Abstract: Drawing a curve on a computer actually involves approximating it by a set of segments. The De
Casteljau algorithm allows to construct these piecewise linear curves which approximate polynomial Bézier
curves using convex combinations. However, for rational Bézier curves, the construction no longer admits
regular sampling. To solve this problem, we propose a generalization of the De Casteljau algorithm that
addresses this issue and is applicable to Bézier curves with mass points (a weighted point or a vector) as
control points and using a homographic parameter change dividing the interval [0,1]into two equal-length
intervals 0,1
2and 1
2,1. If the initial Bézier curve is in standard form, we obtain two curves in standard
form, unless the mass endpoint of the curve is a vector. This homographic parameter change also allows
transforming curves dened over an interval [α, +∞],α∈R, into Bézier curves, which then enables the
use of the De Casteljau algorithm. Some examples are given: three-quart of circle, semicircle and a branch
of a hyperbola (degree 2), cubic curve on [0; +∞]and loop of a Descartes Folium (degree 3) and a loop of
a Bernouilli Lemniscate (degree 4).
Key-Words: De Casteljau algorithms, Bézier curves, mass points, homographic parameter change,
regular sampling iterative construction.
Received: October 16, 2023. Revised: February 18, 2024. Accepted: March 9, 2024. Published: April 3, 2024.
1 Introduction
Bézier curves are the simplest curves dened by
control points and were invented by [1] at Renault
and [2] at Citroën. More details about their uses
in C.A.G.D. can be found in [3], [4], [5], [6], [7], [8],
[9], [10], [11] and [12]. Initially, these curves were
dened as the barycentric locus of weighted points,
with the weights computed using the appropriate
Bernstein polynomials. In a second step, adding
weights to the control points allows for obtaining
more curves such as conic arcs with center, but
there is no way to obtain semi-ellipses [12]. A
mass point that is either a weighted point or a
vector with a weight equal to 0is a generalization
of the concept of barycenter. For example
1−−→
GB −1−→
GA =−−→
GB −−−→
GB +−→
BA=−→
AB
and the barycenter of the weighted points (A, −1)
and (B, 1) does not exist, but the calculus leads
to the vector −→
AB. For any natural number greater
than or equal to 2, the conversion of the paramet-
ric equation curve t, 1
tnon [0,1] from the canon-
ical basis to the Bernstein basis yields a rational
Bézier curve of degree n+ 1 with n−1control
vectors. Furthermore, by converting the paramet-
ric curve t7→ (t, tn)with n > 2, after the variable
change t=u
1−uin the appropriate Bernstein basis,
the null vector −→
0appears n−2times.
To solve the problem of constructing semi-
ellipses or hyperbola branches, it is sucient to
add control vectors by using mass points as con-
trol points. Furthermore, it is possible to model
Descartes Folium or Bernoulli Lemniscate loops.
The De Casteljau algorithm is generalized using
a specic homographic parameter change, which
allows for obtaining regular curve subdivision al-
gorithms that can be applied to quadratic curves
usable in the usual Euclidean space or in the (non-
Euclidean) Minkowski-Lorentz space for Dupin cy-
clides [13] cubic for Descartes Folium, quartic for
Bernoulli Lemniscate.
In 2004 the fractal nature of Bézier curves is
demonstrated in [14], that is, their self-similarities,
based on the works [15], particularly focusing on
the concepts of attractors and iterative processes.
The purpose of this paper is to construct a Bézier
curve iteratively with regular sampling. Figure 1
provides a visual representation of the dierence
between the projective De Casteljau algorithm
A survey of De Casteljau algorithms and regular iterative
constructions of Bézier curves with control mass points
1LIONEL GARNIER, 2JEAN-PAUL BÉCAR, 3LUCIE DRUOTON
1L.I.B., University of Burgundy, B.P. 47870, 21078 Dijon Cedex, FRANCE
2U.P.H.F. - Campus Mont Houy - 59313 Valenciennes Cedex 9, FRANCE
3I.U.T. of Dijon, University of Burgundy, B.P. 47870, 21078 Dijon Cedex, FRANCE
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
216
Volume 23, 2024
[7], [12], Figure 1 and our generalized De Castel-
jau algorithm, Figure 2, to construct a circle arc
as a rational quadratic Bézier. The rst algo-
rithm does not yield a regular distribution of con-
structed points, Figure 1, whereas our generalized
De Casteljau algorithm do, Figure 2.
In Figure 1, the angles are given in the Table 1.
In the Figure 2, the angle, in degree, between two
consecutive constructed points and the center O
of the circle equals to
\
P0OB0=
\
BiOBi+1 =
\
B6OP2= 33.75 = 270
8
where i∈[[0; 5]]. Considering a quarter of a circle,
the calculations are performed to show that the
De Casteljau algorithm does not allow obtaining
a regular construction.
Angle
\
P0OA0
\
A0OA1
\
A1OA2
\
A2OA3
Value 12.82 21.46 38.50 62.23
Angle
\
A3OA4
\
A4OA5
\
A5OA6
\
A6OP2
Value 62.23 38.50 21.46 12.82
Table 1: Angles, in degree, between two consec-
utive points and the center of the circle in the
Figure 1.
Figure 1: Distribution of points constructed by
the projective De Casteljau algorithm.
Figure 2: Distribution of points constructed by
our algorithm based on the change of homographic
parameter.
Moreover, let γbe the rational quadratic Bézier
curve in standard form of control points (P0,1),
P1,−√2
2and (P2,1) representing the three quar-
ters of circle in Figure 1. Then B3=γ1
2. Let
γ1one of the two subcurve of γ, in standard form,
of endpoints (P0,1) and (B3,1). Note that the
tangents to the Bézier curve at P0and B3are
known thanks to the De Casteljau algorithm. Un-
like polynomial curves, in the rational case, it is
necessary to perform an iterative construction be-
cause
γ11
26=γ 1
22!=γ1
4(1)
In [7], it is written, regarding the projective
version of the De Casteljau algorithm: ”The in-
termediate points could also be close to being vec-
tors i.e. having a very small third component.
This may cause numerical problems.”. The use of
mass points allows solving this problem since, in
the case of a rational curve, we do not divide by
the weights when obtaining a vector (including the
null vector −→
0), see Formula (8).
The article is structured as follows. In the
second section, we provide a brief overview of
the classical De Casteljau algorithm for degrees
2 and 3. Before concluding and discussing future
prospects, we extend the De Casteljau algorithm
to rational Bézier curves with control mass points
by using a homographic parameter change func-
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
217
Volume 23, 2024
tion.
2 Classical De Casteljau Algorithm
The De Casteljau algorithm for a polynomial
Bézier curve of degree nis given by the Algo-
rithm 1. The algorithm performs iterative calcu-
lations to obtain the Bézier curve point tab[n][0].
