Four-Dimensional Visual Exploration of the Complex Number Plane

Abstract

A straight line intersects a circle in two, one, or no real points. In the last case, they have two complex conjugate intersecting points. We present their construction by tracing the circle with all lines. To visualize these points, the real plane is extended with the imaginary dimensions to four-dimensional real space. The surface generated by all complex points is orthogonally projected into two three-dimensional subspaces generated by both real and one of the imaginary dimensions. The same method is used to trace and visualize other real and imaginary conics and a cubic curve. Furthermore, we describe a graphical representation of complex lines in the four-dimensional space and discuss the elementary incidence properties of points and lines. This paper provides an accessible method of visualization of the complex number plane.
KeywordsComplex number planeFourth dimensionMulti-dimensional visualizationComplex roots

Full text

Four-Dimensional Visual Exploration of the
Complex Number Plane
Jakub ˇ
Rada1[0000−0003−2273−151X]and Michal Zamboj2[0000−0002−7927−2302]
1Charles University, Faculty of Mathematics and Physics, Mathematical Institute of
Charles University, Czech Republic rada@karlin.mff.cuni.cz
2Charles University, Faculty of Education, Department of Mathematics and
Mathematical Education michal.zamboj@pedf.cuni.cz
https://www2.karlin.mff.cuni.cz/∼zamboj/index en.html
Abstract. A straight line intersects a circle in two, one, or no real
points. In the last case, they have two complex conjugate intersecting
points. We present their construction by tracing the circle with all lines.
To visualize these points, the real plane is extended with the imaginary
dimensions to four-dimensional real space. The surface generated by all
complex points is orthogonally projected into two three-dimensional sub-
spaces generated by both real and one of the imaginary dimensions. The
same method is used to trace and visualize other real and imaginary
conics and a cubic curve. Furthermore, we describe a graphical repre-
sentation of complex lines in the four-dimensional space and discuss the
elementary incidence properties of points and lines. The paper provides
an accessible method of visualization of the complex number plane.
Keywords: Complex number plane ·Fourth dimension ·Multi-dimensional
visualization ·Complex roots.
1 Introduction
Let us have a circle and a line in a real plane. The line intersects the circle in two
real points, touches the circle in the point of tangency, or has no real but two
complex conjugate points on the circle. This could be an elementary exercise in
analytic geometry. However, the geometric construction or visualization of the
last case is not as obvious. In this paper, our focus lies on the visualization of
complex points.
The property of keeping the number of intersecting points of a line and a
conic (or algebraic curve in general) and their geometric construction based on
the polar properties of conics were described in Poncelet’s early texts on the
principle of continuity in the framework of projective geometry. Figure 1, from
the second edition of his comprehensive work — Trait´e des propri´et´es projectives
des figures [5] shows a construction of a secant and non-secant line intersecting an
ellipse. These ideas were thoroughly revisited and presented by Hatton around
one hundred years later (see [4]). In Chapter 6, Hatton described the process
of “tracing of conics” along their conjugate diameters and created their planar
This is a preprint of the following chapter: Řada, J., Zamboj, M. (2023). Four-Dimensional Visual Exploration of the Complex Number Plane. In: Cheng, LY. (eds)
ICGG 2022 - Proceedings of the 20th International Conference on Geometry and Graphics. ICGG 2022. Lecture Notes on Data Engineering and Communications
Technologies, vol 146. Springer, Cham reproduced with permission of Springer Nature Switzerland AG. The final authenticated version is available online at:
http://dx.doi.org/10.1007/978-3-031-13588-0_12.
Errata: p. 3, last line: should be ±i√{k2 − 1} ; p. 4, last 2 expressions: should be k sin φ in the second coordinate; Fig. 7, blue point P1, should be labeled P2.

