Dynamic construction of a family of octic curves as geometric loci

Abstract

We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations, of the given curves. Finally, we highlight the involved mathematical or computational issues arising when reflecting on the outputs obtained in each case.

Full text

http://www.aimspress.com/journal/Math
AIMS Mathematics, 8(8): 19461–19476.
DOI: 10.3934/math.2023993
Received: 29 March 2023
Revised: 17 May 2023
Accepted: 18 May 2023
Published: 09 June 2023
Research article
Dynamic construction of a family of octic curves as geometric loci
Thierry Dana-Picard1,*and Tom´
as Recio2
1Department of Mathematics, Jerusalem College of Technology, Havaad Haleumi Street 21,
Jerusalem 9116011, Israel
2Universidad Antonio de Nebrija, C/Santa Cruz de Marcenado 27, 28015 Madrid, Spain
*Correspondence: Email: ndp@jct.ac.il; Tel: +97226751056; Fax: +97226751285.
Abstract: We explore the construction of curves of degree 8 (octics) appearing as geometric loci
of points defined by moving points on an ellipse and its director circle. To achieve this goal we
develop different computer algebra methods, dealing with trigonometric or with rational parametric
representations, as well as through implicit polynomial equations, of the given curves. Finally, we
highlight the involved mathematical or computational issues arising when reflecting on the outputs
obtained in each case.
Keywords: plane curves; geometric locus; octic curve; interactive construction; dynamic geometry;
GeoGebra; symbolic computation; Maple; implicitization
Mathematics Subject Classification: 14H50, 51M15, 13P10
1. Introduction
In this paper we report on our experimentation, carried out by merging a dynamic geometric system
(DGS) and a computer algebra system (CAS), dealing with plotting the geometric locus described
by a certain family of octic curves, that can be motivated as a mathematical construct related to
modeling planetary orbits. Actually, our final goal is to exhibit the challenging mathematical issues
that are involved in this quite simple context, showing the need for DGS/CAS cooperation, and
the rising of yet unsolved mathematical questions, dealing with the interrelation of specialization
(required sometimes to simplify input) and elimination, or with the appearance in some circumstances
of “strange” components in the locus plot.
Plane curves have been studied for centuries, and large catalogues are available, either printed or
online. Numerous curves are presented as stand-alone examples, such as the Witch of Agnesi, but there
exist also families of curves, depending on one parameter or more. Even so, curves known as stand-
alone appear also in families. For example, the Bernoulli lemniscate appears also as a specialization of

19462
Cassini ovals. The curves, and the families of curves, may be defined through implicit, or parametric,
or polar equations, or in a concurrent way, via more than one of these tools: e.g. the aforementioned
example has both a trigonometric parametric presentation and an implicit equation. The latter is a
polynomial equation of degree 3, making the curve an algebraic curve. Actually it is a rational cubic.
There exist also large families of curves, for which the parametric equations or the implicit equation
contain a parameter. Famous examples are epicycloids, hypocycloids, epitrochoids, hypotrochoids,
given by trigonometric parametric equations.
Two processes in reversed directions may be at work, but not always simultaneously:
•If a plane curve Cis given by a parametric presentation, then implicitization might be a core issue.
For example, if algebraic manipulation transforms the given parametrization into a rational one, a
presentation with polynomials is afforded. These polynomials generate an ideal in a polynomial
ring over the field of coefficients. Using methods based on Gr ¨
obner bases and elimination (see
[4]), an implicit equation may be derived for the curve C. This has been extensively used in
[6]. We must mention that in some cases, even with a powerful CAS and a powerful computer,
implicitization revealed impossible, not for theoretical but for technical reasons [11, 13].
•Deriving a rational parametrization from an implicit equation of a curve Cis a non trivial issue.
In some instances, as the one studied in [5] it is possible, but not in most cases. For a general
presentation of the problem and available methods with a Computer Algebra approach, we refer
to [18].
Technology is a useful tool to explore plane curves. Indeed, visualization of a curve in a screen may
be quite faithful, as the mathematical situation under study is a 2D situation. Moreover, a Dynamic
Geometry System (DGS) offers two main dynamical features:
•Dragging of points [17]: the display can be changed continuously using the mouse. Here the
changes may not be uniform, as the process is interactive.