It starts by linearly interpolating between adja-
cent control points to generate intermediate points
tabP[k]. These intermediate points are then lin-
early interpolated to obtain further intermediate
points tab[j][k]. Finally, the last step involves lin-
early interpolating between the remaining inter-
mediate points to obtain the nal point tab[n][0]
on the Bézier curve.
The use of a table allows for the iterative con-
struction of a Bézier curve by replacing, at each
iteration, a Bézier curve with two Bézier curves
of the same degree. The control points of these
new curves are either the rst elements of each
column or the last elements of each column (the
table forms an upper triangular table). Note that
at each iteration, the tangents at the endpoints of
the Bézier curves are dened either by the rst two
control points or by the last two control points.
Algorithm 1 De Casteljau Algorithm for polyno-
mial Bézier curves of degree n.
Input : let tabP be a table of n+ 1 points and Ω
an other point.
Denition of the table tab of dimension (n+ 1) ×
(n+ 1)
For k= 0 To n
tab[0][k]←tabP[k]
For j= 1 To n
For k= 0 To n−j
−−−−−−−→
Ωtab[j][k]←(1 −t)−−−−−−−−−−→
Ωtab[j−1][k]
+t−−−−−−−−−−−−−→
Ωtab[j−1][k+ 1]
(2)
Output : the point tab[n][0]of the polynomial
Bézier curve with n+ 1 control points in tabP.
The tangent to the curve at the point tab[n][0]is
the line (tab[n−1][0]tab[n−1][1]).
Let us detail the Agorithm 1 for degrees 2and
3by explaining the iterative constructions.
2.1 Quadratic case
2.1.1 De Casteljau algorithm
With n= 2, the Algorithm 1 allows building
a parabola arc CPusing a polynomial quadratic
Bézier curve with control points P0,P1and P2.
To simplify the gures, let us denote tab [1] [0] =
Q0(t),tab [1] [1] = Q1(t)and tab [2] [0] = R0(t)
which is a point of the curve CP, Figure 3 for t=1
2.
2.1.2 Iterative construction
The polynomial quadratic Bézier curve with con-
trol points P0,P1and P2is cut into two polyno-
mial quadratic Bézier curves γ0with control points
P0,Q0(t)and R0(t)on the one hand and γ1with
control points R0(t),Q1(t)and P2on the other
hand, Figure 3. Moreover, the line (Q0(t)Q1(t))
is the tangent to the curves γ0and γ1at the point
R0(t). If Ωequals O, the elements of the array
tab are P0Q0(t)R0(t)
P1Q1(t)
P2
(3)
and the points of the rst line of tab correspond to
the control points of the rst sub-curve of Bézier,
whereas the points on the diagonal of tab corre-
spond to the control points of the second sub-curve
of Bézier. The common point belonging to the two
sub-curves is R0(t)and the tangent to these sub-
curves at R0(t)is the line (Q0(t)Q1(t)).
2.1.3 Comparison between De Casteljau
algorithm and iterative construction
Consider the example of a Bézier curve of con-
trol points P0(−4,0),P1(0,4) and P2(4,0).
The points A−3,7
8,B−2,3
2,C−1,15
8,
D(0,2),E1,15
8,F2,3
2and G3,7
8are
computed by De Casteljau algorithm or the iter-
ative construction, Figures 4. The Table 2 pro-
vides the dierent points constructed by the two
algorithms at dierent steps. Seven iterations are
needed with the De Casteljau algorithm whereas
only three are needed with the iterative method
(with four threads). In the Figure 4:
• the square point designates the points which
is built at rst with De Casteljau algorithm;
• the triangle points designate the points which
are built in the same time with De Casteljau
algorithm and the iterative construction;
• the pentagon points designates the points
which are built at rst with the iterative con-
struction.
In the Figure 4, the point Ais computed by the
De Casteljau algorithm and the point Dis built
using the iterative construction.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
218
Volume 23, 2024
Figure 3: Iterative construction of a Bézier curve based on De Casteljau method, Algorithm 1 with n= 2
and t=1
2,CP=γ0∪γ1and R01
2=γ0∩γ1.
Constructed point on the Bézier curve A B C D E F G
Step of De Casteljau algorithm 1234 5 6 7
Step of the iterative construction 3231323
Table 2: Comparison between the steps : sequential De Casteljau Algorithm or the iterative construction
with four threads.
The point Bis computed by the De Castel-
jau algorithm and by the iterative construction
whereas the point Fis built using the iterative
construction.
The point Cis computed by the De Castel-
jau algorithm and by the iterative construction
whereas the points Eand Gare built using the
iterative construction.
2.2 Cubic case
With n= 3, the Algorithm 1 allows building
a polynomial cubic Bézier curve γwith control
points P0,P1,P2and P3. To simplify the g-
ures, let us notate tab [1] [0] = Q0(t),tab [1] [1] =
Q1(t),tab [1] [2] = Q2(t),tab [2] [0] = R0(t),
tab [2] [1] = R1(t), and tab [3] [0] = S0(t)which
is a point of the curve γ, Figure 5 for t=1
2.
The polynomial cubic Bézier curve with control
points P0,P1,P2and P3is cut into two polyno-
mial cubic Bézier curves γ0with control points P0,
Q0(t),R0(t)and S0(t)on the one hand and γ1
with control points S0(t),R1(t),Q2(t)and P3on
the other hand. Moreover, the line (R0(t)R1(t))
is the tangent to the curves γ0and γ1at the point
S0(t). Let Ωbe the point O, the elements of the
array tab are
P0Q0(t)R0(t)S0(t)
P1Q1(t)R1(t)
P2Q2(t)
P3
(4)
and the points of the rst line of tab correspond to
the control points of the rst sub-curve of Bézier
γ0, whereas the points on the diagonal of tab cor-
respond to the control points of the second sub-
curve of Bézier γ1. The common point belonging
to the two sub-curve is S0(t)and the tangent to
these sub-curves at S0(t)is the line (R0(t)R1(t)).
Unfortunately, regular construction for rational
Bézier curves is not possible using the projective
De Casteljau algorithm. One solution could be to
recalculate the weights at each step [7], which is
computationally heavy and time-consuming. So,
another method, which generalises the usual De
Casteljau algorithm must be developed.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
219
Volume 23, 2024
Figure 4: Comparison between sequential De Casteljau algorithm and iterative construction of a Bézier
curve, third iteration.
Figure 5: Iterative construction of a Bézier curve based on De Casteljau method, Algorithm 1 with n= 3
and t=1
2.
