Abstract: The paper deals with the G2continuity for planar curves. The G2continuity is considered
as a superior quality of curvature, which is often sought after in high-precision designs and industrial
applications. It ensures a perfectly smooth transition between dierent parts of a surface or curve, which
can improve the functionality, aesthetics, and durability of the nished object. This article describes an
algorithm to achieve a G2junction between two sets of data –point, tangent, curvature–. The junction
is based on a rational Bézier curve dened by control mass points. The control mass points generalize
those of classical Bézier curves dened with weighted points with no negative weights. It is necessary as
vectors and points with negative weights are coming out while applying homographic parameter change
on a curve segment or converting any polynomial function into a rational Bézier representation. Here,
from two sets of data –point, tangent and curvature–, a Bézier curve of degree nis built. This curve is
described by control mass points. In most situations, the best degree for G2connection of those two sets
equals 5.
Key-Words: Mass points, Rational Quadratic and Quintic Bézier curves, G2joints, osculating circle.
Received: June 26, 2024. Revised: November 9, 2024. Accepted: December 4, 2024. Published: December 31, 2024.
1 Introduction
Bézier curves are the simplest control point curves.
They were invented in the same time by [1],
at Renault and, [2], at Citroën. Initially, any
point of these curves is the barycentric locus of
a list of weighted points called control points.
These points are weighted by Bernstein polyno-
mials, [3], [4], [5]. In case of the sum of the
weights equals zero, the barycenter no more ex-
ists, [6], [7], [8]. The result of the calculation
provides a vector. The solution that generalizes
the notion of barycenter consists in using mass
points, [9], [10]. Based on this concept and the
help of a homographic parameter change, it is pos-
sible to determine any conic feature, [11]. This
is impossible by using the concept of projective
geometry, [12]. Furthermore, this model is inde-
pendent of the metric or pseudo-metric structure.
This model can be used in the Minkowski-Lorentz
space to represent canal surfaces, [13].This model
oers also to construct Dupin cyclides as subdi-
vided surfaces, [14].Here, the paper deals with the
G2continuity. The joints are built from a ratio-
nal Bézier curve with mass control points. The
G2continuity is considered as a superior quality
of curvature, which is often sought after in high-
precision designs and industrial applications, such
as the production of three-dimensional objects like
molds or automotive parts. It ensures a perfectly
smooth transition between dierent parts of a sur-
face or curve, which can improve the functionality,
aesthetics, and durability of the nished object. In
lighting, G2geometric continuity can be applied to
the design of reectors or lenses that are used to
control the distribution of light emitted by a light
source.
In [15], a parametric cubic spline interpola-
tion scheme for planar curve is given for the con-
struction of C1bicubic parametric spline surfaces.
They are a generalization of any standard Her-
mite interpolation. The Pythagorean Hodograph
curves were introduced in [16], [17]. In [18], the
arc splines that are triarcs interpolate, match
unit tangents and curvatures at the interpolation
points. In [19], the G2blend can be achieved by
the curve dened by a pair of polynomial spiral
segments. In [20], the G2blend is composed of
cubic Bézier spirals. In [21], the G2blend is built
from a quartic rational curve obtained by an in-
version of a hyperbola. In [22], the inversion is
applied to an arc of spiral. In [23], a cubic ratio-
nal Bézier spiral (planar curves of monotonic cur-
Method for &onnection of 7wo G2 'ata 6ets by the Use of a Quintic
Rational Bézier Curve Defined with Mass Control Points
LIONEL GARNIER1, JEAN-PAUL BÉCAR2, LAURENT FUCHS3
1L.I.B., Université de bourgogne, B.P. 47870, 21078 Dijon Cedex, FRANCE
2U.P.H.F. - Campus Mont Houy - 59313 Valenciennes Cedex 9, FRANCE
3X.L.I.M., U.M.R. 7252, University of Poitiers, FRANCE
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Lionel Garnier, Jean-Paul Bécar, Laurent Fuchs
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vature) interpolates end conditions that are posi-
tions, tangents and curvatures.
