THEORETICAL ASPECTS IN THE FORMATION OF
PEDAGOGICAL SCIENCES
International scientific-online conference
177
CENTRAL AND PARALLEL PROJECTIONS AND THEIR PROPERTIES
Abdiqayumov Abduraxim Abdubannob òġli
NamSU, Faculty of Physics and Mathematics
1st-cource student of Mathematics
Dilnoza Xaytmirzayevna Maxmudova
Supervisor:
https://doi.org/10.5281/zenodo.15307538
Abstract
Projection is a fundamental concept in projective geometry and has
significant applications in mathematics, computer graphics, and engineering.
This article explores two principal types of projections: central and parallel
projections. We examine their mathematical definitions, properties, and
differences, supported by relevant formulas. Special attention is given to the
invariance of collinearity, the behavior of lines and planes under projections,
and practical applications in various fields.
Keywords
: Central projection, parallel projection, projective geometry,
perspective, collinearity, invariance.
Introduction
Projection methods are essential tools in both theoretical and applied
mathematics, playing a critical role in projective geometry, computer vision, and
engineering design. At its core, a projection maps points from a three-
dimensional space onto a two-dimensional surface, preserving some of the
geometrical relationships between points and lines. Two major types of
projections dominate the study: central projections (also called perspective
projections) and parallel projections (also known as orthographic projections).
Central projections simulate how humans perceive the world: objects
appear smaller as they get farther from the observer. This principle underpins
perspective drawings in art, photographic imaging, and the algorithms behind
3D rendering engines. Parallel projections, by contrast, preserve true sizes and
shapes along certain directions, making them indispensable in architectural
blueprints, engineering designs, and technical drawings where accuracy is
crucial.
Both types of projections share some fundamental properties, such as
preserving the collinearity of points and the incidence structure between points
and lines. However, they differ significantly in how they handle parallelism,
angles, and ratios of distances. A deep understanding of these projections not
only enriches one's appreciation of classical geometry but also provides vital
THEORETICAL ASPECTS IN THE FORMATION OF
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tools for solving real-world problems in fields as diverse as robotics, astronomy,
and cartography.
In this article, we formally define central and parallel projections using
coordinate geometry, explore their key mathematical properties, derive relevant
formulas, and discuss practical applications with illustrative examples.
Methods
To systematically analyze central and parallel projections, we employ tools
from coordinate geometry, vector calculus, and matrix transformations. The
approach involves setting up a coordinate system, defining projection centers or
directions, specifying target projection planes, and mathematically deriving
projection formulas based on geometric constraints.
First, for central projection, we model the projection as the intersection
between a line joining the center of projection
𝑂
to a point
𝐴
and a projection
plane
π.
The equations governing the line and plane intersection are solved
parametrically, leading to explicit formulas for the projected point's coordinates.
Central projection naturally involves the use of homogeneous coordinates,
allowing a unified treatment of points at infinity and facilitating the study of
projective invariants.
Second, for parallel projection, we consider the set of parallel lines defined
by a fixed direction vector
𝑑.
Each point
𝐴
is mapped along
𝑑
onto a projection
plane
π
. By solving for the intersection between a line through
𝐴
along
𝑑
and the
plane
π
, we derive the coordinates of the projected point. Parallel projections
often involve affine transformations, which can be represented by matrices
preserving parallelism.
The methodology includes:
-Parametric line equations: Modeling rays or lines using parameterizations
based on a direction vector.
-Plane equations: Defining the projection plane via normal vectors and
point-normal form equations.
-Intersection solving: Finding intersections between parametric lines and
planes through algebraic manipulation.
-Homogeneous coordinates: Utilizing a four-dimensional coordinate system
to simplify and generalize projections, especially useful for central projection.
-Matrix representation: Representing projections as matrix multiplications
to formalize and compute projections efficiently, especially in applications like
computer graphics and vision.
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Throughout the study, we verify projection properties such as the
preservation of collinearity and straightness of lines, and analyze the behavior of
angles, distances, and ratios under each projection type. Illustrative examples
are provided to concretely demonstrate these properties and to offer insight into
real-world implications.
Results
In a
central projection
, each point in space is connected to the center of
projection
𝑂
, and its image is the intersection point of the line
𝑂𝐴
with the
projection plane
π
.