3 Homographic parameter change for
the De Casteljau algorithm
The purpose of this section is to construct, iter-
atively and with regular sampling, points on a
Bézier curve. To achieve this, the formula (1) gives
the expression of the irregularity of the subdivision
in the case of a rational Bézier curve, we replace
the original Bézier curve with two Bézier curves of
the same degree. If the initial Bézier curve is in
standard form, we obtain two curves in standard
form, unless the mass endpoint of the curve is a
vector.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
220
Volume 23, 2024
3.1 Rational Bézier curves in e
P
In the following (O;−→
ı;−→
ȷ)designates a direct ref-
erence frame in the usual Euclidean ane plane P
and −→
Pis the set of vectors of the plane. The set
of mass points is dened by
e
P= (P × R∗)∪−→
P × {0}
On the mass point space, the addition, denoted
⊕, is dened as follows
•ω6= 0 =⇒(M;ω)⊕(N;−ω) = ω−−→
NM; 0;
•ω µ (ω+µ)6= 0 =⇒(M;ω)⊕(N;µ) =
Bar (M;ω) ; (N;µ);ω+µwhere
Bar (M;ω) ; (N;µ)denotes the barycenter
of the weighted points (M;ω)and (N;µ);
•(−→
u; 0) ⊕(−→
v; 0) = (−→
u+−→
v; 0);
•ω6= 0 =⇒(M;ω)⊕(−→
u; 0) = T1
ω−→
u(M) ; ω
where T−→
Wis the translation of Pof vector −→
W.
In the same way, on the space e
P, the multipli-
cation by a scalar, denoted , is dened as follows
•ω α 6= 0 =⇒α(M;ω) = (M;α ω)
•ω6= 0 =⇒0(M;ω) = −→
0 ; 0
•α(−→
u; 0) = (α−→
u; 0)
One can note that e
P,⊕,is a vector space
[16]. So, a mass point is a weighted point (M, ω)
with ω6= 0 or a vector (−→
u , 0). The Bernstein
polynomials of degree nare dened by
Bi,n (t) = n
i(1 −t)n−iti(5)
These Bernstein polynomials provide the def-
inition of rational Bézier curve (BR curve) in e
P
given below.
Denition 1 Rational Bézier curve (BR curve) in
e
PLet (Pi;ωi)i∈[[0;n]] n+1 mass points in e
P. Dene
two sets
I={i|ωi6= 0}and J={i|ωi= 0}
Dene the weight function ωfas follows
ωf: [0; 1] −→ R
t7−→ ωf(t) = X
i∈I
ωi×Bi(t)(6)
A mass point (M;ω)or (−→
u; 0) lays to the ratio-
nal Bézier curve dened by the control mass points
(Pi;ωi)i∈[[0;n]] if there is a real t0in [0; 1] such that:
•if ωf(t0)6= 0 then
−−→
OM =1
ωf(t0) X
i∈I
ωiBi(t0)−−→
OPi!
+1
ωf(t0) X
i∈J
Bi(t0)−→
Pi!
ω=ωf(t0)
(7)
•if ωf(t0) = 0 then
−→
u=X
i∈I
ωiBi(t0)−−→
OPi+X
i∈J
Bi(t0)−→
Pi(8)
Such a curve is denoted BR n(Pi;ωi)i∈[[0;n]]o
If J=∅, this denition leads to the usual ra-
tional Bézier curve.
The Algorithm 1 can be generalized to use mass
points: the Equation (2) is replaced by
tab[j][k]←(1 −t)tab[j−1][k]
⊕ttab[j−1][k+ 1](9)
but the issue regarding the weight of the con-
structed point remains.
3.2 Homographic parameter change
To achieve regular constructions, the weights of
mass endpoints equal 0in the case of a vector and
1in the case of a weighted point.
3.2.1 Denition and fundamental theorem
Let αand βbe two distinct reals. The homo-
graphic change allows obtaining, without increas-
ing the degree of the curve, the portion of the curve
γdened on the interval [α;β],[α, +∞]or [−∞, β]
by using the Bézier curve γ◦hover the interval
[0; 1] i.e. h([0,1]) = [α, β],h([0,1]) = [α, +∞]or
h([0,1]) = [−∞, α].
Theorem 1 : Homographic parameter change
Let γbe a Bézier curve of degree nof control
mass points ((Pi;ωi))i∈[[0;n]].
Let hbe the homographic function from Rto
Rdened by
h(u) = a(1 −u) + bu
c(1 −u) + du(10)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
221
Volume 23, 2024
Then γ◦his a Bézier curve of degree nof con-
trol mass points ((Qi;ϖi))i∈[[0;n]] and their expres-
sions depend on the control points ((Pi;ωi))i∈[[0;n]]
and on the values a,b,cand d, see [17].
One can note that his monotone since the sign
of h0(u)is
b−a a
d−c c =
b a
d c =bc −ad 6= 0
In this paper, to obtain a regular construction,
the condition are
h([0,1]) = 0,1
2(11)
and
h([0,1]) = 1
2,1(12)
The restriction of hto the interval 0,1
2is de-
noted as h1, while the restriction of hto the inter-
val 1
2,1is denoted as h2.
Corollary 1 (h1dened by Formula (11) )
Let ((Pi;ωi))i∈[[0;n]] be n+1 control mass points
of a Bézier curve γof degree n.
Let band cbe two non null real numbers. Let
h1be dened by
h1:R−→ R
u7−→ b u
c(1 −u)+2b u
(13)
Then γ◦h1is a Bézier curve of degree nof control
mass points ((Qi;ϖi))i∈[[0;n]] and their expressions
depend on the control points ((Pi;ωi))i∈[[0;n]] and
on the values band c, see Tables 3, 5 and 7.
Moreover, the function h1is monotonically in-
creasing if b×c > 0.
::::::
Proof: :Using the fonction h1dened by the
Equation (10).
h1(0) = a
c= 0 =⇒a= 0
h1(1) = b
d=1
2=⇒d= 2b
Moreover
b a
d c =
b0
2b c =b×c
■
Corollary 2 (h2dened by Formula (12) )
Let ((Pi;ωi))i∈[[0;n]] be n+1 control mass points
of a Bézier curve γof degree n.
Let aand dbe two non null real numbers. Let
h2be dened by
h2:R−→ R
u7−→ a(1 −u) + d u
2a(1 −u) + d u
(14)
Then γ◦h2is a Bézier curve of degree nof control
mass points ((Qi;ϖi))i∈[[0;n]] and their expressions
depend on the control points ((Pi;ωi))i∈[[0;n]] and
on the values aand d, see Tables 4, 6 and 8.
Moreover, the function h2is monotonically in-
creasing if a×d > 0.
::::::
Proof: :Using the fonction h2dened by the
Equation (10).
h2(0) = a
c=1
2=⇒c= 2a
h2(1) = b
d= 1 =⇒b=d
Moreover,
b a
d c =
d a
d2a=
d a
0a=a×d
■
3.2.2 Degree 2case
The Table 3 denes the control mass points using
the function h1dened by the Equation (13) with
a= 0,d= 2band c= 1 to keep the rst control
mass point. The value of bis calculated so that
the weight of the last weighted point is equal to 1
or the vector is unchanged.
The Table 4 denes the control mass points us-
ing the function h2dened by the Equation (14)
with c= 2aand b=d= 1 to keep the last control
mass point. The value of ais calculated so that
the weight of the rst weighted point is equal to 1
or the vector is unchanged.
