lanman, Hirsch, Kim, Raskar, 2010). This autostereoscopic 3D display
allows several students to visualize the projected 3D models at the same time without wearing any
kind of special glasses.
The HoloVision device is capable of creating the illusion of a 30 cm image focused at
approximately 90 cm in front of the monitor, viewable at a 60-degree angle from the center.
The HoloVision device has an in-built minicomputer that allows the storing of 3D content
that later can be played (with a pre-installed media player) if the 3D display is used for commercial
purposes. Since our team created the teaching tools to be interactive, we could not use the device’s
player and connected our laptop to the HoloVision’s VGA port, using it in a similar fashion as a 3D
projector. The laptop our team used had a 17.3 inch display, an Intel Core i3 processor at 2.2 GHz,
4 GB DDR 3 memory and an integrated HD 3000 graphics processor that provided enough
processing resources to render the interactive 3D content.
3. The Applications Developed for Interactive Teaching Tools
In this paragraph we present some of the applications developed based on Java 3D API and
JavaScript (Lucente, 1994; Mustea, Naaji, Herman, 2014). They are related to topics from linear
algebra and analytical geometry (Osmani, 2014; Xie, Zhao, Yang, et al. 2012), namely vectors,
straight lines, planes, and conics. The software was developed in two versions: one for 2D monitors,
suitable for individual study, and the other one for autostereoscopic display. In the first version, the
text fields and tags are created to input data, positioned in the upper side of the monitor, with
buttons for calculations and 3D representation of the images. Likewise, results of calculations and
details regarding geometric objects are displayed in the lower side of the monitor (Figure 3).
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Figure 3. Interface developed for 2D monitor
In the second version for the autostereoscopic device (Figure 4) the area for data input
displayed at the beginning is hidden when 3D objects are represented, thus allowing emphasis of the
pseudo-holographic image. Two types of interfaces were needed because the 3D image cannot be
rendered in the same way on a 2D monitor as on the HoloVision device, where the 3D model keeps
its details even if it is scaled.
Figure 4. Interface developed for holographic display
3.1. The applications developed for representing vectors and elements of vector calculus
The Vector Spaces ITTP comprises vector spaces, Euclidean vector spaces, and free vectors,
emphasizing connections between linear algebra and analytical geometry. Isomorphism between a
vector space and a real space of the same dimension leads to a vector being expressed with the help
of its coordinates relative to a basis. The notions of scalar product of two vectors, vector norm, and
angle between two vectors are introduced, with the aim of constructing an orthonormal vector
system. Definitions have a general character, being valid in an n-dimensional space, but for the
applications to have an intuitive support, spaces of dimensions 2 and 3 are considered to represent
geometric objects. Bounded vectors (oriented segments), equivalence relation on the set of bounded
vectors and free vectors (equivalence classes) are defined and elements of vector calculation are
presented.
One of the applications created allows establishing whether a vector system forms a basis or
not, determining a vector’s coordinates in relation to a base.
Another application represents 3D vectors obtained by multiplication with a scalar, vector
addition and subtraction, vector product. In the version for individual study, on a 2D monitor
(Figure 5), data are introduced in text fields found in the upper side of the monitor. Tags indicate
3D coordinates of vector extremities. Data input for several vectors and their representation is made
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by successively clicking the Draw button. The Delete button allows deleting the last drawn vector,
thus making it possible to successively reduce the number of vectors. The Add button comprises a
submenu which allows selecting the vector addition method, i.e. parallelogram method or triangle
method. The Subtract button works in a similar way. The Multiply button allows calculating the
product between a vector and a scalar and the vector product. Displayed images can be rotated
around the Oy axis to better view the result of the performed operation. In the lower side of the
monitor there is a text area where the coordinates of the resulting vector are shown.
This application was created using Java 3D API.
Figure 5. Interface developed for visualizing vector calculus
3.2 The applications developed for representing straight lines and planes
The Straight lines and planes in 3-dimensional Euclidian space ITTP defines the theoretical
bases related to these topics and presents the application developed for visualizing them.
The vector equation of a line is given when one point on the line and the directing vector of
the line are known or when two points on the line are known. The vector equation of the plane is
given when one point of the plane and one basis are known or when the position vectors of three
points on the plane are known.
In order to give the Cartesian equations of planes and lines, the notion of general (affine)
Cartesian coordinate system is defined. The following are given: the equation of a plane passing
through a point and which is perpendicular to a vector; the vector equation of the plane when one
point and two of its direction vectors are known; the parametric equations of the plane; the equation
of the plane in the form of a determinant; the equation of the plane through cuts. The necessary and
sufficient conditions for two planes to be secant, parallel or to coincide are also presented.