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Rada,Zamboj
graphs called Poncelet figures. We aim to lift this idea into a four-dimensional
space to visualize the complex points of all branches at once.
Fig. 1: Poncelet’s Fig. 6 [5]. The points M, N are real points on the ellipse, while
M′, N ′are complex points on the same ellipse. The points Oand O′are poles
of the polars M′N′and M N with respect to the ellipse.
Source gallica.bnf.fr / Biblioth`eque nationale de France.
In fact, the presently often used and popular visualizations of complex num-
bers founded by Argand and Wessel had appeared only two decades before
Poncelet’s Trait´e (see [8] pp. 438–439 for historical details). To visualize two
complex numbers as coordinates of the complex number plane3C2in a similar
manner, one has to approach a four-dimensional real space. The advancement
of computer graphics brought effective visualization tools in higher dimensions.
Several authors displayed complex elements in separate 3-dimensional spaces
(see [1,9]). Four-dimensional set of points is plotted in (Re(x), Re(y), Im(x)),
(Re(x), Re(y), I m(y)), (Re(x), Im(x), Im(y)), or (Re(y), I m(x), Im(y)). The au-
thor of [9] has also created an iOS application [10] that can display each of these
graphs. Buttler in [3] placed a perpendicular plane with axes (I m(x), I m(y))
at each point of the real plane (Re(x), Re(y)), then the perpendicular plane is
rotated to the real plane such that the axes Re(x) with Im(x) and Re(y) with
Im(y) are parallel at each point. Bozlee in [2] used 3D-printing to create a 3D-
printed model with complex parts of elliptic curves using Silviana’s bertini real
software.
In this paper, we contribute to the topic by the visualization of complex
points on a circle created in a double orthogonal projection into two mutually
perpendicular 3-spaces (4D-DOP, see [12]) and in a four-dimensional perspec-
3Not to be confused with the term “complex plane”, which usually indicates the
(Argand, Wessel, or also Gauss) plane with coordinate axes corresponding to real
and imaginary elements of one complex variable.

Four-Dimensional Visual Exploration of the Complex Number Plane 3
tive (4D-perspective, see [6]). Furthermore, we elaborate the visualization of a
line through two points in C2. All upcoming figures are created in Wolfram
Mathematica.
2 Tracing a circle
(a) (b)
Fig. 2: (a) A point Pwith the coordinates P[(Re(px), I m(px)); (Re(py), Im(py))]
graphically represented in 4D-DOP. P1[Re(px), Re(py), I m(px)] and
P2[Re(px), Re(py),−Im(py)] are its Ξ- and Ω-images in one modeling 3-
space.
(b) The intersections of a circle c:x2+y2= 1 traced by the lines
n:xcos φ+ysin φ=kfor k∈IR and φ= 0 (red), π
6(yellow), π
2(cyan). Points
Pand Pare complex conjugate intersections on the line for φ=π
6.
Let us return to the circle – line problem. Suppose we have a real plane IR2
with the coordinate system (x, y), the circle
c:x2+y2= 1,
and trace it with the line
l:x=k,
for k∈IR parallel to y-axis. For k∈(−∞,−1) ∪(1,∞), the roots of the
corresponding quadratic equation in yare ±i√c2−1. The intersecting points