First example : three-quarters of circle of cen-
ter O(0,0), of radius r= 2
The control mass points of the quadratic Bézier
curve are P0(2,0),ω0= 1,P1(2,2),ω1=−√2
2
and P2(0,0),ω2= 1, Figure 6.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
222
Volume 23, 2024
(Q0;ϖ0) = (P0, ω0)
(Q1;ϖ1) = b((P0, ω0)⊕(P1, ω1))
(Q2;ϖ2) = b2((P0, ω0)⊕2(P1, ω1)⊕(P2, ω2))
(15)
Table 3: Control mass points for the quadratic case for h1.
(Q0;ϖ0) = a2((P0, ω0)⊕2(P1, ω1)⊕(P2, ω2))
(Q1;ϖ1) = a((P1, ω1)⊕(P2, ω2))
(Q2;ϖ2) = (P2, ω2)
(16)
Table 4: Control mass points for the quadratic case for h2.
First iteration, function h1
Directely, (Q0; 1) = (P0; 1), Figure 6. The
value of bmust be determined such that the last
weight is equal to 1which leads to the equation
b2 1−2×√2
2+ 1!= 1
and the positive solution is
b=s1
2−√2=s2 + √2
2
Let I1be the midpoint of the segment [P0P2].
(P0,1) ⊕2 P1,−√2
2!⊕(P2,1) =
(I1,2) ⊕P1,−√2=G1,2−√2where
G1−√2,−√2and then
(Q2;ϖ2) = (G1; 1)
(P0, ω0)⊕(P1, ω1) = (P0,1) ⊕
P1,−√2
2!= G2,2−√2
2!with the point
G22; −2√2+1and then
(Q1;ϖ1) =
G2;q2−√2
2
First iteration, function h2
The points of this curve are changed into the
points R0,R1and R2instead of Q0,Q1and Q2.
The value of amust be determined such that the
rst weight is equal to 1which leads to the equa-
tion
a2 1−2×√2
2+ 1!= 1
and the positive solution1is
a=b
and then
(R0;ϖ0) = (G1; 1)
Concerning the point P1:
(P1, ω1)⊕(P2, ω2)
= P1,−√2
2!⊕(P2,1)
= G3,2−√2
2!
with the point G3−2√2+1,2and then
(R1;ϖ1) =
G3;q2−√2
2
and (R2;ϖ2) = (P2; 1), Figure 6.
Second iteration
The quadratic Bézier curve with control mass
points (Qi, ωi),i∈[[0,2]], is split into two
quadratic Bézier curves with control mass points
(Si, ϖi),i∈[[0,2]] on the one hand, and (Ti, υi),
i∈[[0,2]] on the other hand, Figure 6.
• The value of bis the positive solution of the
equation
b2
1+2×q2−√2
2+ 1
= 1
1Always, ais equal to b.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
223
Volume 23, 2024
which leads to
b=v
u
u
u
t2−q2−√22−√2
2'0.601
The other coecients of the homographic pa-
rameter change function equal to a= 0, c = 1
equal to d'1.203.
The points are S0(2,0),S1(2,;−1.336) and
S2(0.765,−1.848). The weights are ϖ0= 1,
ϖ1= 0.831 and ϖ2= 1.
• The coecients of the homographic param-
eter change function equal to a'0.542,
b=d= 1,c'1.085. The points are
T0(0.765,−1.848),T1(−0.469,−2.359) and
T2(−1.414,−1.414). The weights are υ0= 1,
υ1= 0.831 and υ2= 1.
The quadratic Bézier curve with control mass
points (Ri, ωi),i∈[[0,2]], is separeted into two
quadratic Bézier curves with control mass points
(Ui, ϖi),i∈[[0,2]] on the one hand, and (Vi, υi),
i∈[[0,2]] on the other hand, Figure 6.
• The coecients of the homographic parame-
ter change function equal to a= 0,b'0.601,
c= 1 d'1.203.
The points are U0(−1.414,−1.414),
U1(−2.359,−0.469) and
U2(−1.848,0.765). The weights are ϖ0= 1,
ϖ1= 0.831 and ϖ2= 1.
• The coecients of the homographic pa-
rameter change function equal to a'
0.601,d=b= 1 c'1.203. The
points are V0(−1.848,0.765),V1(−1.336,2)
and V2(0,2). The weights are υ0= 1,υ1=
0.831 and υ2= 1.
Second example : semi-circle of center
O(0,0), of radius r= 1
The control mass points of the quadratic Bézier
curve are chosen as P0(1,0),ω0= 1,−→
P1(0,1),
ω1= 0 and P2(−1,0),ω2= 1, Figure 7.
First iteration, function h1
Directely, (Q0; 1) = (P0; 1), Figure 7. The
value of bmust be determined such that the last
weight is equal to 1which leads to the equation
b2(1 + 2 ×0 + 1) = 1
and the positive solution equals to
b=√2
2
Let I1be the midpoint of the segment [P0P2].
(P0,1) ⊕2−→
P1,0⊕(P2,1)
= (I1,2) ⊕2−→
P1,0
= (G1,2)
where G1(0,1) and then
(Q2;ϖ2) = (G1; 1)
(P0, ω0)⊕(P1, ω1) = (P0,1) ⊕−→
P1,0= (G2,1)
where G2(1,1) and then
(Q1;ϖ1) = G2;√2
2!
First iteration, function h2
The points of this curve are changed into R0,
R1and R2instead of Q0,Q1and Q2. The value
of a=bleads to
(R0;ϖ0) = (G1; 1)
(P1, ω1)⊕(P2, ω2) = −→
P1,0⊕(P2,1) = (G3,1)
where G3(−1; 1) and then
(R1;ϖ1) = G3;√2
2!
and (R2;ϖ2) = (P2; 1), Figure 7.
Second iteration
The quadratic Bézier curve with control mass
points (Qi, ωi),i∈[[0,2]], is decomposed into two
quadratic Bézier curves with control mass points
(Si, ϖi),i∈[[0,2]] on the one hand, and (Ti, υi),
i∈[[0,2]] on the other hand, Figure 7.
• The value of bis the positive solution of the
equation
b2(1 + 2 ×0 + 1) = 1
which leads to
b=√2
2
The other coecients of the homographic pa-
rameter change function are a= 0, c = 1 and
d=√2. The points are S0(1,0),S1(1,0.414)
and S2(0.707,0.707). The weights are ϖ0=
1,ϖ1'0.924 and ϖ2= 1.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
224
Volume 23, 2024
Figure 6: Second step of an iterative construction of a rational quadratic Bézier curve with control weighted
points with homographic parameter change based on De Casteljau algorithm for a three-quarters of circle.
• The coecients of the homographic parame-
ter change function are a=√2
2,b=d= 1,
c=√2. The points are T0(0.707,0.707),
T1(0.414,1) and T2(0,1). The weights are
υ0= 1,υ1'0.924 and υ2= 1.