Given:
Center of projection
𝑂 = (0,0,0),
Projection plane
π: 𝑧 = 𝑑 𝑤ℎ𝑒𝑟𝑒 𝑑 ≠ 0,
A point
𝐴 = (𝑥, 𝑦, 𝑧)
in space.
The parametric equation of the line passing through
𝑂 𝑎𝑛𝑑 𝐴
is:
𝑟(𝑡) = 𝑡(𝑥, 𝑦, 𝑧), 𝑡 ∈ 𝑅.
Setting the
𝑧
-coordinate of
𝑟(𝑡)
equal to
𝑑
(since the plane is at
𝑧 = 𝑑):
𝑧𝑡 = 𝑑 ⇒ 𝑡 =
𝑑
𝑧
.
Substituting back, the coordinates of the projection
𝐴
′
are:
Important properties of central projection:
-Straightness preservation: Lines project to lines.
-Collinearity preservation: If points
𝐴, 𝐵
, and
𝐶
are collinear, their
projections
𝐴
′
, 𝐵
′
, 𝑎𝑛𝑑 𝐶
′
are also collinear.
-Parallelism distortion: Parallel lines in space may meet at a vanishing point
on the plane.
-Distance and angle distortion: Lengths and angles are generally not
preserved.
-Homogeneous coordinates interpretation: Central projection can be
interpreted as mapping via homogeneous coordinate scaling:
(𝑥, 𝑦, 𝑧, 1) ↦ (𝑑𝑥, 𝑑𝑦, 𝑑𝑧, 𝑧),
which after dehomogenization (dividing by
𝑧
) gives:
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Vanishing Points: In central projection, sets of parallel lines in space often
meet at a single point on the projection plane, called the
vanishing point
,
reinforcing the sense of depth.
In a
parallel projection
, all projection lines are parallel to a given direction
vector
𝑑 = (𝑑
𝑥
, 𝑑
𝑦
, 𝑑
𝑧
).
Given:
Projection direction
𝑑,
Projection plane
π
with normal
𝑣𝑒𝑐𝑡𝑜𝑟 𝑛 = (𝑛
𝑥
, 𝑛
𝑦
, 𝑛
𝑧
)
and distance
𝑝
from the origin.
For a point
𝐴 = (𝑥, 𝑦, 𝑧),
the projection
𝐴
′
is determined by solving:
𝐴
′
= 𝐴 + 𝑡𝑑
where
𝑡
satisfies:
𝑛 ⋅ (𝐴
′
− 𝐴) = 0.
Expanding:
𝑛 ⋅ (𝐴 + 𝑡𝑑 − 𝐴) = 𝑡(𝑛 ⋅ 𝑑) = 0,
thus:
-Orthographic projection: Projection direction is perpendicular to the
projection plane, usually along
𝑧
-axis.
For
𝑑 = (0,0, −1)𝑎𝑛𝑑 𝜋: 𝑧 = 0
, the projection simplifies to:
𝐴
′
= (𝑥, 𝑦, 0).
Important properties of parallel projection:
-Linearity preservation: Lines project to lines.
-Parallelism preservation: Parallel lines in space remain parallel after
projection.
-Length preservation (partial): Distances are preserved along directions
parallel to the projection plane.
-Angle preservation (in special orthographic cases): If the projection
direction is perpendicular to the projection plane, then true angles between
lines on the plane are preserved.
Comparative summary
Property
Central projection
Parallel projection
Straight lines Preserved
Preserved
Collinearity Preserved
Preserved
THEORETICAL ASPECTS IN THE FORMATION OF
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Property
Central projection
Parallel projection
Parallelism Not preserved (lines may meet) Preserved
Lengths
Distorted
Partially preserved
Angles
Distorted
Preserved
(in
orthographic)
Depth effect Strong
None
Example 1: Central projection of a cube
Consider a cube centered at
(0,0,5)
with sides of length
2
. Its central
projection onto the plane
𝑧 = 2 𝑓𝑟𝑜𝑚 𝑂 = (0,0,0)
results in a depiction where
edges seem to converge toward vanishing points, creating a realistic 3D
perspective.
Example 2: Parallel projection of the same cube
Projecting the same cube along the
𝑧
-axis onto
𝑧 = 0
preserves the true
shape and relative proportions, with no convergence of lines, typical for
technical drawings.