In [24], using complex numbers, the use
of Pythagorean-Hodograph quintics of monotone
curvature dened in [25], is simplied. In [26], the
G2blend is computed using planar interpolation
with minimum strain energy. For a G2conncetion,
in [27], one circle is inside the other circle. In [28],
a Pythagorean-Hodograph Bézier curve, which is
not always a spiral, is used. In [29], the G2join is
built using curvature variation minimization. In
[30], the G2join is built with at most spiral seg-
ments. In [31], the authors deal with the reachable
regions for a single segment of parametric rational
cubic Bézier spiral. These curves are computed
from the given G2Hermite data. This article fo-
cuses on the G2junction between two curves using
Bézier curve of degree ndened by mass control
points. Mainly, a quintic rational Bézier curve of
that type is one anwer. In our construction, in
addition to the G2junction, we can enhace the C2
junction, meaning that velocity is taken into ac-
count at the endpoints. When both degenereted
circles are straight lines, the null vector −→
0can be
used to decrease the degree of the Bézier curve.
The section 2 details the background of mass
points and rational Bézier curve with mass control
points.
After a presentation of some results on dier-
ential properties - tangent vector and curvature- of
any rational Bézier curve with control mass points
at t= 0 and t= 1, the section 3, provides a
method for G2joints. This method takes two sets
of G2data -point, tangent and curvature for input
and returns the mass control points of a Bézier
curve of degree 5for output. The section 4 gives
some application examples of the method. A G2
connection between a circle and a straight line is
proposed. In some situation, the curve degree may
be down to 4. An example of a G2connection be-
tween to circles is detailed. Two straight lines are
also G2connected by the use of the algorithm.
The section ends with the joints rst G2and C1
and second G2et C2doing a connection between a
Bernouilli Lemniscate loop and a Descartes Folium
as they present particular examples: some control
mass-point are vectors, including the null vector.
Conclusion and perspectives are given in section 5.
2 Rational Bézier curves in f
P
In the following (O;−→
ı;−→
ȷ)denotes a direct refer-
ence frame in the usual Euclidean ane plane P
and −→
Pis the set of vectors of the plane. The set
of mass points is dened by
e
P= (P × R∗)∪−→
P × {0}
On the mass point space, the addition, de-
noted ⊕, is dened 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 dened 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, [9], [10], [11]. So, a mass point is a weighted
point (M, ω)with ω6= 0 or a vector (−→
u , 0).
The Bernstein polynomials of degree n
Bi,n (t) = n
i(1 −t)n−iti
dene rational Bézier curve of degree n.
Denition 1 [Rational Bézier curve (BR curve)
in e
P]
Let (Pi;ωi)i∈[[0;n]] n+ 1 mass points in e
P.
Dene two sets
I={i|ωi6= 0}and J={i|ωi= 0}
Dene the weight function ωfas follows
ωf: [0; 1] −→ R
t7−→ ωf(t) = X
i∈I
ωi×Bi(t)
A mass point (M;ω)or (−→
u; 0) lays to the ratio-
nal Bézier curve dened by the control mass points
(Pi;ωi)i∈[[0;n]] if there is a real t0in [0; 1] such that:
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•if ωf(t0)6= 0 then
ω=ωf(t0)
−−→
OM =1
ω X
i∈I
ωiBi(t0)−−→
OPi!
+1
ωf(t0) X
i∈J
Bi(t0)−→
Pi!(1)
•if ωf(t0) = 0 then
−→
u=X
i∈I
ωiBi(t0)−−→
OPi+X
i∈J
Bi(t0)−→
Pi
Such a curve is denoted BR n(Pi;ωi)i∈[[0;n]]o
Using ⊕and , the mass point (M;ω)is writ-
ten as
(M;ω) = ⊕
X
i∈I∪J
Bi(t)(Pi;ωi)
where ⊕
X
i∈I∪J
denotes a sum of ⊕.
