E4D Geometry: Straight Lines in Four-Dimensional Euclidean Space (E4D) and Their Orthogonal Projections onto Lower-Dimensional Subspaces
Objectives
General Objective
Definition of Characteristic Points
Characteristic points of a line include intersection points of the line and its projections, trace points, intersections with axes and/or with coordinate planes and hyperplanes of the space under consideration. Points in projections preserve essential geometric properties of the main line and are useful for tracing its projections in lower-dimensional spaces.
Analogous to the “traces” in classical Descriptive Geometry in E³D (intersections with coordinate planes), in E4D these points are usually chosen as the intersections of the line with two different principal coordinate hyperplanes (for example, the hyperplane w = 0 and/or y = 0, or any convenient pair). This choice greatly simplifies the calculation of parametric equations and the direct obtaining of all successive orthogonal projections.
Specific Objectives
Our ultimate “macro” objective is to make known a technique that allows the tracing of simple and complex varieties in E⁴D space. This blog post is part of a series of articles published with the aim of illustrating the tracing of one of the simplest geometric varieties in the complex world of fourth-dimensional varieties: the straight line in 4D.
We seek to extend the systematically developed knowledge of a methodology designed to trace E⁴D straight lines. This blog is only the tip of a complete encyclopedia already developed and tested since 2014 (this is not speculation) on the topic — see reference [3].
The methodology I will use in this post is simple, pedagogical, reproducible, easy to understand, and useful for tracing and visualizing 4D varieties. First, we apply the methodology to a classic 3D example (so the reader recognizes and keeps in mind the familiar procedure), and then we extend the concepts to the space of interest (4D). To develop the exercises, I will use two complementary support tools: the “E4D Graphicator” software (developed by the author) and the free version of “Desmos 3D” (to demonstrate the methodology in the illustrative 3D example).
Illustrative Examples
Illustrative Example 01 (E3D)
Determine the equation of the 3D straight line passing through the points P₁ = (7.5, 5, 10) and P₂ = (10, 15, 15), trace the line and its projections onto the coordinate planes.
Solution: As previously mentioned, we first apply the methodology to this illustrative example in 3D and then extend the concepts to 4D.Direction vector: D = P₂ − P₁ Parametric form: P(x, y, z) = P₂ + t · D Line equation:
P(x, y, z) = (10, 15, 15) + t · (2.5, 10, 5) Parametric equations:
The two characteristic points and the trace of L are highlighted.
We then proceed analytically and graphically:
Projection onto XY plane
P₁xy = (7.5, 5, 0), P₂xy = (10, 15, 0)
Parametric equations: x = 10 + 2.5t, y = 15 + 10t, z = 0 Projection onto XZ plane
P₁xz = (7.5, 0, 10), P₂xz = (10, 0, 15)
Parametric equations: x = 10 + 2.5t, y = 0, z = 15 + 5t Projection onto YZ plane
P₁yz = (0, 5, 10), P₂yz = (0, 15, 15)
Parametric equations: x = 0, y = 15 + 10t, z = 15 + 5t
Illustrative Example 02 (E4D)
Determine the equation of the 4D straight line passing through P₁ = (7.5, 5, 10, 15) and P₂ = (10, 15, 15, 20), trace the line and its projections onto E³D and E²D subspaces.
Solution: The methodology is analogous to Example 01. Direction vector: D = P₂ − P₁ = (2.5, 10, 5, 5) Parametric equations:
Figure 05. Characteristic points P₁ and P₂ in E4D space.
Figure 07. Orthogonal projections of P₁ and P₂ onto the four E3D subspaces (XYZ, XYW, XZW, YZW).Colors: navy blue (XYZ), violet (XYW), red (XZW), apple green (YZW).
Projection onto XYZ subspace
P₁xyz = (7.5, 5, 10, 0), P₂xyz = (10, 15, 15, 0)
Parametric equations: x = 10 + 2.5t, y = 15 + 10t, z = 15 + 5t, w = 0 Projection onto XYW subspace
P₁xyw = (7.5, 5, 0, 15), P₂xyw = (10, 15, 0, 20)
Parametric equations: x = 10 + 2.5t, y = 15 + 10t, z = 0, w = 20 + 5t Projection onto XZW subspace
P₁xzw = (7.5, 0, 10, 15), P₂xzw = (10, 0, 15, 20)
Parametric equations: x = 10 + 2.5t, y = 0, z = 15 + 5t, w = 20 + 5t Projection onto YZW subspace
P₁yzw = (0, 5, 10, 15), P₂yzw = (0, 15, 15, 20)
Parametric equations: x = 0, y = 15 + 10t, z = 15 + 5t, w = 20 + 5t
Figure 08. Joint tracing of the E4D line L and its four E3D projections (E4D Graphicator).
The next step — projecting line L onto the E2D subspaces of E4D — is left as an exercise for our valued readers (analogous procedure to the 3D case).