•Slider bars: the construction of an object depending on a parameter can be analyzed when the
parameter changes in a uniform way, i.e. receiving values in a pre-defined arithmetic sequence.
The two features offer different kinds of man-and-machine interaction. Moreover, some Dynamic
Geometry Systems include the direct computation and/or display of geometric loci, involving both
“mover” and “tracer” points, see [1] for a summary description of loci computation in DGS. But the
obtained display does not have to be, in general, topologically reliable, as achieving such requirement
involves complex symbolic computation algorithms, see the recent work [12] and the references
therein, pointing out to over 30 years of research on this issue. Computing the topology of families
of curves depending on a parameter – as it is often the case when dealing with locus – is even a more
challenging issue, addressed, for example, in [2, 3], but its applicability in our context is very limited,
since the kind of curves we will work with here is not included in the first reference, and, for profiting
from the second one we would have to handle implicit equations, something that often turns out to be
too involved.
Indeed, let us point out that the algebraic manipulations requested to switch between parametric
equations and implicit equations may be performed by hand just in very simple cases, but they are
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19463
generally performed using the Gr¨
ober package implemented in a Computer Algebra System (CAS).
At the beginning, CAS and DGS were conceived as different systems. Finding ways towards a dialog,
a collaboration between the two kinds of systems has been explored for a long time; see [15, 16, 9].
Some time ago, a CAS has been implemented into the GeoGebra DGS system [14]. Nevertheless, it is
sometimes not powerful enough for implicitization, as we will show in subsection 2.2.
Therefore, in what follows, we use both GeoGebra and Maple for the exploration of algebraic
varieties in R2and in R3. Of course, when working in 3D, visualization rises specific problems, and in
this case our main contribution here is devoted to algebraic work, whence to the CAS, as in [7].
In [8], we described a simple model of planetary orbits and mathematical constructs based on them,
namely the trajectory of the midpoint of two planets. This midpoint has no true physical meaning,
but its dynamics can be a trigger to study some epi and hypocycloids, letting some parameters vary
according the precision of the orbital data in use. An extension has been made also when the two
mobile objects are moving in opposite directions (a situation which occurs when a satellite is launched
on a retrograde orbit). The movements of the two planets have different angular velocity.
The present paper is another contribution to the construction of plane curves given by trigonometric
parametric equations. This time, we consider (depending on a parameter k) an ellipse x2+k2y2=1, its
director circle, and one point on each with the same angular velocity. In the next section 2 we approach
the locus of the corresponding midpoints, in two different ways: using a trigonometric parametrization
of the ellipse and director circle, and using a rational parametrization of the same curves, and then
implicitizing. Figure 4 shows already some surprising facts about the path followed by the locus
point in terms of the values of the parameter u. On the other hand subsection 2.2 deals already with
some more involved (some of them not yet well understood) issues, dealing with the existence of
three absolutely irreducible components for each octic, with only one of them coinciding with the one
displayed in the previous subsection.
Aiming to clarify this situation, section 3 starts with a strict polynomial equation setting, through
the implicit equations of the ellipse and director circle, but recalling that they depend on a parameter k.
Now the construction of the implicit equation of the locus for each particular value of kcan be achieved
through different alternatives, such as including kas a variable in the input ideal, computing the locus,
and then specializing the result for specific values of k; or specializing the input ideal for values of k
and then obtaining the locus. We will check that, in this particular case, both approaches lead to the
same, absolutely irreducible, equation of degree 8 in the locus coordinates (with kalso of degree 8):
the octic.
The paper ends with a short summary of the different results and open problems that this apparently
simple context has yield.
2. An algorithmic exploration leading to the discovery of an octic curve
2.1. The parametric construction
We consider an ellipse Ewhose equation is x2+k2y2=1, where kis a positive real parameter. Its
director circle Dhas equation x2+y2=1+1/k2. Recall that the director circle is the orthoptic curve
of the ellipse, i.e. the geometric locus of points through which passes a pair of tangents to Ewhich
are perpendicular. This can be easily checked graphically using the DGS, as in Figure 1. The slider
enables an automated check that the construct is valid for any k>0, and dragging the point Aalong
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19464
the circle Dshows perpendicularity of the tangents from any point A.
