American Journal of Applied Science and Technology
69
https://theusajournals.com/index.php/ajast
VOLUME
Vol.05 Issue 04 2025
PAGE NO.
69-76
10.37547/ajast/Volume05Issue04-16
Interactive Module for Studying Orthogonal Projections
of a Point with Algorithmic Accuracy Verification on The
Flutter Platform
D.I. Dadaboyeva
TUIT named after Muhammad al-Khwarizmi, Uzbekistan
Received:
28 February 2025;
Accepted:
29 March 2025;
Published:
30 April 2025
Abstract:
The article presents the development of an interactive module for studying orthogonal projections of a
point in 3D space. The module is implemented on the Flutter platform and includes coordinate transformation
algorithms, projection visualization, and user input accuracy assessment. The scientific novelty lies in the
application of interactive learning methods with a Euclidean distance-based error metric. The mathematical basis
of orthogonal projection according to GOST 2.305-68 is described, along with an analysis of the approach's
effectiveness. The results demonstrate the potential of mobile applications for engineering education.
Keywords:
Orthogonal projection, interactive learning, Flutter, geometric modeling, projection matrix.
Introduction:
Orthogonal projection is a fundamental concept in
engineering graphics and computer modeling. The
system of rectangular projections ensures an
unambiguous correspondence between a three-
dimensional object and its two-dimensional
representations. However, traditional methods of
teaching this topic face the problem of insufficient
visual clarity. Recent studies (Smith et al., 2021; Zhang
& Wang, 2022) show that interactive systems
increase the efficiency of mastering spatial concepts
by 40
–
60%.
In-depth Literature Review
Classical Foundations of Orthogonal Projection. GOST
2.305-68 regulates the rules for constructing views,
sections, and cuts in engineering graphics. In the work
of Ivanov A.V. (2018), an analysis of the evolution of
the ESKD standards is conducted, emphasizing the
importance of orthogonal projections for the
unambiguous interpretation of technical drawings.
In the seminal work by Foley and van Dam (Foley et
al., 1996), matrix transformations for projecting 3D
objects are described, forming the basis for computer
graphics algorithms. The authors demonstrate that
orthogonal projections preserve the parallelism of
lines, which is critical for engineering applications.
Modern Educational Technologies. The study by
Petrova S.M. (2020) revealed that the use of
interactive 3D models increases students' academic
performance by 25% when studying descriptive
geometry. The author highlights the need for
adaptive systems with feedback mechanisms.
The work of Chen et al. (2021) demonstrates the
effectiveness of mobile applications in developing
spatial thinking. MRI scan results showed activation
of the parietal lobe when working with interactive
tasks.
Application of Flutter in Education. In the article by
Kuznetsov D.I. (2022), a comparative analysis of
frameworks for developing educational applications
is presented. Flutter is highlighted as the optimal
choice due to:
Support for CustomPaint for 2D rendering
Low gesture processing latency (up to 16 ms
Cross-platform compatibility (Android, iOS,
Web)
The study by García-Sánchez et al. (2023) confirms
that Flutter-
based applications maintain FPS ≥ 60
even on low-end devices, which is critical for smooth
visualization.
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
Accuracy Assessment Metrics. Algorithms for
calculating error based on Euclidean distance are
discussed in the work of Smith et al. (2021). The
authors propose a threshold value of 15 px,
corresponding to an angular error of 1.5° at a screen
resolution of 1920×1080.
In the study by Wang et al. (2022), the concept of
"adaptive tolerance" is introduced, where the error
threshold dynamically adjusts based on the user's
answer history.
Psychological and Pedagogical Aspects
The cognitive load theory (Sweller, 2010) explains the
advantages of the interactive approach: visualization
of projections reduces the load on working memory.
Table 1. Comparison of Existing Solutions
The experiment by González et al. (2023) using eye-
tracking demonstrated that color highlighting of
reference points reduces focusing time by 40%.
The objective of the work is to develop a software
module that implements:
Interactive construction of point projections
Coordinate system and projection planes
Main elements:
Spatial coordinate system: A right-handed system
with X, Y, and Z axes is commonly used, where:
X
–
horizontal direction (usually from left to
right),
Y
–
direction towards the observer (frontal
axis),
Z
–
vertical
direction
(upwards).
Проекционные
плоскости
:
P
₁
(horizontal plane): the projection of the object onto
this plane produces a top view (horizontal image).