Note: For guidance on tracing straight lines in E4D, consult similar posts in this series.
Observations
Through the worked example with points P₁ = (7.5, 5, 10, 15) and P₂ = (10, 15, 15, 20), we have demonstrated in a practical and fully reproducible way how this method naturally and elegantly extends the classical tools of Descriptive Geometry to the four-dimensional domain. The analogy with traces in E2D is immediate and powerful: what used to be done with ruler and compass in three dimensions can now be achieved with the same precision in four.
The combined use of the E4D Graphicator (software developed by the author) and Desmos 3D allows the reader to visualize in real time the perfect consistency between the parametric equations and the graphical representations, making accessible a topic traditionally considered abstract and difficult to grasp.
This technique is not speculation. It is part of a complete and systematic encyclopedia developed since 2014, published in the book Geometry E4D (2016), and continuing to grow through this series of articles. With this first step we aim to open the door to an international audience eager to understand and visualize the fourth dimension in a rigorous and reproducible way. We invite readers worldwide to continue with the upcoming installments, where we will address parallel lines, intersections, planes, and more complex varieties in E4D. The fourth dimension is no longer merely a mathematical abstraction: it can now be drawn, understood, and explored step by step. Thank you for joining us on this journey toward high-dimensional geometric visualization!
Bibliography
Saturday, February 28, 2026
By: Dr. Carlos M. Martínez M.
cmmm7031@gmail.com (mailto:cmmm7031@gmail.com)
Objectives
General Objective
The main objective of this blog post is to trace straight lines in four-dimensional Euclidean space (E4D) using two arbitrary known characteristic points that define their equation. Then, using the information contained in these two characteristic points and the line equation, proceed to trace their orthogonal projections onto the three-dimensional (E3D) and two-dimensional (E2D) subspaces.
Definition of Characteristic Points
We define as characteristic points of a straight line in E4D those notable points on the line that are essential for its tracing and for the tracing of its projections, due to the information they provide. Characteristic points may or may not lie on the line itself. Those that belong to the line uniquely determine its position and direction in four-dimensional space; those that do not belong to it allow the tracing of its projections onto lower-dimensional spaces. The concept extends to any number of dimensions.
Characteristic points of a line include intersection points of the line and its projections, trace points, intersections with axes and/or with coordinate planes and hyperplanes of the space under consideration. Points in projections preserve essential geometric properties of the main line and are useful for tracing its projections in lower-dimensional spaces.
Analogous to the “traces” in classical Descriptive Geometry in E³D (intersections with coordinate planes), in E4D these points are usually chosen as the intersections of the line with two different principal coordinate hyperplanes (for example, the hyperplane w = 0 and/or y = 0, or any convenient pair). This choice greatly simplifies the calculation of parametric equations and the direct obtaining of all successive orthogonal projections.
Specific Objectives
- Determine the equations of the E4D straight line passing through the two given characteristic points.
- Trace the E4D straight line, highlighting the central role of the characteristic points in the line equation and in the geometric construction.
- Identify the orthogonal projections of the two given characteristic points onto the E3D and E2D subspaces.
- Determine the equations of the projected lines in the E3D and E2D subspaces.
- Illustrate the joint tracing of the E4D straight line and its projected lines in the E3D and E2D subspaces.
Our ultimate “macro” objective is to make known a technique that allows the tracing of simple and complex varieties in E⁴D space. This blog post is part of a series of articles published with the aim of illustrating the tracing of one of the simplest geometric varieties in the complex world of fourth-dimensional varieties: the straight line in 4D.
We seek to extend the systematically developed knowledge of a methodology designed to trace E⁴D straight lines. This blog is only the tip of a complete encyclopedia already developed and tested since 2014 (this is not speculation) on the topic — see reference [3].
The methodology I will use in this post is simple, pedagogical, reproducible, easy to understand, and useful for tracing and visualizing 4D varieties. First, we apply the methodology to a classic 3D example (so the reader recognizes and keeps in mind the familiar procedure), and then we extend the concepts to the space of interest (4D). To develop the exercises, I will use two complementary support tools: the “E4D Graphicator” software (developed by the author) and the free version of “Desmos 3D” (to demonstrate the methodology in the illustrative 3D example).
Illustrative Examples
Illustrative Example 01 (E3D)
Determine the equation of the 3D straight line passing through the points P₁ = (7.5, 5, 10) and P₂ = (10, 15, 15), trace the line and its projections onto the coordinate planes.
Solution: As previously mentioned, we first apply the methodology to this illustrative example in 3D and then extend the concepts to 4D.Direction vector: D = P₂ − P₁ Parametric form: P(x, y, z) = P₂ + t · D Line equation:
P(x, y, z) = (10, 15, 15) + t · (2.5, 10, 5) Parametric equations:
x = 10 + 2.5t
y = 15 + 10t
z = 15 + 5t
Figure 01. Tracing of line L in E3D space (using E4D Graphicator).
The two characteristic points and the trace of L are highlighted.