Figure 1. An ellipse and its director circle.
A general point Eon Ehas coordinates
(xE,yE)= cos u,1
ksin u!,u∈R.(2.1)
Let Obe the origin. The line (OE) intersects the circle Dat 2 points Fand Gwhose coordinates are
as follows:











xF=q(k2+1) cos2u
k2cos2u−cos2u+1
yF=sin u
(kcos u)q(k2+1) cos2u
1+(k2−1) cos2u)
(2.2)
and











xG=−q(k2+1) cos2u
k2cos2u−cos2u+1
yG=−sin u
(kcos u)q(k2+1) cos2u
1+(k2−1) cos2u)
(2.3)
Note that the formulas express the fact that Fand Gare symmetric about the origin.
Now consider the midpoints Mand Nof the segment EF and EG respectively; their coordinates
are given by:
(xM,yM)= 1
2(xE+xF),1
2(yE+yF)!
(xN,yN)= 1
2(xE+xG),1
2(yE+yG)!(2.4)
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19465
A snapshot of a GeoGebra animation is displayed in Figure 2. It is possible to obtain both
components looking continuous by changing the parameters of the animation (we mean choosing a
lower speed, and a smaller increment on the values of the parameter). Changes in other parameters,
enabled with slider bars, provide other shapes for the desired locus. In order for the display for new
values of the parameters to be readable, the user has to erase the previous plots.
Figure 2. A dynamic construction of the loci of the midpoints.
For this example, the automated Locus command of GeoGebra did not work; therefore, we could
construct the locus only using the animation based on the slider for the parameter u. Trying other
values of the parameter krequires running the animation again and again.
In order to look for an implicit equation (if possible) of the locus of Fand the locus of Gwhen u
varies, we transfer the work to the CAS. A surprise is waiting with observation of the plots. It has to
do with the choice of the parametrization of the ellipse and its director circle. Here is the source code
for the beginning of the Maple session:
ellps := kˆ2*yˆ2 + xˆ2 = 1;
dircrcl := xˆ2 + yˆ2 = 1 + 1/kˆ2;
x_E := cos(u);
y_E := sin(u)/k;
l_E := y = y_E*x/x_E;
pts := solve({dircrcl, l_E}, {x, y});
parampts := allvalues(%);
x_F := simplify(rhs(parampts[1][1]));
y_F := simplify(rhs(parampts[1][2]));
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19466
x_G := simplify(rhs(parampts[2][1]));
y_G := simplify(rhs(parampts[2][2]));
The solve command produces generally an output containing the place holder RootOf, which is
resolved into different components by allvalues. Even then, the expressions may be quite complicated,
whence the usage of simplify. In the present case, the resulting expressions are not really simpler; they
are built on sine and cosine instead of tangent. In order to display the construct, we use the following
code:
for p from 1/2 by 1/2 to 6 do
p;
plot({[subs(k = p, x_E), subs(k = p, y_E), u = 0 .. 2*Pi],
[subs(k = p, x_F), subs(k = p, y_F), u = 0 .. 2*Pi],
[subs(k = p, x_G), subs(k = p, y_G), u = 0 .. 2*Pi],
[subs(k = p, x_M), subs(k = p, y_M), u = 0 .. 2*Pi],
[subs(k = p, x_N), subs(k = p, y_N), u = 0 .. 2*Pi]});
end do
Figure 3 shows the output for some values of the parameter. The case k=1 is very special, it
provides 2 concentric circles. In the other cases, the output presents 2 components with different
topologies. It is necessary to compute an implicit equation for the obtained curve and to (hopefully)
factorize the obtained polynomial in order to distinguish algebraically the two components.