DISCUSSION
The results obtained from the analysis of central and parallel projections
provide deep insights into the structural behavior of geometric figures under
different types of mappings. Understanding these distinctions is crucial, not only
from a theoretical standpoint in projective geometry but also for practical
applications across various fields.
The mathematical derivations clearly show that while both central and
parallel projections preserve basic geometric properties such as collinearity and
straightness of lines, they diverge significantly in how they handle parallelism,
angles, and ratios of distances:
Central projection, being dependent on a fixed center, inherently
introduces perspective distortion, leading to the convergence of parallel lines at
vanishing points. This mimics human visual perception, making it ideal for
artistic rendering and 3D computer graphics.
Parallel projection, conversely, maintains the parallelism of lines and
allows accurate scaling along specific directions. This characteristic is
fundamental in technical drawing, mechanical engineering, and architectural
planning, where dimensions must remain true.
Thus, the choice between these two types of projections is not arbitrary but
is guided by the intended purpose: realism versus precision.
THEORETICAL ASPECTS IN THE FORMATION OF
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The differences between central and parallel projections have direct
implications in multiple disciplines:
In computer vision and robotics, central projection models the behavior of
cameras and is used in algorithms for 3D reconstruction from 2D images.
In cartography, different types of map projections often use parallel
projection principles to represent Earth's surface while minimizing distortions
over selected regions.
In architecture and engineering, orthographic (parallel) projections are
fundamental in creating blueprints and machine parts drawings where accurate
measurements must be preserved.
Furthermore, understanding how projections distort or preserve various
properties aids in error analysis and correction algorithms, for example,
rectifying images distorted by perspective.
From a theoretical perspective, central projection naturally leads to the
study of projective spaces and introduces the concept of points at infinity.
Parallel projection, on the other hand, aligns more closely with affine geometry,
where parallelism is a preserved property but concepts like "ideal points" are
absent.
The mathematical frameworks established through these projections also
pave the way toward more advanced topics:
Homogeneous coordinates offer a unifying treatment for both types of
projections and simplify the formulation of geometric transformations.
Matrix transformations illustrate how projections can be efficiently
computed and manipulated in higher dimensions, which is foundational in
computational geometry and computer graphics.
While the article focuses on classical plane projections, real-world scenarios
often involve more complex settings:
Curved surfaces and nonlinear projections (e.g., fisheye lenses) introduce
additional challenges.
Multiple centers of projection (as in stereo vision) complicate simple
central projection models.
Future work could explore generalized projections, including perspective-
n-point problems, 3D-to-2D correspondences, and non-Euclidean projections for
advanced visualization techniques.
Moreover, integrating machine learning approaches with geometric
projection models holds promise for enhancing object recognition, scene
understanding, and autonomous navigation systems.
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CONCLUSION
In this study, we have rigorously examined the fundamental properties and
mathematical formulations of central and parallel projections. Through detailed
derivations and comparisons, it was demonstrated that while both projection
types preserve straightness and collinearity, they diverge in preserving
parallelism, distances, and angles.
Central projection, with its inherent perspective effects and vanishing
points, is crucial for realistically representing three-dimensional scenes on two-
dimensional surfaces. It forms the mathematical backbone of fields such as
computer graphics, photography, and fine arts. On the other hand, parallel
projection, particularly its orthographic variant, is indispensable for technical
applications where accurate measurements and true shapes must be
maintained, such as in engineering drawings and architectural plans.
Understanding the mathematical structure of these projections enables a
deeper grasp of geometric transformations in higher-dimensional spaces. It also
provides essential tools for practical applications, from camera modeling and 3D
visualization to machine vision and robotics. The use of homogeneous
coordinates and matrix transformations in representing projections bridges
pure mathematics with computational techniques, showcasing the
interdisciplinary nature of the topic.
Future research could expand upon these classical projections to study
generalized mappings, explore their behavior under non-Euclidean geometries,
and integrate them into modern technologies such as augmented reality and
autonomous systems. By building on the foundational knowledge of projection
systems, new pathways for innovation in both theoretical and applied domains
can be opened.
Ultimately, the study of projections is not merely a technical exercise; it is a
gateway to understanding how we model, interpret, and interact with the
multidimensional world around us.
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