The vector equation of a line is re-written in the form of parametric equations of the line
when one point on the line and the direction vector are known, i.e. Cartesian or canonic equations.
The canonic equation of a line is also given when we know the coordinates of two points on the
line.
One of the developed applications allows generating the image of straight lines and planes.
As stated above, two versions of the application were created, the one for individual study
being shown in Figure 6. The application has tags and text fields in the upper side for inputting data
for two planes and two lines. By using buttons, 2 planes or 2 lines or one line and one plane can be
selected for viewing. If the two planes or the two lines are parallel or if the line is parallel to the
plane, the application displays this information. If the objects intersect each other, the application
displays the angle between them in radians and the coordinates of the point or line of intersection.
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Figure 6. Interface developed for visualizing straight lines and planes
The notion of common perpendicular to two non-coplanar lines is defined in order to give
the formula for distance between two non-coplanar lines.
The ITTP was thought out by the authors so as the students would assimilate the formulas
presented as easily as possible.
The version for the autostereoscopic device does not display the results of calculations, but
creates a pseudo-holographic image for perceiving the presented notions.
The applications were created using Java 3D API.
3.3 The applications developed for representing conics
The Hyperquadrics ITTP is structured into three parts, namely: sphere and circle, conics,
and quadrics. With regard to sphere, the definition of a sphere and the implicit Cartesian equation is
presented, as well as the general Cartesian equation and parametric equations of the sphere. A
classification of points with respect to a sphere, position of a sphere to a straight line, a plane or
another sphere is also presented. The circle is defined in 3-dimensional space as the intersection
between a sphere and a plane, as well as between two spheres, case which can be reduced to the
first one by considering one of the spheres and the radical plane of them.
One application is related to the calculus of the power of a point with respect to a sphere. In
the version for individual study (Figure 3), in the upper side of the monitor there are tags and text
fields for data input and the Draw button displays the image of the corresponding sphere. By
clicking on the surface of the monitor, the application randomly draws a half line starting from that
point, which intersects the sphere in two points. At the same time, it calculates and displays the
power the point with respect to the sphere in the lower side of the monitor.
For two spheres, the data are entered successively and the radical plane is displayed (Figure
4). By clicking on the plane, two half lines are randomly drawn, starting from that point and
intersecting the spheres. The application calculates and displays the power of the point with respect
to the two spheres.
A radical axis is drawn for three spheres (Figure 3). By clicking on a point on this line, three
half lines which intersect the spheres are randomly drawn. The power of the point with respect to
the three spheres is calculated and displayed.
A radical point is obtained for four spheres. By clicking on the point, four half lines which
intersect the spheres are randomly drawn and the power of the point with respect to the spheres is
determined.
For better viewing, the Rotate button rotates the entire set of spheres around the Oy axis.
Another application presents the intersection between a sphere and a straight line, a sphere
and a plane and between two spheres (Figure 7).
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Figure 7. Interface developed for visualizing intersections between a sphere with a line,
a plane or a second sphere
For the individual study version, the coordinates of the center and radius of the circle
obtained as intersection are calculated and displayed in the lower side of the monitor (Ciobanu,
Naaji, Virag, Dascal, 2016).
To have a better image over the 3D model and to see it from different angles, the application
allows rotating it.
These applications were also made using Java 3D API.
With regard to conics, the theoretical part of ITTP presents them by canonical and general
equations as well as the intersection between a conical surface and a plane.
An application was developed using Java, to represent conics starting from their general
equations.
Orthogonal invariants are defined and the center or vertex of the conic and the axes (or axis)
of symmetry are determined. In the version for individual study, all results, including the nature and
genus of the conic, are displayed in the left side of the monitor (Figure 8).
Figure 8. Interface developed for visualizing conics given by their general equations
In another application, the software module used for explaining the conics and the
intersection between a cone surface and a plane was written in JavaScript. The 3D model can be
rotated with a regular 2D mouse or with a gyromouse that has built-in gyroscopes which allows for
three degrees of movement freedom, enhancing the user experience. For a better view of the conic
section, the 3D model can be rotated. Different conics such as: circle, ellipse, hyperbola or parabola
can be obtained by the movement of the plane on the Ox and Oy axes, or its rotation around the Ox
axis. (Figure 9) (Ciobanu, Naaji, Virag, Dascal, 2016; Ciobanu, Ploscar, Dascal, Virag, Naaji,
2015).