The quadratic Bézier curve with control mass
points (Ri, ωi),i∈[[0,2]], is decomposed into two
quadratic Bézier curves with control mass points
(Ui, ϖi),i∈[[0,2]] on the one hand, and (Vi, υi),
i∈[[0,2]] on the other hand, Figure 7.
• The coecients of the homographic param-
eter change function are a= 0,b'0.541,
c= 1 d'1.082. The points are U0(0,1),
U1(−0.414; 1) and U2(−0.707; 0.707). The
weights are ϖ0= 1,ϖ1'0.924 and ϖ2= 1.
• The coecients of the homographic parame-
ter change function are a'0.541,d=b= 1
c'1.082. The points are V0(−0.707; 0.707),
V1(−1; 0.414) and V2(−1; 0). The weights are
υ0= 1,υ1'0.924 and υ2= 1, Figure 7.
Third example : a branch of a hyperbola
The control mass points of the quadratic Bézier
curve are −→
P0(1,1),ω0= 0,P1(0,0),ω1= 1 and
−→
P2(1,−1),ω2= 0, Figure 8. The point P1is the
center of the hyberbola, the directions of the vec-
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
225
Volume 23, 2024
Figure 7: Iterative construction of a rational quadratic Bézier curve with control mass points with homo-
graphic parameter change based on De Casteljau algorithm for a semicircle.
tors −→
P0and −→
P2are the directions of the asymptotic
lines to the hyperbola.
Function h1
Directely, −→
Q0; 0=−→
P0; 0, Figure 8. The
value of bmust be determined such that the last
weight is equal to 1which leads to the equation
b2(0 + 2 + 0) = 1
and the positive solution is
b=√2
2
−→
P0,0⊕2(P1,1)⊕−→
P2,0=−→
P0+−→
P2,0⊕
(P1,2) = (2−→
ı , 0) ⊕(P1,2) = (G1,2) where
G1(1,0) and then
(Q2;ϖ2) = (G1; 1)
(P0, ω0)⊕(P1, ω1) = −→
P0,0⊕(P1,1) = (G2,1)
where G2(1,1) and then
(Q1;ϖ1) = G2;√2
2!
Function h2
The points of this curve are changed into R0,
R1and R2instead of Q0,Q1and Q2. The value
of a=bleads to
(R0;ϖ0) = (G1; 1)
(P1, ω1)⊕(P2, ω2) = (P1,1) ⊕−→
P2,0= (G3,1)
where G3(1; −1) and then
(R1;ϖ1) = G3;√2
2!
and (R2;ϖ2) = (P2; 1), Figure 8. The tangents to
the curves at the point Q0=R0is the line (Q1R1).
This last property will always hold true in future
constructions.
Second iteration
The quadratic Bézier curve with control mass
points (Qi, ωi),i∈[[0,2]], is decomposed into two
quadratic Bézier curves with control mass points
(Si, ϖi),i∈[[0,2]] on the one hand, and (Ti, υi),
i∈[[0,2]] on the other hand, Figure 8.
• The value of bis the positive solution of the
equation
b2 0+2×√2
2+ 1!= 1
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
226
Volume 23, 2024
which leads to
b=q√2−1'0.644
The other coecients of the homographic pa-
rameter change function are a= 0, c = 1
and d'1.287. The points are −→
S0(1; 1),
S1(2.414; 2.414) and S2(1.414,1.000). The
weights are ϖ0= 1,ϖ1'0.455 and ϖ2= 1.
• The coecients of the homographic parame-
ter change function are
a=q√2−1'0.644,b=d= 1,c'1.287.
The points are T0(1.414,1),T1(1,0.414) et
T2(1; 0). The weights are υ0= 1,υ1'1.099
and υ2= 1.
The quadratic Bézier curve with control mass
points (Ri, ωi),i∈[[0,2]], is decomposed into two
quadratic Bézier curves with control mass points
(Ui, ϖi),i∈[[0,2]] on the one hand, and (Vi, υi),
i∈[[0,2]] on the other hand, Figure 8.
• The coecients of the homographic param-
eter change function are a= 0,b'0.644,
c= 1 d'1.287. The points are U0(1,0),
U1(1,−0.414) and U2(1.414,−1.000). The
weights are ϖ0= 1,ϖ1= 1.099 and ϖ2= 1.
• The coecients of the homographic pa-
rameter change function are a'0.644,
d=b= 1 c'1.287. The points
are V0(1.414; −1.000),V1(2.414; −2.414) and
−→
V2(1; −1). The weights are υ0= 1,υ1=
0.455 and υ2= 1, Figure 8.
Applications in 5-dimensional
Minkowski-Lorentz space
The 5-dimensional Minkowski-Lorentz space is
a generalization of the space of relativity used by
A. Einstein. In this Minkowski-Lorentz space, a
Dupin cyclide is represented by two conics [13]:
a circle that appears as an ellipse or as a hyper-
bola, or an isometric parabola with respect to a
line. The points on the curve are spheres with
the Dupin cyclide being their envelope. The tan-
gent to the conic at a given point denes a sphere,
known as the derived sphere, and the intersec-
tion of these two spheres is a circle of curvature
of the Dupin cyclide [13]. The singular points of a
Dupin cyclide correspond to isotropic vectors. By
using homographic parameter transformations, it
is possible to iteratively construct circles of curva-
ture for Dupin cyclides and patches of these sur-
faces [18], [19].
3.2.3 Degree 3case
For quadratic Bézier curves with control points
(P0,1),(P1, ω1), and (P2,1), the concept of regular
construction arises from the fact that the weighted
constructed point (R0,1) lies on the median of the
triangle formed by P1. This means that we replace
a Bézier curve in standard form with two Bézier
curves in standard form. From degree 3, the no-
tion of regularity means that the standard form is
preserved.
The Table 5 denes the control mass points us-
ing the function h1dened by the Equation (13)
with a= 0,d= 2b, and c=−1to keep the rst
control mass point.
The Table 6 denes the control mass points us-
ing the function h2dened by the Equation (14)
with c= 2aand b=d=−1to keep the rst
control mass point.
First example : function x7→ x3on R+
The homographic transformation allows us to
use Bézier curves dened over the interval [0,1] in-
stead of a curve parameterized over an unbounded
interval, where one of the bounds is −∞ or +∞.
First, the conversion from the canonical basis
to the appropriate Bernstein basis is performed,
where at least one control mass point is a vector.
Then, the homographic parameter change is ap-
plied, resulting in control mass points. Finally,
we use our generalized version of the De Casteljau
algorithm.
For example, the control mass points of the cu-
bic Bézier curve which represents the curve t, t3,
t∈[0,+∞], using the changement of parameter
t=u
1−u
are P0(0,0) with ω0= 1,−→
P11
3,0with ω1= 0,
−→
P2=−→
0with ω2= 0 and −→
P3(0,1) with ω3= 0.
The generalized De Casteljau algorithm is applied
to this Bézier curve, Figures 9 and 11.