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Rada,Zamboj
[k;i√k2−1] and [k;−i√k2−1] are on cand l, but not in the plane IR2. Let us
extend the real plane with the imaginary components such that the real coordi-
nates xand ywill be denoted Re(x) and Re(y) and the imaginary parts I m(x)
and Im(y). A point Pin the complex number plane has coordinates P[px, py] in
C2and P[(Re(px), Im(px)); (Re(py), I m(py))] in IR4, for px= (Re(px), Im(px))
and py= (Re(py), Im(py)). Consequently, we identify the complex number plane
C2with IR4with the orthogonal system of axes (Re(x), I m(x), Re(y), Im(y)).
For example, the above-mentioned complex intersecting points of land chave co-
ordinates [(k, 0); (0,√k2−1)] and [(k, 0); (0,−√k2−1)] in IR4(while the purely
real points for k∈ ⟨−1,1⟩are [(k, 0); (√1−k2,0)] and [(k, 0); (−√1−k2,0)]).
For visualization, we use the 4D-DOP method, which is a generalization
of Monge’s projection. Each point P[(Re(px), I m(px)); (Re(py), Im(py))] is or-
thogonally projected into the reference 3-spaces Ξ(Re(x), I m(x), Re(y)) and
Ω(Re(x), Re(y), I m(y)) with the common plane π(Re(x), Re(y)) (see Fig. 2a).
Both 3-spaces Ξand Ωare represented in one modeling 3-space such that a
perpendicular line to the plane π(Re(x), Re(y)) creates axes I m(x) and I m(y)
with the opposite orientations (Im(x) upwards, Im(y) downwards).
Let us have a closer look at the example above. Observe the locus of inter-
secting points of cand l, in 4D-DOP (Fig. 2b). For k∈ ⟨−1,1⟩,I m(x) and
Im(y)-coordinates are zero, the real parts are obviously related by the equation
Re(x)2+Re(y)2= 1,
representing the circle in the plane (Re(x), Re(y)). However, for k∈(−∞,−1) ∪
(1,∞), equations
Im(x) = 0; Re(y) = 0; Re(x)2−Im(y)2= 1
represent a hyperbola in the plane (Re(x), Im(y)) and hence also in the 3-space
Ω(Re(x), Re(y), I m(y)). Both branches of this hyperbola are projected into two
rays in the 3-space Ξ(Re(x), Re(y), Im(x)).
Tracing the circle cwith a line
m:y=k,
for k∈IR, parallel with the x-axis (back in π(Re(x), Re(y))), we obtain the
points of intersection [(0,±√k2−1); (k, 0)] for k∈(−∞,−1) ∪(1,∞) and
[(±√1−k2,0); (k, 0)] for k∈ ⟨−1,1⟩in C2. Apart from the same circle in
the plane π(Re(x), Re(y)), the complex points lie on a hyperbola in the plane
(Im(x), Re(y)) and so in Ξ(Re(x), I m(x), Re(y)). The Ω-image of the hyperbola
consists of two rays in (Re(x), Re(y), I m(y)).
For a general case, assume a line ngiven by the following equation
n:xcos φ+ysin φ=kfor φ∈ ⟨0,2π), k ∈IR.
Its intersection points with care
[kcos φ−q(1 −k2) sin2φ;kcos φ+ cot φq(1 −k2) sin2φ],
[kcos φ+q(1 −k2) sin2φ;kcos φ−cot φq(1 −k2) sin2φ].

Four-Dimensional Visual Exploration of the Complex Number Plane 5
Fig. 3: The surface of a real circle cgenerated by its complex points visual-
ized in 4D-DOP. Views in special positions are on the right side. The circle
is shifted in the directions Im(x) and Im(y) so that images in the 3-spaces
(Re(x), Im(x), Re(y)) and (Re(x), Re(y), I m(y)) do not overlap in the figure.
For k∈ ⟨−1,1⟩the points must lie on the circle in π(Re(x), Re(y)). For k∈
(−∞,−1) ∪(1,∞) the values of coordinates always contain an imaginary ele-
ment, and they represent a hyperbola rotated along the circle and twisted in
C2. Orthogonal images of the surface generated in Fig. 3 are created by the
extraction of the real and imaginary parts of the intersection points. At last, the
image in 4D-perspective is in Fig 4.

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Fig. 4: The surface of a real circle cgenerated by its complex points visualized
in 4D-perspective.
2.1 Further issues
The method used in the previous section is theoretically applicable for any alge-
braic curve over IR. At first, the curve is traced by all real lines in the real plane
to obtain complex intersections. Next, we extract the real and imaginary parts
of the complex points of intersection and plot the final image embedded in IR4.
The surfaces corresponding to some other conics: a hyperbola, parabola, imagi-
nary regular conic; and a cubic are depicted in Figs. 5a–5d. However, raising the
order of the curve, the computational complexity (equation solving, plotting)
increases rapidly.
3 Lines in CIP2
We have been constructing complex points of real curves and lines until now. On
top of that, we can construct any point with coordinates in C2. In this section,
we will move a little further and explore the construction of an arbitrary line in
C2. Since our visualizations are created in the four-dimensional real space, we
should be aware that images of lines in C2will behave differently from the real
lines. For example, one linear equation represents a hyperspace in IRn. While this
holds well for lines in IR2, one equation in IR4represents a 3-space. Therefore,
each line in C2will generates a 3-space in IR4. Furthermore, the lines Re(x)=0