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Figure 9. Interface for visualizing a conic section
The source code for the applications is based on standards like HTML5, CSS, the jQuery
API (jQuery API Documentation, 2013) for the user interface, the Raphael library (The Raphaël
JavaScript library home page) for working with vector graphics and the Conics3D JavaScript
extension written by Lodewijk Bogaards (Bogaards, 2016) for the visualization of conic sections.
4. Conclusions
The interactive teaching tools packages (ITTPs) are designed to present some topics from
mathematical disciplines, in order to facilitate the understanding of these abstract subjects. Through
our applications, for topics from linear algebra and analytical geometry, geometric objects can be
visualized with help of an autostereoscopic display where the 3D model can be observed and
rotated. The display does not require the use of any kind of special glasses. The applications were
developed in two versions, one for the autostereoscopic display, and one for regular 2D flat screen.
The resulting ITTPs are freely available on the involved partner universities e-learning
platforms and project web pages.
The preliminary results and feedback from the target groups are promising and future testing
is required to gain insights regarding the execution times of the various ITTPs.
Acknowledgment:
The results presented here were financially supported by the Hungary-Romania Cross-
Border Co-operation Programme 2007-2013, under the grant no. HURO/1001/040/2.3.1., entitled:
Development of Open Source Software and Interactive Teaching Tools for Mathematics –
SOFTMAT (http://softmat.uvvg.ro). The project was implemented by College of Nyiregyhaza,
Hungary as lead partner, and “Vasile Goldis” Western University of Arad, Romania.
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Monica Ciobanu (b. August 15, 1973) received her BSc in Mathematics (1996),
MSc in Operator models, variety of applications and optimal control (1997), PhD
in Mathematics (2003) from West University of Timisoara, Romania. Now she is
lecturer of mathematics in Department of Engineering and Computer Science,
Faculty of Economics, Engineering and Computer Science, “Vasile Goldis”
Western University of Arad, Romania. Her current research interests include
different aspects of linear algebra, differential geometry, dynamical systems,
graph’s algorithms and she focused on applications in biology and medical
informatics. She has (co-) authored 2 books and more than 25 papers, more than 40 conferences
participation. She participated in more than 5 projects, funded by EU, national or international
research authorities. Within the Hungarian - Romanian cross-border project, Development of Open
Source Software and Interactive Teaching Tools for Mathematics Mathematics – SOFTMAT, she
has been project assistant and coordinator of the project for the “Vasile Goldis” Western University
of Arad, Romania. She is a member of Tensor Society, Japan, of Balkan Society of Geometers,
Romania, and SRIM (Romanian Society for Medical Informatics).
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Antoanela Naaji (b. February, 1963) received her MSc in Automation and
Computers (1987), PhD in Mechanical Engineering (2005) from “Politehnica”
University of Timisoara, Romania. Now she is associate professor of computer
science in Department of Engineering and Computer Science, Faculty of
Economics, Engineering and Computer Science, “Vasile Goldis” Western
University of Arad, Romania. Her current research interests include web
technologies (e-learning), biomechanics (modeling of biological systems) and
medical informatics. She has (co-) authored and (co-) edited 13 books and more
than 90 papers. Her preoccupations in the above-mentioned fields are reflected in a number of over
20 projects, as manager or member, funded by EU, national or international research authorities or
county council. She is a senior member of IEEE, member of EAEEIE (European Association for
Electrical Engineer and Information Engineering), AGIR (General Association of Romanian
Engineers) and SRIM (Romanian Society for Medical Informatics).
Ioan Dascal (b. April 15, 1963) received his BSc in Informatics (2008), MSc in
Web Development Technologies (2010) from „Vasile Goldis” Western University
of Arad. He also received his MSc in Chemical Engineering (1989) from "Babes-
Bolyai" University of Cluj-Napoca. Now he is C++ programmer. His main area of
interest is programming in C, C++, Java, JavaScript, F#, HTML, XML, CSS, PHP,
MySQL, etc. His current research interests include functional programming. He has
(co-) authored 4 papers.
Ioan Virag (b. March 30, 1977) received his BSc in Informatics (2009) and MSc
in Web Development Technologies (2012) from „Vasile Goldis” Western
University of Arad. Since 2013 he is PhD candidate in Information Technology
and Computers at the Politehnica University of Timisoara, Automation and
Computer Science Faculty. His current research activities include Web-Based
Medical Imaging Applications and Gesture-Based Interaction in Biomedical
Interfaces. He has (co-) authored more than 10 papers and attended numerous
international conferences.