The surprise comes from a trial to master the colors of the different plots, by inserting a color option
into the commands of the loop above, as follows:
for p from 1/2 by 1/2 to 6 do
p;
ellpsplot := plot([subs(k = p, x_E), subs(k = p, y_E), u = 0 .. 2*Pi],
color = blue, thickness = 3);
plotF := plot([subs(k = p, x_F), subs(k = p, y_F), u = 0 .. 2*Pi], color = red);
plotG := plot([subs(k = p, x_G), subs(k = p, y_G), u = 0 .. 2*Pi], color = green);
plotM := plot([subs(k = p, x_M), subs(k = p, y_M), u = 0 .. 2*Pi], color = orange);
plotN := plot([subs(k = p, x_N), subs(k = p, y_N), u = 0 .. 2*Pi], color = yellow);
display(ellpsplot, plotF, plotG, plotM, plotN);
end do;
The plots have been named separately and the final output obtained by a display command, in order
to follow precisely after what happens, i.e. from which line comes which part of the plot. The original
ellipse is in blue, the director circle is divided into two parts, as it is plotted as the geometric locus of
Fand of G.
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19467
Figure 3. Several plots from the Maple exploration.
As a whole, it was expected that the resulting curves will be symmetric about the coordinate axes,
the specific symmetries being a result of the role exchange between Fand Gwhen passing from the
left half-plane to the right half-plane, and vice versa. Indeed, the x-coordinate of Fis always taken as
the positive root of some expression, while its y-coordinate is positive if both sin(u),cos(u) have the
same sign, and negative otherwise. Likewise, the x- coordinate of Gis always taken as negative, while
its y-coordinate is negative if both sin(u),cos(u) have the same sign, and positive otherwise. Thus,
when angle utakes point Efrom the left half-plane to the right half-plane, suddenly Fis replaced by
G(as the x-coordinate becomes negative) and Fjumps to the symmetrical position of Gwith respect
to the origin.
Some of the outputs are displayed in Figure 4.
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19468
Figure 4. Several plots from the Maple exploration with prescribed colors.
2.2. Rationalization and implicitization
Recall that for every u∈R, there exist v∈Rsuch that cos u=1−v2
1+v2and sin u=2v
1+v2. By substitution
into Eq (2.4), we obtain:











xM=1
2(v2+1) (v2+1) q(v2−1)2(k2+1)
(v2−1)2k2+4v2−(v2−1)
yM=−v
k(v4−1) (v2+1) q(v2−1)2(k2+1)
(v2−1)2k2+4v2−(v2−1)(2.5)
By symmetry, we obtain similar equations for N. In order to transform these equations into
polynomials, we used the following code:
x*denom(xx_M) + vˆ2 - 1 = numer(xx_M) + vˆ2 - 1:
simplify(%ˆ2):
p1 := numer(lhs(%) - rhs(%)):
y*denom(yy_M) - v*(vˆ2 - 1) = numer(yy_M) - v*(vˆ2 - 1):
simplify(%ˆ2):
p2 := numer(lhs(%) - rhs(%)):
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19469
The computed polynomials are the following:
P1(x,y,k,v)=4k2v8x2+4k2v8x−8k2v6x−4k2v6−8k2v4x2−v8+16v6x2+16v6x+8k2v4+4v6
+32v4x2+8k2v2x−4k2v2+4k2x2−6v4+16v2x2−4k2x−16v2x+4v2−1
P2(x,y,k,v)=k4v12y2−2k4v10 y2−2k3v11y−k4v8y2+4k2v10 y2+6k3v9y+4k4v6y2−4k3v7y
−8kv9y−k4v4y2−4k2v8−8k2v6y2−v10 −4k3v5y+8kv7y−2k4v2y2+8k2v6+4v8
+6k3v3y+8kv5y+k4y2−4k2v4+4k2v2y2−6v6−2k3vy −8kv3y+4v4−v2
(2.6)
The polynomials P1(x,y,v,k) and P2(x,y,v,k) generate an ideal J⊂R[x,y,v,k]. Elimination of
parameter valone did not work with the CAS, but it is possible to eliminate vfor various values of
the parameter k, to provide some hints about the elimination ideal (bearing in mind that, in general,
the specialization of the elimination, for some values of the parameter, does not necessarily have to
coincide with the elimination of the specialized ideal, but it is an ideal contained on it). Here is the
Maple code:
for p from 1/2 to 10 do
J[p] := <subs(k = p, p1), subs(k = p, p2)>;
JE[p] := EliminationIdeal(J[p], {x, y});
gen[p] := Generators(JE[p])[1];
cc := implicitplot({subs(k = p, dircrcl) = 0, subs(k = p, ellps) = 0},
x = -4 .. 4, y = -4 .. 4, color = navy);
fac[p] := evala(AFactor(gen[p])); #absolute factorization
facts[p] := factors(gen[p]); #factorization over the rationals
comp[p][1] := facts[p][2][1][1];
comp[p][2] := facts[p][2][2][1];
comp[p][3] := facts[p][2][3][1];
c[1]:= implicitplot(subs(k = p, comp[p][1]) = 0, x = -4 .. 4, y = -4 .. 4,
color = magenta);