Four-Dimensional Visual Exploration of the Complex Number Plane 7
(a) Hyperbola: x2
−y2= 1 (b) Parabola: y=x2
(c) Imaginary regular conic: x2+y2=−1 (d) Cubic: y=x3
Fig. 5: Surfaces of curves generated by their complex points visualized in 4D-
DOP. All the surfaces are shifted in I m(x) and I m(y) directions so that they do
not overlap.

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and Im(x) = 0 are equivalent in C2, due to multiplication by a constant i, but
they seem distinct in IR4. To avoid such confusion, we approach lines through
the projective extension CIP2.
A point Pin CIP2has homogeneous coordinates P(p1;p2;p0)= (0; 0; 0) for
p1;p2;p0∈C such that (p1;p2;p0)∼(λp1;λp2;λp0) for λ∈C\ {0}. Expanding
real and imaginary parts of the point P, the coordinates will be in the form
P((Re(p1), Im(p1)); (Re(p2), I m(p2)); (Re(p0), Im(p0))). For the sake of visual
representation, we always factorize the coordinates by the last nonzero coordi-
nate. Therefore, proper points in C2will be represented by points with coor-
dinates ((Re(px), Im(px)); (Re(py), I m(py)); (1,0)) and directions or improper
points as ((Re(px), Im(px)); (1,0); (0,0)) or ((1,0); (0,0); (0,0)). Conveniently
using the duality in projective spaces, the same holds for the coordinates of
lines. Let ((Re(lx), Im(lx)); (Re(ly), I m(ly)); (1,0)) be (factorized) coordinates
of a line l, then its equation in the expanded form in IR4is
Re(lx)Re(x) + Im(lx)Im(x) + Re(ly)Re(y) + Im(ly)Im(y) + 1 = 0.
Similarly for lines with coordinates ((Re(lx), I m(lx)); (1,0); (0,0))
Re(lx)Re(x) + Im(lx)Im(x) + Re(y)=0
or for ((1,0); (0,0); (0,0))
Re(x)=0.
Such equations represent 3-spaces in IR4. To visualize 3-space in orthogonal
projection, we construct its traces, i.e., intersecting planes with the 3-spaces
Ξ(Re(x), Im(x), Re(y)) and Ω(Re(x), Re(y), I m(y)). Substituting Im(y) = 0
and Im(x) = 0 into the equation of the line, we obtain the respective Ξ- and Ω-
traces (see also [11] for synthetic constructions of traces of 3-spaces). As a conse-
quence, the real part of the line is its intersection with the plane π(Re(x), Re(y))
obtained by vanishing the terms with I m(x) and I m(y).
Let us examine the visual representations of lines with several examples
Fig. 6.
1. A line lwith coordinates ((1,0); (−1,0); (1,0)) and the equation
Re(x)−Re(y) + 1 = 0
is depicted in Fig. 6a. Observe, that the intersection of lwith the plane
π(Re(x), Re(y)) does not change the equation. Furthermore, it is arbitrary
in Im(x) and Im(y). The extension of the line in the directions Im(x) in
the 3-space Ξ(Re(x), Im(x), Re(y)) and in Im(y) in Ω(Re(x), Re(y), I m(y))
generates the trace planes of the 3-space of l. Therefore, the trace planes are
perpendicular to πin the modeling 3-space. Additionally, we should remind,
that due to equivalence, the same representation will have all lines multiplied
by a nonzero complex scalar, e.g.:
Im(x)−Im(y) + i= 0 ∼
Re(x)−Im(x)−Re(y) + Im(y)+1−i= 0 . . . .