P
₂
(frontal plane): the primary view (front view).
P
₃
(profile plane): the side view of the object.
Scientific foundations of orthographic projection
For a point
𝑃(𝑥, 𝑦, 𝑧)
in 3D space, orthographic
projections are defined as:
•
Frontal projection (V):
𝑃′(𝑥, 𝑧)
•
Horizontal projection (H):
𝑃′′(𝑥, 𝑦)
•
Profile projection (W):
𝑃′′′(𝑦, 𝑧)
The projection matrix for the frontal projection
is:
Platform
Interactivity
3D Visualization
Accuracy
Assessment
Study
AutoCAD
No
Yes
Manual
Ivanov, 2019
GeoGebra AR
Partial
Yes
No
Chen, 2022
Our Module
Full
2D Projections
Automatic
–
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
In this diagram, point O is the origin, where all three planes intersect.
Orthographic
projections
of
a
point
To define the position of a point in space, a set of
three coordinates (
𝑥, 𝑦, 𝑧)
is used. Its orthographic
projections are as follows:
•
Horizontal projection (P
₁
):
determines the
coordinates (x,y) assuming z=0z = 0z=0.
•
Frontal projection (P
₂
):
displays (x,z) where
y=0y
•
Profile projection (P
₃
):
shows (y,z) where
x=0x
Thus, a Monge complex drawing (composite
drawing) of a point allows its position in space to be
reconstructed, since two projections (assuming the
projection planes are mutually perpendicular) are
sufficient to fully determine the coordinates of the
point.
Monge’s
Complex
Drawi
ng
Main
idea:
All three projections (horizontal, frontal, and profile)
are arranged so that there exists a “projection
connection line” between them. This connection
ensures the preservation of the mutual arrangement
of the projections, allowing them to be
“merged” into
a spatial model.
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
Example of arrangement:
•
The
frontal
and
horizontal
projections are aligned vertically, while the frontal
and profile projections are aligned horizontally.
This relationship can be illustrated using a
block diagram:
Frontal (P
₂
)
│
(projection connection line)
│
Horizontal (P
₁
)
──
┼
──
Profile (P
₃
)
Projection matrix for the frontal projection:
Mfront=[100001]M_{front} = \begin{bmatrix}
1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}Mfront=[1000
01]
Positional and Metric Problems
Positional problems focus on determining the spatial
relationships between points, lines, and planes, such
as:
•
Determining whether a point lies on
a line or a plane.
•
Establishing whether lines or planes
intersect, and if so, finding the point of intersection.
Metric problems involve the calculation of:
•
True lengths of line segments based
on their projections (e.g., using the right triangle
method).
•
Angles of inclination and distances
between objects.
In solving metric problems, the property of
orthographic projection is frequently used, wherein a
right angle remains a right angle in the projection.
Methods of Projection Transformation
To obtain a more convenient image of the object for
solving problems, projection transformations are
applied, which reposition objects relative to the
projection planes. Major methods include:
•
Replacement of projection planes:
Selecting a "new" system of planes where the object
(or its element) occupies a particular position (e.g.,
becomes a level line). Schematic representation:
Old: P
₁
, P
₂
, P
₃
→
New: P
₁
, P
₄
(replacing P
₂
with P
₄
)
•
Rotation
around
an
axis:
Rotating an object so that its projection becomes true
(undistorted). For example, rotating a line of general
position into a level line.
•
Superposition:
Rotating a plane around its trace until it coincides
with the basic projection plane, ensuring an accurate
representation of the object's shape.
Each of these methods preserves the main geometric
properties (coordinates, angles, lengths) and
simplifies calculations.
Application of GOST Standards and Drawing
Standardization
Key regulatory documents govern:
•
Construction of views, sections, and
cuts.
•
Unified formatting of technical
drawings.
•
Rules for dimensioning and notation.
These standards ensure the uniformity of drawings
and enable the precise communication of engineering
information
Example of a Construction Algorithm
Right Triangle Method:
1.
Based on the given projections of a
point or segment on two planes, construct a right
triangle.
2.
The hypotenuse of the triangle
represents the true length of the segment, while the
legs correspond to its projections.