c[2]:= implicitplot(subs(k = p, comp[p][2]) = 0, x = -4 .. 4, y = -4 .. 4,
color = green);
c[3]:= implicitplot(subs(k = p, comp[p][3]) = 0, x = -4 .. 4, y = -4 .. 4,
color = brown);
display(cc, c[1], c[2],c[3]);
end do
Of course, the values of the local variable pcan be modified. We ran this code not only for the values
of pappearing here, but for other ones as it is clear from the illustrations). We should mention that for
p=1 the computation fails. The factors command provides more information than the regular factor
command. For every positive value of k, the elimination ideal JE is generated by a unique polynomial,
denoted above by gen, of degree 19. The output of the factors command says that over the rationals
this polynomial is the product of a factor x3and 2 factors of degree 8. We call the corresponding curves
octic curves, or simply octics.
This is not enough to check that the obtained varieties (the y-axis, and the two degree 8 curves) are
irreducible over the complexes. For this, another command has to be applied: the AFactor command,
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19470
for absolute factorization, i.e. over the field C, is a place holder, and its output is resolved by the
command evala. Thus, we have addressed this (technical) issue, which requires human observation, in
order to check validity of the output, by comparing the outputs of the rows
fac[p] := evala(AFactor(gen[p])); #absolute factorization
facts[p] := factors(gen[p]); #factorization over the rationals
The conclusion is that both outputs yield the same factorization (obviously, a very particular result,
just holding in this case, as both factorizations are, in principle, different), namely, the two factors
of degree 8 and the factor x3. But the output of the factor command does not keep the order of the
factors for different values of the parameter. The factor x3can appear in any position in the list. The 3
components can be plotted simultaneously, using what is performed in the 3 subsequent rows, and the
user has to identify each component. Comparing with the outputs of the work in parameterized setting
(in previous section), it appears that in each case, only one plot of an octic is relevant. Identifying
a pattern may be possible, but not useful, as here we show examples for pbeing either an integer or
half-integer, but other cases exist of course. In any case, one component only coalesces with what has
been obtained, either with the GeoGebra exploration or with Maple in the parametric setting.
Figure 5 shows the output for some values of k. The different colors are a consequence of the order
of the factors obtained by the command factors. Note also that for k=15, more irreducible components
seem to exist. The algebraic computations, in the next section, show that this is not the true situation.
What appears on the display is a consequence of the fact that a software has often difficulties to plot
close to a singular point. This plotting issue has been discussed in [19]. In conclusion: a detailed
comparison between this output with the output of the previous subsection, and the interpretation of
coincidences/differences, seems to deserve further, future work.
Figure 5. Several plots from the Maple exploration after specialization-elimination.
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19471
3. A construction based on implicit equations only
3.1. A particular case
Aiming to obtain the general implicit equation of this family of curves, in this Section we will
approach this goal starting with an alternate description of the construction, using implicit equations.
In order to make things easier for the non-familiar reader, we begin with the special case k=1, i.e. we
consider the circles Cand Dwhose respective equations are : x2+y2−1=0 and x2+y2−2=0. A
point E(e1,e2) on the circle Cverifies the condition e2
1+e2
2−1=0, the line OE (where Odenotes the
origin) has equation e2x−e1y=0. We denote by P(x,y) the point of intersection of the line OE with
the circle D. The midpoint F(f,g) of EP is determined by







f=1
2(x+e1)
g=1
2(y+e2)(3.1)
Actually, Equation (3.1) represent the two midpoints from Eto the intersections F,Gof the line OE
with the circle D. Define an ideal
H=⟨2f−x−e1,2g−y−e2,−e1y+e2x,e2
1+e2
2−1,x2+y2−2⟩.