First iteration, function h1to obtain the
curve γQ
Directly, (Q0; 1) = (P0; 1), Figure 9. The value
of bmust be determined such that the last weight
is equal to 1which leads to the equation
−b3(1 + 3 ×0+3×0 + 0) = 1
and the solution is
b=−1
(P0; 1) ⊕−→
P1,0= (G1,1) where G11
3,0and
then
(Q1;ϖ1) = (G1; 1)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
227
Volume 23, 2024
Figure 8: Iterative construction of a rational quadratic Bézier curve with control mass points with homo-
graphic parameter change based on De Casteljau algorithm for a branch of a hyperbola.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
228
Volume 23, 2024
(Q0;ϖ0) = (P0;ω0)
(Q1;ϖ1) = −b((P0;ω0)⊕(P1;ω1))
(Q2;ϖ2) = b2((P0;ω0)⊕2(P1;ω1)⊕(P2;ω2))
(Q3;ϖ3) = −b3((P0;ω0)⊕3(P1;ω1))
⊕ −b3(3 (P2;ω2)⊕(P3;ω3))
(17)
Table 5: Control mass points for the cubic case for h1.
(Q0;ϖ0) = −a3((P0;ω0)⊕3(P1;ω1))
⊕ −a3(3 (P2;ω2)⊕(P3;ω3))
(Q1;ϖ1) = a2((P1;ω1)⊕2(P2;ω2)⊕(P3;ω3))
(Q2;ϖ2) = −a((P2;ω2)⊕(P3;ω3))
(Q3;ϖ3) = (P3;ω3)
(18)
Table 6: Control mass points for the cubic case for h2.
(P0,1)⊕2−→
P1,0⊕−→
P2,0= (P0,1)⊕(−→
u , 0)
where −→
u2
3,0and then −→
P1,0⊕(−→
u , 0) = (G2,1)
where G22
3,0and then
(Q2;ϖ2) = (G2; 1)
= (P0,1) ⊕3−→
P1,0⊕3−→
P2,0⊕−→
P3,1
= (P0,1) ⊕3−→
P1+−→
P3,0
= (P0,2) ⊕(−→
v , 0)
where −→
v(1,1) and then (P0,1) ⊕(−→
v , 0) = (G3,1)
where G3(1,1) and then
(Q3;ϖ3) = (G3; 1)
First iteration, function h2to obtain the
curve γR
The points of this curve are changed into R0,
R1,R2and R3instead of Q0,Q1,Q2and Q3. The
value of a=bleads to (R0; 1) = (Q3; 1), Figure 9.
−→
P1,0⊕2−→
P2,0⊕−→
P3,0=−→
P3,0⊕
−→
P1,0=−→
G3,0where −→
G31
3,1and then
−→
R1;ϖ1=−→
G3; 0
−→
P2,0⊕−→
P3; 0=−→
P3; 0and then
−→
R2;ϖ2=−→
P3; 0
Second iteration
The cubic Bézier curve with control mass
points (Qi, ωi),i∈[[0,3]], is decomposed into
two cubic Bézier curves with control mass points
(Si, ϖi),i∈[[0,3]] on the one hand, and (Ti, υi),
i∈[[0,3]] on the other hand, Figure 10.
• The coecients of the homographic param-
eter change function are a= 0, b =−1
2,
c=−1and d=−1. The points are S0(0,0),
S1(0.167,0),S2(0.333,0) and S3(0.5,0.125).
The weights are ϖ0= 1,ϖ1= 1,ϖ2= 1 and
ϖ3= 1.
• The coecients of the homographic parame-
ter change function are a=−1
2,b=c=
d=−1. The points are T0(0.5,0.125),
T1(0.667,0.25),T2(0.833,0.5) and T3(1,1).
The weights are υ0= 1,υ1= 1,υ2= 1 and
υ3= 1.
The cubic Bézier curve with control mass
points (Ri, ωi),i∈[[0,3]], is decomposed into
two cubic Bézier curves with control mass points
(Ui, ϖi),i∈[[0,3]] on the one hand, and (Vi, υi),
i∈[[0,3]] on the other hand, Figure 11.
• The coecients of the homographic param-
eter change function are a= 0,b=−1,
c=−1d=−2. The points are U0(1,1),
U1(1.333,2),U2(1.667,4) and U3(2,8). The
weights are ϖ0= 1,ϖ1= 1,ϖ2= 1 and
ϖ3= 1
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
229
Volume 23, 2024
Figure 9: Iterative construction of a rational cubic Bézier curve with control mass points with homographic
parameter change based on De Casteljau algorithm for the cubic curve x7→ x3on [0,+∞].
• The coecients of the homographic param-
eter change function are a=−1,d=
b=−1c=−2. The points are V0(2; 8),
−→
V1(0.333,4),−→
V2(0; 2) and −→
V3(0; 1). The
weights are υ0= 1,υ1= 0,υ2= 0 and υ3= 0.
Second example : loop of a Descartes
Folium
The generalized De Casteljau algorithm is ap-
plied to the loop of the Descartes Folium with
parameter a= 2. This loop is modeled by the
cubic rational Bézier curve γwith control mass
points P0(0; 0),ω0= 1,−→
P1(2; 0),ω1= 0,−→
P2(0; 2),
ω2= 0 and P3=P0,ω3= 1, Figure 12.
First iteration, function h1
Directely, (Q0; 1) = (P0; 1), Figure 12.
The value of bmust be determined such that
the last weight is equal to 1which leads to the
equation
−b3(1 + 3 ×0+3×0 + 1) = 1
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
230
Volume 23, 2024
Figure 10: Second iterative construction of the
rational cubic Bézier curve γQwith control mass
points with homographic parameter change based
on De Casteljau algorithm for the function x7→ x3
on [0,+∞].
and the solution is
b=−1
3
√2
(P0; 1) ⊕−→
P1,0= (G1,1) where G1(2,0) and
then
(Q1;ϖ1) = G1;−1
3
√2
(P0,1)⊕2−→
P1,0⊕−→
P2,0= (P0,1)⊕(−→
u , 0)
where −→
u(4,2) and then −→
P1,0⊕(−→
u , 0) = (G2,1)
where G2(4,2) and then
(Q2;ϖ2) = G2;1
3
√4
(P0,1) ⊕3−→
P1,0⊕3−→
P2,0⊕(P3,1) =
(P0,2) ⊕3−→
P1+ 3−→
P2,0=(P0,2) ⊕(−→
v , 0) where
−→
v(6,6) and then (P0,2) ⊕(−→
v , 0) = (G3,2) where
G3(3,3) and then
(Q3;ϖ3) = (G3; 1)
Figure 11: Second iterative construction of the
rational cubic Bézier curve γRwith control mass
points with homographic parameter change based
on De Casteljau algorithm for the function x7→ x3
on [0,+∞].