If J=∅, this denition leads to the usual ra-
tional Bézier curve.
3 Method to build a G2junction
curve
The section highlights some tools used to compute
control mass points of the junction curve. The k-
level line of a determinant and formal expressions
of velocity and curvature are applied in the G2
building method. All calculation are detailed in
the Appendix.
3.1 Tools
3.1.1 The k-level line for a determinant
The following result is used for the computation
of the control points.
Proposition 1 : Considering a determinant
det (−→
u , −→
v)
of two vectors −→
uand −→
v, if one is xed, we get a
function of the other vector whose k-level curves
are straight lines.
Proof : Let (0,ı, ȷ)be an orthoromal frame, k
a real number and M(x, y)a point in the plane
such that det ı,
OM=k. Then,
det ı,
OM= det (ı, yȷ) = y=k
Conversely, any point of the straight line pass-
ing by Hsuch that −−→
OH =kȷ and directed by ı is
convenient, see Figure 1.
Figure 1: k-level line of a determinant.
3.1.2 Curvature and joints
Denition 2 (a curve curvature)
Let γbe a plane curve dened by a regular C2
parametrisation γ. The curvature ρ(t0)in a point
γ(t0)of γsatises
ρ(t0) =
det
−→
dγ
dt (t0) ; −−→
d2γ
dt2(t0)
−→
dγ
dt (t0)
3
The osculating circle to a curve γ, at a point
M0where the curvature not vanishes is the best
approximation, [7], [32], of curve arc in the neigh-
bourhood of M0. It is the only tangent circle to
γat M0possessing three common points with γ.
The curvature radius R(t0)equals the inverse of
curvature ρ(t0)to this curve at that point.
If the curvature ρ(t0)equals zero thus the cur-
vature radius R(t0)is equal to innity. It implies
that the center of osculating circle is moved at in-
nity, [33]. In the case of −−→
d2γ
dt2(t0)is non-collinear
to −→
dγ
dt (t0), the osculating curvature center Ω (t0)
at M0=γ(t0)is given by
−−−−−→
M0Ω (t0) = R(t0)−→
np(t0)
where −→
np(t0)is the main unit normal vector to γ
at point γ(t0)witch veries
−→
np(t0)•−−→
d2γ
dt2(t0)>0
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Denition 3 (geometrical joint)
Let γ0and γ1be two plane curves respectively
dened on [a;b]and [c;d].
the geometrical joint between γ0and γ1is:
•G0if γ0(b) = γ1(c);
•G1if γ0(b) = γ1(c)and −−→
dγ0
dt (b) = λ−−→
dγ1
dt (c),
with λ∈R∗, thus the curve share the same
tangent line at the contact point;
•G2if the joint is G1ργ0(b) = ργ1(c)and
−→
np
γ0(t0) = −→
np
γ1(t0)thus the curve share the
same osculating circle at the contact point.
3.1.3 Velocity, acceleration and curvature at
t= 0 for a rational Bézier curve with
mass control points
Let nbe an integer greater or equal to 3. Let
ω0be non null real number. Let γbe Bézier
curve with control mass points (P0;ω0),(P1;ω1),
··· ,(Pn;ωn).
Let us dene the function χby
χ:R−→ R∗
07−→ 1
t7−→ t
(2)
For any k, the vector f
Pk=−−−→
P0Pkif ωk6= 0
or f
Pk=−→
Pkif ωk= 0, using the function of For-
mula (2), the equation (1) is thus written
g
M0=1
n
X
k=0
ωkBk,n (t)
n
X
k=1
χ(ωk)Bk,n (t)f
Pk
with g
M0=−−→
P0Mif
n
X
k=0
ωkBk,n (t)6= 0 and g
M0=
−→
M0otherwise.