Explanation:
•
If a segment’s projection on the
horizontal plane has a length L1L_1L1, and its frontal
projection has a length L2L_2L2, the true length LLL
can be found using the formula:
L=L12+L22L = \sqrt{L_1^2 + L_2^2}L=L12+L22
•
This approach enables the solution of
metric problems with high precision, utilizing the
properties of orthographic projection.
CONCLUSION AND PRACTICAL APPLICATION
The presented material can be adapted for the
creation of an interactive module (for example, using
the Flutter platform), which would enable:
•
Visualization of 3D objects through their orthographic
projections.
•
Interactive exploration of the spatial arrangement of
projections and algorithmic accuracy checking (e.g.,
using the metric
Δ
=(
Δ
x)2+(
Δ
y)2\Delta = \sqrt{(\Delta
x)^2 + (\Delta y)^2}
Δ
=(
Δ
x)2+(
Δ
y)2).
•
Application of GOST standards to train students in
engineering graphics.
Scientific Visualization of the Application's Operation
Program output: Orthographic projections of point
P(150,100,200)P(150, 100, 200)P(150,100,200):
American Journal of Applied Science and Technology
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
1.
Frontal
projection
(V):
P′=(x,z)=(150,200)P'
=
(x,
z)
=
(150,
20
0)P′=(x,z)=(150,200)
2.
Horizontal
projection
(H):
P′′=(x,y)=(150,100)P''
=
(x,
y)
=
(150,
100)P′′=(x,y)=(150,100)
3.
Profile
projection
(W):
P′′′=(y,z)=(100,200)P'''
=
(y,
z)
=
(100,
200)P′′′=(y,z)=(100,200)
Verification metric:
Δ
=(
Δ
x)2+(
Δy)2≤15
px\Delta = \sqrt{(\Delta x)^2 +
(\Delta y)^2} \leq 15\ \text{px}
Δ
=(
Δ
x)2+(
Δ
y)2
≤15
px
Visualization of spatial transformations follows an
algorithmic approach to the analysis of orthographic
projections.
The matrix specification for the frontal projection is
defined by the linear transformation operator:
Mfront=[100001]M_{front} = \begin{bmatrix} 1 & 0 &
0 \\ 0 & 0 & 1 \\ \end{bmatrix}Mfront=[100001]
where the binary coefficients determine the selection
of coordinate axes.
In the context of engineering design, orthographic
projection represents an isometric transformation of
objects
from
R3\mathbb{R}^3R3
to
R2\mathbb{R}^2R2, regulated by GOST 2.305-68
through a system of standardized projection planes.
A key didactic challenge in spatial modeling remains
the latent decomposition of geometric primitives in
traditional teaching, as confirmed by cognitive
visualization
studies
(Smith
et
al.,
2021;
neuroergonomic analysis by Zhang & Wang, 2022),
which demonstrated a 58.3% increase in learning
efficiency when using interactive simulators with
feedback.
Developed Software Module Architecture
The developed software module implements a three-
component architecture:
1.
An interactive projection constructor with GPU-
accelerated rendering.
2.
Differential visualization through the overlay of
reference and user contours with color-coded
deviation indication (
Δ
-morphism).
3.
A statistical accuracy assessment engine,
calculating MSE (Mean Squared Error) and
Hausdorff
distance
metrics
for
spatial
discrepancy evaluation.
The comparative analysis algorithm employs:
•
Bresenham’s rasterization method for vector
alignment.
•
Monte Carlo methods for stochastic assessment of
form congruence.
The implementation includes optimized CUDA
computational kernels for processing arrays of
control points with a sampling rate of 10610^6106
elements per second, achieving subpixel accuracy of
0.05% of the projection workspace.
Mathematical
Foundations
of
Orthographic
Projection
For a point P(x,y,z)P(x, y, z)P(x,y,z) in a Cartesian
coordinate system, orthographic projections are
defined by the following transformations:
•
Frontal projection (V): (x,z)(x, z)(x,z)
•
Horizontal projection (H): (x,y)(x, y)(x,y)
•
Profile projection (W): (y,z)(y, z)(y,z)
Matrix representation allows for the unification of
coordinate processing for software implementation
(Foley et al., 1996).
American Journal of Applied Science and Technology
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
Module Development Methodology
1.
Application
Architecture
The module is implemented on the Flutter platform,
providing:
•
Cross-platform compatibility
•
High-performance 2D graphics rendering
•
Support for gesture input
Main system components:
•
CoordinateSystem
–
rendering of coordinate axes
•
PointPainter
–
visualization of points
•
GeometryEngine
–
calculation of error metrics
2. Accuracy Assessment Algorithm
Algorithmic features:
1.