First iteration, function h2
The points of this curve are changed into R0,
R1,R2and R3instead of Q0,Q1,Q2and Q3. The
value of a=bleads to (R0; 1) = (Q3; 1), Figure 12.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
231
Volume 23, 2024
−→
P1,0⊕2−→
P2,0⊕(P3,1) = (P3,1)⊕(−→
v , 0)
where −→
v(2,4) and then (P3,1) ⊕(−→
v , 0) = (G3,1)
where G3(2,4) and then
(R1;ϖ1) = G3;1
3
√4
−→
P2,0⊕(P3; 1) = (G4,1) where G4(0,2) and
then
(R2;ϖ2) = G4;−1
3
√2
Second iteration
The cubic Bézier curve with control mass
points (Qi, ωi),i∈[[0,3]], is decomposed into
two cubic Bézier curves with control mass points
(Si, ϖi),i∈[[0,3]] on the one hand, and (Ti, υi),
i∈[[0,3]] on the other hand, Figure 12.
• The coecients of the homographic param-
eter change function are a= 0, b '0.542,
c= 1 and d'1.085. The points are
S0(0,0),S1(0.885,0),S2(1.770,0.392) and
S3(2.443,1.081). The weights are ϖ0= 1,
ϖ1'0.973,ϖ2'0.946 and ϖ3= 1.
• The coecients of the homographic pa-
rameter change function are a'0.542,
b=d= 1,c'1.085. The
points are T0(2.443,1.081),T1(3.153,1.808),
T2(3.386,2.614) and T3(3; 3). The weights
are υ0= 1,υ1'0.898,υ2'0.884 and υ3= 1.
The cubic Bézier curve with control mass
points (Ri, ωi),i∈[[0,3]], is decomposed into
two cubic Bézier curves with control mass points
(Ui, ϖi),i∈[[0,3]] on one hand, and (Vi, υi),
i∈[[0,3]] on the other hand, Figure 12.
• The coecients of the homographic parame-
ter change function are a= 0,b' −0.542,
c=−1d' −1.085. The points are
U0(3; 3),U1(2.614,3.386),U2(1.808,3.153)
and U3(1.081,2.443). The weights are ϖ0=
1,ϖ1'0.884,ϖ2'0.898 and ϖ3= 1.
• The coecients of the homographic param-
eter change function are a' −0.542,
d=b=−1c' −1.085. The
points are V0(1.081,2.443),V1(0.391,1.770),
V2(0; 0.885) and V3(0; 0). The weights are
υ0= 1,υ1= 0.946,υ2= 0.973 and υ3= 1.
3.2.4 Degree 4case
The Table 7 denes the control mass points using
the function h1dened by the Equation (13) with
a= 0,d= 2band c= 1 to keep the rst control
mass point.
The Table 8 denes the control mass points us-
ing the function h2dened by the dened by the
Equation (14) with c= 2aand b=d= 1 to keep
the rst control mass point.
The generalized De Casteljau algorithm is ap-
plied to the loop of the Bernouilli Lemniscate.
This loop is modeled by the quartic rational Bézier
curve γwith control mass points P0(0; 0),ω= 1,
−→
P11
4;1
4,ω1= 0,−→
P2=−→
0,ω2= 0,−→
P31
4;−1
4,
ω3= 0 and P4=P0,ω4= 1, Figure 13.
First iteration
Function h1
Directely, (Q0; 1) = (P0; 1), Figure 13.
The value of bmust be determined such that
the last weight is equal to 1which leads to the
equation
b4(1 + 4 ×0+6×0+4×0 + 1) = 1
and the positive solution is
b=1
4
√2
(P0; 1) ⊕−→
P1,0= (G1,1) where G11
4;1
4and
then
(Q1;ϖ1) = G1;1
4
√2
(P0,1)⊕2−→
P1,0⊕−→
P2,0= (P0,1)⊕(−→
u , 0)
where −→
u1
2;1
2and then −→
P1,0⊕(−→
u , 0) = (G2,1)
where G21
2;1
2and then
(Q2;ϖ2) = G2;1
√2
= (P0,1) ⊕3−→
P1,0⊕3−→
P2,0⊕−→
P3,0
= (P0,1) ⊕3−→
P1+−→
P2,0
= (P0,1) ⊕(−→
v , 0)
where −→
v1,1
2and then (P0,1)⊕(−→
v , 0) = (G3,1)
where G31,1
2and then
(Q3;ϖ3) = G3;1
4
√8
Then
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
232
Volume 23, 2024
Figure 12: Second iterative construction of a rational cubic Bézier curve with control mass points with
homographic parameter change based on De Casteljau algorithm for the loop of a Descartes Folium.
(P0,1) ⊕4−→
P1,0⊕6−→
P2,0
⊕4−→
P3,0⊕(P0,1)
= (P0,2) ⊕3−→
P1+ 3−→
P3,0
= (P0,2) ⊕(2−→
ı , 0)
= (G4,1)
where G4(1,0) and then
(Q4;ϖ4) = (G4; 1)
Function h2
The points of this curve are changed into R0,
R1,R2,R3and R4instead of Q0,Q1,Q2,Q3and
Q4. Then
a=1
4
√2
We have (R0; 1) = (Q4; 1), Figure 13.
−→
P1,0⊕3−→
P2,0⊕3−→
P3,0⊕(P4,1) =
(P4,1) ⊕(−→
v , 0) where −→
v1,1
2and then (P3,1) ⊕
(−→
v , 0) = (G5,1) where G51,1
2and then
(R1;ϖ1) = G5;1
4
√8
−→
P2,0⊕2−→
P3,0⊕(P4; 1) = (G6,1) where
G61
2,−1
2and then
(R2;ϖ2) = G6;1
√2
−→
P3,0⊕(P4; 1) = (G7,1) where G71
4,−1
4
and then
(R3;ϖ3) = G7;1
4
√2
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
233
Volume 23, 2024
(Q0;ϖ0) = (P0;ω0)
(Q1;ϖ1) = b((P0;ω0)⊕(P1;ω1))
(Q2;ϖ2) = b2((P0;ω0)⊕2(P1;ω1)⊕(P2;ω2))
(Q3;ϖ3) = b3((P0;ω0)⊕3(P1;ω1)⊕3(P2;ω2))
⊕b3(P3;ω3)
(Q4;ϖ4) = b4((P0;ω0)⊕4(P1;ω1)⊕6(P2;ω2))
⊕b4(4 (P3;ω3)⊕(P4;ω4))
(19)
Table 7: Control mass points for the quartic case for h1.
(Q0;ϖ0) = a4((P0;ω0)⊕4(P1;ω1)⊕6(P2;ω2))
⊕a4(4 (P3;ω3)⊕(P4;ω4))
(Q1;ϖ1) = a3((P1;ω1)⊕3(P2;ω2)⊕3(P3;ω3)⊕)
⊕a3(P4;ω4)
(Q2;ϖ2) = a2((P2;ω2)⊕2(P3;ω3)⊕(P4;ω4))
(Q3;ϖ3) = a((P3;ω3)⊕(P4;ω4))
(Q4;ϖ4) = (P4;ω4)
(20)
Table 8: Control mass points for the quartic case for h2.
and
(R4;ϖ4) = (P4; 1)
Second iteration
The quartic Bézier curve with control mass
points (Qi, ωi),i∈[[0,4]], is decomposed into
two quartic Bézier curves with control mass points
(Si, ϖi),i∈[[0,4]] on the one hand, and (Ti, υi),
i∈[[0,4]] on the other hand, Figure 13.