Velocity at t= 0
If ω1= 0 the velocity at t= 0 equals
n
ω0
−→
P1
If ω16= 0 the velocity at t= 0 equals
nω1
ω0
−−→
P0P1
Curvature at t= 0
The acceleration vector at t= 0 equals
d2−−→
P0γ
dt2(0) = 2nω0−nω1
ω2
0χ(ω1)f
P1
+1
ω0
n(n−1) χ(ω2)f
P2
(3)
The curvature ρ(0) at t= 0 is given by
ω0(n−1) χ(ω2) det
1
f
P1
f
P1;f
P2
nχ (ω1)2
f
P1
2
The curvature at t= 0 equals zero if
det f
P1;f
P2= 0
3.1.4 Velocity, acceleration and curvature at
t= 1 for a rational Bézier curve with
mass control points
Let ω0be non null real number. Let γbe Bézier
curve with mass control points (P0;ω0),(P1;ω1),
··· ,(Pn;ωn).
Let dene f
Pk=−−−→
PnPkif ωk6= 0 and f
Pk=−→
Pk
if ωk= 0, using the function of Formula (2), the
Equation (1) is written
g
Mn=1
n
X
k=0
ωkBk,n (t)
n
X
k=0
χ(ωk)Bk,n (t)f
Pk
with g
Mn=−−−→
PnMif
n
X
k=0
ωkBk,n (t)6= 0 and g
Mn=
−→
Mnotherwise.
Velocity at t= 1
• if ωn−1= 0, the velocity at Pnis given by
d
dt−−−−→
Oγ (1) = −n
ωn
−−→
Pn−1
• if ω16= 0, the velocity at Pnis given by
d
dt−−−−→
Oγ (1) = −nωn−1
ωn
−−−−→
PnPn−1
The acceleration vector at t= 1 is given by
d2−−→
Pnγ
dt2(1) = −2nωn−nωn−1
ω2
nχ(ωn−1)
]
Pn−1
+1
ωn
n(n−1) χ(ωn−2)
]
Pn−2
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Curvature at t= 1
Let nbe an integer greater or equal to 3. Let
ω0be a non null real number. Let γbe Bézier
curve with control mass points (P0;ω0),(P1;ω1),
··· ,(Pn;ωn)
The curvature ρ(1) at t= 1 equals
αndet
1
]
Pn−1
]
Pn−1;
]
Pn−2
nχ (ωn−1)2
]
Pn−1
2
where
αn=ωn(n−1) χ(ωn−2)
The curvature equals zero if
det ]
Pn−1;
]
Pn−2= 0
3.2 Method description
Input:
(P0, ω0),(P1, ω1)two rst mass control points
and ρ0the curvature at t= 0.
(Pn−1, ωn−1),(Pn, ωn)two last mass control
points and ρ1the curvature at t= 1.
Output :
(P2, ω2),(Pn−2, ωn−2)two mass control points for
the Bézier curve of degree n.
Description :
The algorithm provides the two mass control
points that dene a Bézier curve of degree nthat
joins the data at t= 0 and t= 1. The best degree
equals 5otherwise the other mass control points
can be arbitrary chosen.
Begin G2connection
Step 1 : P2computation
Case ρ06= 0
Let us dene βn=ω0(n−1).
• if ω1=ω2= 0,P2satises :
ρ0=
βndet
1
−→
P1
−→
P1;−→
P2
n
−→
P1
2
• if ω16= 0 and ω2= 0,P2satises :
ρ0=
βndet
1
−−→
P0P1
−−→
P0P1;−→
P2
nω2
1
−−→
P0P1
2
• if ω1= 0 and ω26= 0,P2satises :
ρ0=
βnω2det
1
−→
P1
−→
P1;−−→
P0P2
n
−→
P1
2
• if ω1ω26= 0,P2satises :
ρ0=
βnω2det
1
−−→
P0P1
−−→
P0P1;−−→
P0P2
nω2
1
−−→
P0P1
2‘
case ρ0= 0
• if ω1=ω2= 0, both vectors −→
P1and −→
P2are
collinear vectors.
• if ω16= 0 and ω2= 0, the vector −→
P2equals
−→
0or a direction vector of the straight line
(P0P1).