Now we apply commands from Maple’s PolynomialIdeals package. We obtain
HilbertDimension(H)=1 and EliminationIdeal(H,e2)=⟨0⟩. So the construction is of dimension 1,
ruled by one coordinate of E(say, e2) on the circle C. Now the locus of P(f,g) is a quartic (i.e. , a
curve of degree 4): Applying once again elimination, we obtain:
EliminationIdeal(H,f,g)=⟨16 f4+32 f2g2+16g4−24 f2−24g2+1⟩.
Denote by Q(x,y) the generator of this ideal. Using absolute factorization with the AFactor command,
we have:
Q(x,y)=16 f4+32 f2g2+16g4−24 f2−24g2+1
=(4 f2+4g2−2RootO f (Z2−2) −3) (4 f2+4g2−2RootO f (Z2−2) −3).(3.2)
The variety V(Q) is the union of two circles, concentric with Cand D, with respective radii
approximately 0.2071067816 and 1.207106781. The output is displayed in Figure 6 (we used the
algcurve package and the plot real curve command).
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19472
Figure 6. The loci of the midpoints for k=1.
3.2. The general case
Now we deal with a family of loci, depending on parameter k. There are two ways. One, less
satisfactory, is to convert all input into polynomials in all the variables, including k(thus, requiring the
elimination of denominators). So the ideal now is of dimension 2 (say, taking e1and kas free variables)
and the elimination of all the variables except the coordinates f1,f2of the midpoint F, and k, provides
a collection of f1,f2-curves depending on k. Here is the Maple code of the session:
HH:=<e_1ˆ2+kˆ2*e_2ˆ2-1,kˆ2*xˆ2+kˆ2*yˆ2-kˆ2-1,x*e_2-y*e_1,f_1-(x+e_1)/2,f_2-(y+e_2)/2 >;
HilbertDimension(HH);
EliminationIdeal(HH, {e_1,k});
EHH:=EliminationIdeal(HH,{f_1,f_2,k});
The Hilbert dimension is equal to 2, and the 1st elimination ideal is equal to ⟨0⟩. Finally we obtain that
the ideal EHH is generated by the polynomial
Lock=16 f4
1f4
2k8+32 f2
1f6
2k8+16 f8
2k8+32 f6
1f2
2k6+64 f4
1f4
2k6+32 f2
1f6
2k6−8f2
1f4
2k8−8f6
2k8
+16 f8
1k4+32 f6
1f2
2k4+16 f4
1f4
2k4−24 f4
1f2
2k6−40 f2
1f4
2k6−16 f6
2k6+f4
2k8−16 f6
1k4
−40 f4
1f2
2k4−24 f2
1f4
2k4−8f6
1k2−8f4
1f2
2k2+2f2
1f2
2k4+f4
1.
(3.3)
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19473
Let us compute and plot the curves for k=1/2,3,15. We have:
Loc1/2=f8
1+5
2f6
1f2
2+33
16 f4
1f4
2+5
8f2
1f6
2+1
16 f8
2−3f6
1−39
8f4
1f2
2−69
32 f2
1f4
2−9
32 f6
2+f4
1
+1
8f2
1f2
2+1
256 f4
2
Loc3=1296 f8
1+25920 f6
1f2
2+152928 f4
1f4
2+233280 f2
1f6
2+104976 f8
2−1368 f6
1−20808 f4
1f2
2
−83592 f2
1f4
2−64152 f6
2+f4
1+162 f2
1f2
2+6561 f4
2
Loc15 =810000 f8
1+366120000 f6
1f2
2+41736060000 f4
1f4
2+82377000000 f2
1f6
2
+41006250000 f8
2−811800 f6
1−275401800 f4
1f2
2−20959965000 f2
1f4
2
−20685375000 f6
2+f4
1+101250 f2
1f2
2+2562890625 f4
2
(3.4)
In all the cases – including the generic one – the obtained polynomial is absolutely irreducible (use
the AFactor command), saying that the two components of the obtained curve cannot be distinguished
by algebraic means. The resulting plots are displayed in Figure 7. It is interesting to compare with the
plots in Figure 3, supporting the idea that in both cases these are, indeed, the true “octics”.