• The coecients of the homographic param-
eter change function are a= 0, b '0.537,
c= 1 and d'1.075. The points are
S0(0; 0),S1(0.114,0.114),S2(0.228,0.228),
S3(0.366,0.319) and S4(0.529,0.346). The
ve weights are ϖ0= 1,ϖ1'0.989,ϖ2'
0.979,ϖ3'0.969 and ϖ4= 1.
• The coecients of the homographic pa-
rameter change function are a'0.537,
b=d= 1,c'1.075. The
points are T0(0.529,0.346),T1(0.706,0.376),
T2(0.878,0.327),T3(1; 0.186) and T4(1; 0).
The ve weights are υ0= 1,υ1'0.892,
υ2'0.837,υ3'0.857 and υ4= 1.
The quartic Bézier curve with control mass
points (Ri, ωi),i∈[[0,4]], is decomposed into
two quartic Bézier curves with control mass points
(Ui, ϖi),i∈[[0,4]] on the one hand, and (Vi, υi),
i∈[[0,4]] on the other hand, Figure 13.
• The coecients of the homographic parame-
ter change function equals to
a= 0,b'0.538,c= 1 and d'
1.075. The points are U0(1,0),U1(1,−0.186),
U2(0.879,−0.327),U3(0.706,−0.376) and
U4(0.529,−0.346). The weights are ϖ0= 1,
ϖ1'0.857,ϖ2'0.837,ϖ3'0.892 and
ϖ3= 1
• The coecients of the homographic pa-
rameter change function are a'0.537,
d=b= 1 and c'1.075. The points
are V0(0.530,−0.346),V1(0.366,−0.319),
V2(0.228,−0.228),V3(0.114,−0.114) and
V4(0; 0). The weights are υ0= 1,υ1'0.967,
υ2'0.979,υ3'0.989 and υ4= 1.
4 Conclusion and outlook
In this article, rstly, we generalized the De
Casteljau algorithm to (rational) Bézier curves
with control mass points. Secondly, we utilized
a homographic change theorem to subdivide a
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
234
Volume 23, 2024
Figure 13: Iterative construction of a rational quartic Bézier curve with control mass points with ho-
mographic parameter change based on De Casteljau algorithm for the loop of a Bernouilli Lemniscate.
Bézier curve into two Bézier curves of the same
degree. To achieve this, we mapped the inter-
val [0,1] to 0,1
2and 1
2,1, respectively. We
applied these subdivisions to centered conics, the
Descartes Folium loop, and the Bernoulli Lemnis-
cate loop, with one weighted control point being
the null vector.
In the future, we plan to work on the kinemat-
ics of Bézier curves by controlling velocity vectors
at the endpoints of the curve using a quadratic
parameter change. Additionally, we intend to
explore Bézier curves in the plane with complex
weights.
References:
[1] P. Bézier. Courbe et surface, volume 4. Her-
mès, Paris, 2ème edition, Octobre 1986.
[2] P. De Casteljau. Mathématiques et CAO.
Volume 2 : formes à pôles. Hermes, 1985.
[3] Gerald Farin. From conics to nurbs: A tuto-
rial and survey. IEEE Comput. Graph. Appl.,
12(5):78–86, September 1992.
[4] E.T.Y. Lee. The rational Bézier represen-
tation for conics. In G. Farin (ed.), edi-
tor, In Geometric Modeling, Algorithms and
New Trends, SIAM, pages 3–19, Philadelphia,
1985.
[5] G. Farin. Curves And Surfaces. Academic
Press, 3ème edition, 1993.
[6] G. Farin. Curves and Surfaces for Computer
Aided Geometric Design. Academic Press,
San Diego, 4 edition, 1997.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
235
Volume 23, 2024
[7] G. Farin. NURBS from Projective Geometry
to Pratical Use. A K Peters, Ltd, 2 edition,
1999. ISBN 1-56881-084-9.
[8] J. Peters. Geometric continuity. In Handbook
of Computer Aided Geometric Design, pages
193–229. Elsevier, 2002.
[9] G. Casciola, S. Morigi, and J. Sánchez-Reyes.
Degree elevation for p-bézier curves. Com-
puter Aided Geometric Design, 15(4):313–
322, 1998.
[10] Javier Sánchez-Reyes. Conics in rational cu-
bic bézier form made simple. Computer Aided
Geometric Design, 108:102266, 2024.
[11] Javier Sánchez-Reyes. The uniqueness of the
rational bézier polygon is unique. Computer
Aided Geometric Design, 96:102118, 2022.
[12] L.A. Piegl and W. Tiller. The NURBS
book. Monographs in visual communication.
Springer, 1995.
[13] Lionel Garnier, Jean-Paul Bécar, and Lucie
Druoton. Canal surfaces as Bézier curves us-
ing mass points. Computer Aided Geometric
Design, 54:15–34, 2017.
[14] R. Goldman. The Fractal Nature of Bézier
Curves. Geometric Modeling and Processing,
0:3, 2004.
[15] M.F. Barnsley. Fractals Everywhere. Aca-
demic Press Professional, Inc., San Diego,
CA, USA, 1988.
[16] Lionel Garnier and Jean-Paul Bécar. Mass
points, Bézier curves and conics: a survey. In
Eleventh International Workshop on Auto-
mated Deduction in Geometry, Proceedings
of ADG 2016, pages 97–116, Strasbourg,
France, June 2016. http://ufrsciencestech.u-
bourgogne.fr/∼garnier/publications/adg2016/.
[17] Lionel Garnier. Résultat sur les courbes de
Bézier à points massiques de contrôle. Tech-
nical report, Université de Bourgogne, March
2023. working paper or preprint.
[18] Lionel Garnier, Lucie Druoton, Jean-Paul Bé-
car, Laurent Fuchs, and Géraldine Morin.
Subdivisions of Ring Dupin Cyclides Using
Bézier Curves with Mass Points. WSEAS
TRANSACTIONS ON MATHEMATICS,
20:581–596, 11 2021.
[19] Lionel Garnier, Lucie Druoton, Jean-Paul Bé-
car, Laurent Fuchs, and Géraldine Morin.
Subdivisions of Horned or Spindle Dupin Cy-
clides Using Bézier Curves with Mass Points.
WSEAS TRANSACTIONS ON MATHE-
MATICS, 20:756–776, 12 2021.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.25
Lionel Garnier, Jean-Paul Bécar, Lucie Druoton
E-ISSN: 2224-2880
236
Volume 23, 2024