• if ω1= 0 and ω26= 0, the vector −→
P1a direction
vector of the straight line (P0P2).
• if ω1ω26= 0, the points P0,P1and P2lay on
a same straight line.
Step 2 : Pn−2calculation
Case ρ16= 0
Let us dene
δn=ωn(n−1)
and
σn=ωn(n−1) ω2
• if ωn−1=ωn−2= 0,
ρ1=
δndet
1
−−→
Pn−1;
−−→
Pn−1;−−→
Pn−2
n
−−→
Pn−1
2
• if ωn−16= 0 and ωn−2= 0
ρ1=
det
δn
−−−−→
PnPn−1
−−−−→
PnPn−1;−−→
Pn−2
nω2
n−1
−−−−→
PnPn−1
2
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• if ω1= 0 and ω26= 0,
ρ1=
det
σn
−−→
Pn−1
−−→
Pn−1;−−−−→
PnPn−2
n
−−→
Pn−1
2
• if ωn−1ωn−26= 0,
ρ1=
det
σn
−−−−→
PnPn−1
−−−−→
PnPn−1;−−−−→
PnPn−2
nω2
n−1
−−−−→
PnPn−1
2
Case ρ1= 0
• if ωn−1=ωn−2= 0, the vectors −−→
Pn−1and
−−→
Pn−2are collinear vectors.
• if ωn−16= 0 and ωn−2= 0, the vector −−→
Pn−2
equals −→
0or a direction vector of the line
(PnPn−1).
• if ωn−1= 0 and ωn−26= 0, the vector −−→
Pn−1is
one direction vector of the line (PnPn−2).
• if ωn−1ωn−26= 0, the points Pn,Pn−1and
Pn−2lay on the same straight line.
End G2 connection
Proof :
For any integer k≥3, and ω06= 0,f
Pkis dened
as follows:
• if ωk6= 0 then f
Pk=−−−→
P0Pk;
• if ωk= 0 then f
Pk=−→
Pk.
In the case of a non null curvature, P2is calcu-
lated from
ρ0=
ω0(n−1) χ(ω2) det f
P1;f
P2
nχ (ω1)2
f
P1
3
otherwise from
det f
P1;f
P2= 0
The four cases are coming out from the denition
of the χfunction. The algorithm returns the six
points (Pi, ωi)where i∈ {0,1,2, n −2, n −1, n}
as output data where nequals the rational Bézier
curve degree. The lower degree can be obtained
when the next control mass point P3starting from
P0equals Pn−2thus n= 5.
From Formula (3.2), it yields
det
1
f
P1
3f
P1;f
P2
=ρ0n χ (ω1)2
f
P1
2
(n−1) ω0χ(ω2)
and the computation of f
P2depends on the curva-
ture ρ0of the circle, the weights ω0,ω1and ω2,
the degree nof the curve and the norm of the vec-
tor f
P1.
4 Examples
4.1 G2joints between two circles
In this example, both circles match their osculat-
ing circle at any point of the circles. The input
data are a circle, a point of the circle and a point
of the tangent line at the previous point of the
circle. The rst set is composed by the points
P0(−2,2),P1(−1,2) and ρ0=4
9see the magenta
circle on Figure 2. The second set is composed by
the points P5(3,−1),P4(3,0) and ρ1=4
3see the
red circle on Figure 2. All weights are xed to 1.
The algorithm computes P2(0,3) and P3(2,0).
The weight ωiwhere i∈[[0,5]] can be chosen for
the mass control points P0,P1,P2,P3,P4and P5.
They dene the quintic Bézier curve in blue on
Figure 2 that G2connects both circles.