Four-Dimensional Visual Exploration of the Complex Number Plane 9
(a) l((1,0); (−1,0); (1,0))
Re(x)−Re(y) + 1 = 0
(b) l((1,−1); (0,0); (1,0))
Re(x)−Im(x) + 1 = 0
(c) l((1,0); (0,−1); (1,0))
Re(x)−Im(y) + 1 = 0
(d) l((1,−1); (1,−1); (1,0))
Re(x)−Im(x) + Re(y)−Im(y) + 1 = 0
Fig. 6: Lines in CIP2represented as 3-spaces in IR4in 4D-DOP. The 3-spaces
are given by their intersections with reference 3-spaces Ξ(Re(x), I m(x), Re(y))
(red) and Ω(Re(x), Re(y), I m(y)) (blue).
2. See Fig. 6b for l((1,−1); (0,0); (1,0)) with the equation
Re(x)−Im(x) + 1 = 0.
Apparently, the line Re(x) + 1 = 0 is the intersection with π. The Ξ-image
could be reconstructed from the image in the plane (Re(x), I m(x)), and the
Ω-image is, again, perpendicular to πin the modeling 3-space.

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3. See Fig. 6c for l((1,0); (0,−1); (1,0)) with the equation
Re(x)−Im(y) + 1 = 0.
The situation is similar to the previous case. Now, the Ξ-image is perpen-
dicular to π.
4. See Fig. 6d for l((1,−1); (1,−1); (1,0)) with the equation
Re(x)−Im(x) + Re(y)−Im(y) + 1 = 0.
In this case, none of the trace planes are perpendicular to π. The traces
could be generated separately by vanishing the imaginary components in
respective 3-spaces.
3.1 Joins and intersections
In complex homogeneous coordinates in CIP2, a point P(p1;p2;p0) lies on a line
l(l1;l2;l0) if
p1l1+p2l2+p0l0= 0.
Using the dot product
P·l= 0.
Another point Qlies on lif
Q·l= 0.
Hence
l=P×Q.
Dually, a point Pis the intersection of distinct lines pand q, only if
P=p×q
(see [7], Chapter 3 for details).
Graphical representation in IR4of lines and points in C2will work slightly
differently, too. This is because the multiplication of imaginary components
changes sign. For example, the dot product of a point P(p1;p2;p0) and a line
l(l1;l2;l0)∈CIP2is
p1l1+p2l2+p0l0.
However, after the expansion into real and imaginary components, we have
P((Re(p1), Im(p1)); (Re(p2), I m(p2)); (Re(p0), Im(p0)))·
l((Re(l1), Im(l1)); (Re(l2), I m(l2)); (Re(l0), Im(l0))) =
Re(p1)Re(l1)−Im(p1)Im(l1) + Re(p2)Re(l2)−Im(p2)Im(l2)+
Re(p0)Re(l0)−Im(p0)Im(l0).
Therefore, in the visualizations in IR4the point Pwill not lie in the 3-space
representing line l. On the other hand, the complex conjugate
¯
P((Re(p1),−Im(p1)); (Re(p2),−Im(p2)); (Re(p0),−Im(p0))

Four-Dimensional Visual Exploration of the Complex Number Plane 11
of the point Plies on the 3-space of the line lthrough P. And oppositely, the
point Plies in the 3-space representing the complex conjugate ¯
lof the line lin
IR4. In the last figure 7, the line lhas coordinates
l=P×Q,
but the complex conjugate points ¯
Pof Pand ¯
Qof Qlie on the 3-space of the
line l. This is also verified in the figure by the construction of the plane in the
3-space of lthrough Pparallel to Ω(Re(x), Re(y), I m(y)).
Fig. 7: A line lin CIP2passing through points P, Q represented as 3-space in IR4.
The complex conjugates ¯
P , ¯
Qof P, Q lie on the 3-space, which is given by its
planar intersections with Ξ(Re(x), Im(x), Re(y)) and Ω(Re(x), Re(y), Im(y))
in IR4using the 4D-DOP method.
4 Conclusion
We have revisited the method of finding complex points on a circle by tracing
the circle with a line. Intersecting points generate a surface in the 4-dimensional
space (Re(x), Im(x), Re(y), I m(y)). The final visualization of the images was
plotted in a double orthogonal projection into 3-spaces (Re(x), I m(x), Re(y))
and (Re(x), Re(y), I m(y)) and in four-dimensional perspective projection. The
method was applied to visualize complex points of other conics and a cubic
curve. Moreover, it can be used for many other real curves; however, it is very