Figure 7. Three examples of the locus of midpoints.
This way is less satisfactory since it means that one could be losing values of ksuch as k=0
(but in this case this has no sense). The other, more canonical way, is to consider the ideal GH, with
polynomials in all variables, but with kin the denominator, and to consider the field of coefficients
Q(k). It is like specializing the polynomial ideal to a – say – “numerical” value k. Thus, we have
to bear in mind what we have previously commented about the relation between specialization and
elimination. The ideal has now dimension 1, with e1as only free variable, for example. The Maple
code for the session is as follows:
GH:=<e_1ˆ2+kˆ2*e_2ˆ2-1,xˆ2+yˆ2-1-1/kˆ2,x*e_2-y*e_1,f_1-(x+e_1)/2,
f_2-(y+e_2)/2, variables={e_1,e_2,x,y,f_1,f_2}>;
HilbertDimension(GH);
EliminationIdeal(GH, {e_1});
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19474
EliminationIdeal(GH,{f_1,f_2});
GL:=Generators(GH)[1];
simplify(GL-Loc_k);
Here the Hilbert dimension is 1 and the 1st elimination ideal is ⟨0⟩. The polynomial GL is given by:
GL =16 f4
1f4
2k8+32 f2
1f6
2k8+16 f8
2k8+32 f6
1f2
2k6+64 f4
1f4
2k6+32 f2
1f6
2k6−8f2
1f4
2k8
−8f6
2k8+16 f8
1k4+32 f6
1f2
2k4+16 f4
1f4
2k4−24 f4
1f2
2k6−40 f2
1f4
2k6−16 f6
2k6
+f4
2k8−16 f6
1k4−40 f4
1f2
2k4−24 f2
1f4
2k4−8f6
1k2−8f4
1f2
2k2+2f2
1f2
2k4+f4
1.
(3.5)
Finally, we have verified that, in this case, the locus result GL is equal to Lock, confirming the
previous results!
4. Discussion and conclusions
As we have described along the paper, this apparently simple geometric locus gives rise, through
different approaches: Implicit, parametric, specialized, etc., to a series of challenging mathematical
issues, that require the use of CAS tools for their analysis and interpretation. Thus, we have shown (and
explained) in subsection 2.1 the surprising behavior of the parametric-tracer points for the geometric
locus under consideration, splitting the obtained curves in two halves, divided by the y-axis. Then,
subsection 2.2, presented the failure to implicitize the generic equation of the locus family, although
it succeeded in different specialized cases. It also described some involved issues concerning the
presence of three different factors of such equation (while only one of them should be considered as
the true “octic”), and the need of further study to identify and clarify this fact.
Finally, Section 3 dealt with a pure algebraic geometry approach, but here also considering two
different possibilities (taking the parameter as a variable or as an element of the field of coefficients).
We have shown that both approaches do well in this particular case, in obtaining the general implicit
equation of the family of curves, showing also its absolute irreducibility and allowing the visual and
symbolic verification of the coincidence of the outputs in both the implicit and parametric approaches
to the octics.
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
The first author has been partially supported by the Roland and Astrid Chair (CEMJ) at JCT.
Second author partially supported by a grant PID2020-113192GB-I00 (Mathematical Visualization:
Foundations, Algorithms and Applications) from the Spanish MICINN.
The authors wish to thank Zoltan Kov´
acs for help with GeoGebra Discovery.
Conflict of interest
The authors report that for this work, there is no conflict of interest.
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19475
References
1. M. Abanades, F. Botana, A. Montes, T. Recio, An algebraic taxonomy for locus
computation in dynamic geometry, Comput. Aided Design,56 (2014), 22–33.
https://doi.org/10.1016/j.cad.2014.06.008
2. J. G. Alcazar, On the shape of rational algebraic curves depending on one parameter, Comput.
Aided Geom. D.,27 (2009), 162–178. https://doi.org/10.1016/j.cagd.2009.11.004
3. J. G. Alcazar, J. Schicho, J. R. Sendra, A delineability-based method for Computing
Critical Sets of Algebraic Surfaces, J. Symb. Comput.,42 (2007), 678–691.
https://doi.org/10.1016/j.jsc.2007.02.001
4. D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational
Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, NY:
Springer, (1992).