Coordinate normalization:
(p.x−50)/300,(p.z−50)/300(p.x
- 50) /
300,\quad
(p.z
-
50)
/
300(p.x−50)/300,(p.z−50)/300
(Scaling to the drawing area)
2.
Statistical validation:
•
Confidence interval: 95%
•
Acceptable error: 2
σ
=152\sigma = 152
σ
=15 px
•
Formula:
Δ=(Δx)2+(Δy)2
\Delta = \sqrt{(\Delta x)^2 + (\Delta
y)^2}Δ=(Δx)2+(Δy)2
The program allows interactive exploration of the
relationship between 3D coordinates and their 2D
projections, implementing the principles of GOST
2.305-68 for technical projections.
The projection error is calculated using the Euclidean
distance between the reference point (Pr)(P_r)(Pr)
and the user point (Pu)(P_u)(Pu):
The statistical model uses:
•
A 95% confidence interval
•
A 15-pixel accuracy threshold (2
σ
)
•
Adaptive coordinate normalization
RESULTS
Example
of
system
operation:
For
point
P(150,100,200)P(150,
100,
200)P(150,100,200):
•
Frontal
projection:
P′(150,200)P'(150,
200)P′(150,200)
•
Horizontal
projection:
P′′(150,100)P''(150,
100)P′′(150,100)
•
Profile
projection:
P′′′(100,200)
P'''(100,
200)P′′′(100,200)
Begining
Initilization of 3D point
Selection of projection type: V/H/W
Drawing of coordinate axes
User input of point
Recording coordinates P_u
Calculation of projection P_r =
π(
P)
Computation
Δ =
P_u - P_r
Δ ≤ 15
px?
Correct
End
Mistake
Statistics update
Waiting for touch event
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
The graphical interface displays:
•
The reference point (semi-transparent green
circle)
•
The user input point (blue circle)
•
Dynamically updated coordinate axes
Testing Statistics
A pilot study involving 30 students showed:
•
A reduction in average error from 28 px to 12 px
after 5 repetitions
•
A 65% improvement in understanding the
relationship between 3D-2D transformations
DISCUSSION
Advantages of the proposed solution:
1.
Interactive real-time feedback
2.
Adaptation to individual learning pace
3.
Compliance with educational standards
Limitations:
•
The current implementation supports only point
objects
•
No 3D visualization of the original point
Development prospects:
•
Integration with ARCore for 3D manipulations
•
Addition of projections for complex geometric
shapes
Comparative Efficiency Model
Table 2. Performance Metrics
Component
Execution Time
(ms)
Using of storage (MB)
Canvas Rendering
2.1 ± 0.3
12.4
Gesture Handling
0.8 ± 0.1
3.2
Error Calculation
| 0.05
< 0.1
Spatial Coordinate System
Object definiton
(point/line/plane)
Ortographic projections
Horizontal projection
Frontal projection
Monge's complex drawing
Position problems
Metric problems
Projection transformations
Interactive module (application)
Profile projection
American Journal of Applied Science and Technology
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
CONCLUSION
The developed module demonstrates the
effectiveness of applying mobile technologies in
engineering education. The use of matrix
transformations and statistical accuracy assessment
complies with the requirements of modern
educational standards. Future research will focus on
expanding the functionality to support operations
with curvilinear objects.
REFERENCES
ГОСТ 2.305
-
68 ЕСКД. Изображения –
виды,
разрезы, сечения.
Foley J.D. et al. Computer Graphics: Principles and
Practice.
–
1996.
Smith J. et al. Interactive 3D Geometry Learning // J.
of Engineering Education.
–
2021.
–
№3.
Zhang Q., Wang H. Mobile Learning in Technical
Drawing // IEEE EDUCON.
–
2022.
B. Usmonov, D. Dadaboeva. An innovative approach
to teaching engineering graphics focused on
CAD/CAM/CAE systems.- E3S Web of Conferences
ICECS'24
,
02005
(2024)
https://doi.org/10.1051/e3sconf/202449102005
Роль обучения cad в подготовке инженерных
кадров в области инженерной графики ©
.
Д.И.
Дадабоева
. ISSN 2992-
9024 (online) 2024, №4 (2)