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Rada,Zamboj
limited by computational complexity. A further possibility of application is, for
instance, in finding graphical solutions of complex intersections of real curves.
Furthermore, through a projective extension, we have described how to visu-
alize a complex straight line as a three-dimensional subspace of a four-dimensional
real space. We have also discussed how to verify the incidence of a point and
a line and how to visualize the join of two points. These concepts are easily
extendable and applicable for further research in visualizing a complex number
plane identified with a four-dimensional real space.
References
1. Banchoff, T.F.: Complex Function Graphs. http://www.tombanchoff.com/
complex-function-graphs.html. Last accessed 20 February 2020.
2. Bozlee, S., Amethyst, S.V.: Visualizing complex points of elliptic curves. https://im.
icerm.brown.edu/portfolio/visualizing-complex-points-of- elliptic-curves/. Last ac-
cessed 20 February 2020.
3. Butler, D.: Where the complex points are. https://blogs.adelaide.edu.au/
maths-learning/2016/08/05/where-the-complex- points-are. Last accessed 20 Febru-
ary 2020.
4. Hatton, J.L.S.: The Theory of the Imaginary in Geometry: Together with
the Trigonometry of the Imaginary. Cambridge University Press (1920).
https://doi.org/10.1017/CBO9780511708541
5. Poncelet, J.V.: Trait´e des propri´et´es pro jectives des figures: ouvrage utile `a ceux qui
s’ occupent des applications de la g´eom´etrie descriptive et d’op´erations g´eom´etriques
sur le terrain. Second edition, vol. 1, Gauthier-Villars, Paris (1865).
6. ˇ
Rada, J., Zamboj, M.: 3-Sphere in a 4-Perspective. In Jeli, Z. (ed). Proceedings
moNGeometrija 2020, Belgrade, Serbia. Planeta Print, Belgrade, pp. 52–61 (2021).
7. Richter-Gebert, J.: Perspectives on projective geometry: a guided tour through real
and complex geometry. Heidelberg: Springer (2011). https://doi.org/10.1007/978-
3-642-17286-1
8. Scriba, C., Schreiber, P.: 5000 Years of Geometry. Springer Basel (2015).
https://doi.org/10.1007/978-3-0348-0898-9
9. Avitzur, R.: Visualizing Functions of a Complex Variable. http://www.nucalc.com/
ComplexFunctions.html. Last accessed 20 February 2020.
10. Avitzur, R. Pacific tech graphing calculator for iOS. https://apps.apple.com/us/
app/pacific-tech-graphing- calculator/id1135478998?ls=1. Last accessed 20 Febru-
ary 2020.
11. Zambo j, M.: Double Orthogonal Projection of Four-Dimensional Objects onto Two
Perpendicular Three-Dimensional Spaces. Nexus Network Journal 20(1), pp. 267-
–281 (2018). https://doi.org/10.1007/s00004-017-0368-2
12. Zambo j, M.: Visualizing Objects of Four-Dimensional Space: From Flatland to
the Hopf Fibration (invited talk). In: Szarkov´a, D., Richt´arikov´a, D., Pr´aˇsilov´a, M.
(eds.). Proceedings of the 19th Conference on Applied Mathematics Aplimat 2020.
Bratislava, Slovakia. Slovak University of Technology, Bratislava, 1140–1164 (2020).