Figure 02. Orthogonal projections of points P₁ and P₂ onto the coordinate planes (XY, XZ, YZ).
Projections highlighted in navy blue (XY), red (XZ), and cyan (YZ). We then proceed analytically and graphically:
Projection onto XY plane
P₁xy = (7.5, 5, 0), P₂xy = (10, 15, 0)
Parametric equations: x = 10 + 2.5t, y = 15 + 10t, z = 0 Projection onto XZ plane
P₁xz = (7.5, 0, 10), P₂xz = (10, 0, 15)
Parametric equations: x = 10 + 2.5t, y = 0, z = 15 + 5t Projection onto YZ plane
P₁yz = (0, 5, 10), P₂yz = (0, 15, 15)
Parametric equations: x = 0, y = 15 + 10t, z = 15 + 5t
Figure 03. Joint tracing of line L and its projections in E³D (E4D Graphicator).
Figure 04. Same tracing in Desmos 3D (free version).
Illustrative Example 02 (E4D)
Determine the equation of the 4D straight line passing through P₁ = (7.5, 5, 10, 15) and P₂ = (10, 15, 15, 20), trace the line and its projections onto E³D and E²D subspaces.
Solution: The methodology is analogous to Example 01. Direction vector: D = P₂ − P₁ = (2.5, 10, 5, 5) Parametric equations:
x = 10 + 2.5t
y = 15 + 10t
z = 15 + 5t
w = 20 + 5t
Figure 05. Characteristic points P₁ and P₂ in E4D space.
Figure 06. Tracing of the straight line L in E4D (E4D Graphicator).
Figure 07. Orthogonal projections of P₁ and P₂ onto the four E3D subspaces (XYZ, XYW, XZW, YZW).Colors: navy blue (XYZ), violet (XYW), red (XZW), apple green (YZW).
Projection onto XYZ subspace
P₁xyz = (7.5, 5, 10, 0), P₂xyz = (10, 15, 15, 0)
Parametric equations: x = 10 + 2.5t, y = 15 + 10t, z = 15 + 5t, w = 0 Projection onto XYW subspace
P₁xyw = (7.5, 5, 0, 15), P₂xyw = (10, 15, 0, 20)
Parametric equations: x = 10 + 2.5t, y = 15 + 10t, z = 0, w = 20 + 5t Projection onto XZW subspace
P₁xzw = (7.5, 0, 10, 15), P₂xzw = (10, 0, 15, 20)
Parametric equations: x = 10 + 2.5t, y = 0, z = 15 + 5t, w = 20 + 5t Projection onto YZW subspace
P₁yzw = (0, 5, 10, 15), P₂yzw = (0, 15, 15, 20)
Parametric equations: x = 0, y = 15 + 10t, z = 15 + 5t, w = 20 + 5t
Figure 08. Joint tracing of the E4D line L and its four E3D projections (E4D Graphicator).
The next step — projecting line L onto the E2D subspaces of E4D — is left as an exercise for our valued readers (analogous procedure to the 3D case).
Note: For guidance on tracing straight lines in E4D, consult similar posts in this series.
Observations
- All graphics in this post were created in R⁴ using the “E4D Graphicator” software.
- The axis distribution system used is MaDMaI, the author’s own 4D coordinate system (detailed in the 2nd edition of the book Geometría E4D).
- Some wording suggestions from Grok AI were incorporated to optimize the post.
- Constructive feedback and criticism are always welcome and appreciated by the author.
Conclusions
By the end of this article, we have fully achieved the general objective: to identify two characteristic points that define the trace of a straight line in four-dimensional Euclidean space (E4D) and to perform its complete tracing, together with all its successive orthogonal projections onto the E3D and E2D subspaces.
The combined use of the E4D Graphicator (software developed by the author) and Desmos 3D allows the reader to visualize in real time the perfect consistency between the parametric equations and the graphical representations, making accessible a topic traditionally considered abstract and difficult to grasp.
This technique is not speculation. It is part of a complete and systematic encyclopedia developed since 2014, published in the book Geometry E4D (2016), and continuing to grow through this series of articles. With this first step we aim to open the door to an international audience eager to understand and visualize the fourth dimension in a rigorous and reproducible way. We invite readers worldwide to continue with the upcoming installments, where we will address parallel lines, intersections, planes, and more complex varieties in E4D. The fourth dimension is no longer merely a mathematical abstraction: it can now be drawn, understood, and explored step by step. Thank you for joining us on this journey toward high-dimensional geometric visualization!
Bibliography
- Lehmann, C. H., & Sors, M. S. (1953). Geometría analítica. Unión Tipográfica Editorial Hispano-Americana.
- Leithold, L. (1998). El cálculo. Oxford University Press.
- Martínez, Carlos (2016). Geometría E⁴D. 1st edition, ISBN: 978-980-12-8563-2. DOI:10.13140/RG.2.1.2103.2720, ASIN: B01C1LRGT8.
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