5. Th. Dana-Picard, Automated study of a regular trifolium, Math. Comput. Sci.,13 (2018), 57–67.
6. Th. Dana-Picard, Safety zone in an entertainment park: Envelopes, offsets and a new construction
of a Maltese Cross, Electronic Proceedings of the Asian Conference on Technology in Mathematics
ACTM 2020; Mathematics and Technology (2020), ISSN 1940–4204 (online version).
7. Th. Dana-Picard, Exploration of envelopes of parameterized families of surfaces in a technology-
rich environment, Electron. J. Math. Technol., (2023), In press.
8. Th. Dana-Picard, S. Hershkovitz, From Space to Maths And to Arts: Virtual Art in Space with
Planetary Orbits, Electron. J. Math. Technol., (2023), In press.
9. Th. Dana-Picard, Z. Kov´
acs (2021), Networking of technologies: a dialog between CAS and DGS,
Electronic J. Math. Technol.,15 (2021), 43–59. https://doi.org/10.1515/econ-2021-0004
10. Th. Dana-Picard, Z. Kov´
acs, Dynamic and automated constructions of plane curves, Maple
Transactions, (2023), In press.
11. Th. Dana-Picard, A. Naiman, W. Mozgawa, V. Ci´
eslak, Exploring the isoptics of Fermat curves in
the affine plane using DGS and CAS, Math. Comput. Sci.,14 (2020), 45–67.
12. K. Jin, J. Cheng, On the Complexity of Computing the Topology of Real Algebraic Space
Curve, (2023) (preprint). Available from: https://www.researchgate.net/publication/
330726079_On_the_Complexity_of_Computing_the_Topology_of_Real_Algebraic_
Space_Curves
13. Z. Kov´
acs, Th. Dana-Picard, Inner isoptics of a parabola, Conference: 5th Croatian Conference
on Geometry and Graphics, Dubrovnik, September 4-8, 2022. Available from: https://www.
researchgate.net/publication/363263393_Inner_isoptics_of_a_parabola
14. Z. Kov´
acs, B. Parisse, Giac and GeoGebra-improved Gr¨obner basis computations, In Gutierrez, J.,
Schicho, J., Weimann, M. (eds.), Computer Algebra and Polynomials, Lecture Notes in Computer
Science 8942, (2015), 126–138. Springer. https://doi.org/10.1007/978-3-319-15081-9 7
15. E. Roanes-Lozano, E. Roanes-Mac´
ıas, M. Villar-Mena, A bridge between dynamic geometry and
computer algebra, Math. Comput. Model., 37 (2003), 1005–1028. https://doi.org/10.1016/S0895-
7177(03)00115-8
AIMS Mathematics Volume 8, Issue 8, 19461–19476.

19476
16. E. Roanes-Lozano, N. van Labeke, E. Roanes-Mac´
ıas, Connecting the 3D DGS
Calques3D with the CAS Maple, Math. Comput. Simulat., 80 (2010), 1153–1176.
https://doi.org/10.1016/j.matcom.2009.09.008
17. M. Selakovi´
c, V. Marinkovi´
c, P. Janiˇ
ci´
c, New dynamics in dynamic geometry: Dragging
constructed point, J. Symb. Comput.,97 (2020), 3–15.
18. J. R. Sendra, F. Winkler, S. P´
erez-D´
ıaz, Rational Algebraic Curves: A Computer Algebra
Approach, NY: Springer, (2008).
19. D. Zeitoun, Th. Dana-Picard, Accurate visualization of graphs of functions of two real variables,
Int. J. Comput. Math. Sci., 4(2010), 1–11.
©2023 the Author(s), licensee AIMS Press. This
is an open access article distributed under the
terms of the Creative Commons Attribution License
(http://creativecommons.org/licenses/by/4.0)
AIMS Mathematics Volume 8, Issue 8, 19461–19476.