4.2 G2joints between a ciruclar arc and a
straight line
In this case, the circle arc matches the osculating
circle at any point of the arc. At any point of a
straight line the curvature of a straight line equals
zero as the radius of curvature is innity. The rst
set of data are dened by P0on the circle arc, P1
on the tangent line to the circle at P0. The curva-
ture ρ0equals the inverse of the radius arc. The
second set of data is composed by the two points
P4and P5and a null curvature ρ1at P5. The par-
ticular situation oers the choice of all weights ωi,
i∈[[0,5]]. After computing the points P2and P3,
the algorithm achieves the control points (Pi;ωi),
i∈[[0,5]].
Detailed data for the Figure 3.
1. At t= 0, the curvature is ρ0=√2
2and the
control mass points are given in the Table 1;
2. At t= 1, the curvature is ρ1= 0 and the
control mass points are given in the Table 2.
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Figure 2: G2blends between two circles by a ra-
tional Bézier curve of degree 5.
Table 1: Control mass points of the rationel Bézier
curve in Figure 3 at t= 0.
Point P0(1,−1) P1(3,1) P243
16,21
16
Weight ω0= 2 ω1=1
2ω2= 2
Table 2: Control mass points of the rationel Bézier
curve in Figure 3 at t= 1.
Point P39
2,3
2P4(4,1) P5(3,0)
Weight ω3= 2 ω4= 3 ω5= 1
The circle arc is dened by the equation:
γ0(t) = √2 cos (t) ; √2 sin (t), t ∈3π
4;7π
4
The two points P0and P1satisfy P0=γ07π
4
and P1=P0+ 2−−→
dγ0
dt 7π
4. The point Hdened
in the preamble equals H11
16,−11
16.
Figure 3: G2joints between a half of circle and a
segment based on a quintic rational Bézier curve
4.3 G2joints between a circle arc and a
straight line changing weights
P0(−2; 2),P1(−1; 2) and ρ0=1
2are chosen. The
Figure 4 shows the position of the line (HP2)
changing the weights, see Table 3 which gives the
coordinates of the point H. The other points
are P35
2; 1,P411
4; 1and P5(3; 1) and the other
weights value equals 1.
Table 3: The position of the point Hdepends on
the weight ω0,ω1and ω2, see Figure 4.
ω0ω1ω2H
1 1 1 (−2; 2.625)
2 3 2 −2; 109
32 = 3.40625
1 2 1 −2; 9
2
The same result is obtained at t= 1.
4.4 G2joints between half a circle and a
segment based on a quartic rational
Bézier curve
In the Figure 5 the straight line dened by Hand
−−→
P0P1cuts the segment [P5P4].
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Figure 4: Three G2blends with Bézier curves with
several values of weights.
The choice of P2=P3at the intersection makes
the curve degree down to 4. From the input data
P0(1,−1)),ω0= 2,P1(3,1)),ω0= 0.5,ρ0=√2
2
P4(3,0)),ω4= 3,P5(3,−1)),ω5= 1,ρ1= 0,
the algorithm provides H2
3,−2
3),P23,5
3and
P3=P2.
4.5 G2joints on two parallel segments
Five mass control points dening a quartic can
be chosen. A null vector is added in the Bézier
representation.
In the Figure 6 the parallel lines are G2con-
nected by a quartic curve. A mass control point
is replaced by a null vector. P0(1,1),ω0= 1,
P1(1,0),ω1= 1,−→
P2=−→
0,ω2= 0,P3(2,0),
ω3= 1,P4(2,−1) and ω4= 1.
The data for the Figure 7 follow P0(1,1),ω0=
1,−→
P10,−1
2,ω1= 0,−→
P2=−→
0,ω2= 0,−→
P30,1
2,
ω3= 0,P4(2,−1) and ω4= 1.
These results can be used to dene tube con-
nections. As illustrated on Figure 8 and Figure 9
where the tubes are revolution surfaces obtained
from the curves of Figure 6 and Figure 7.
4.6 G2between a loop of Descartes Folium
and a Bernouilli Lemniscate.
4.6.1 Bézier representation of both loops
On the Figure 10 the Lemniscate loop γPis mod-
eled by the ve mass control points: (P0; 1),
−→
P1; 0,−→
P2; 0,−→
P3; 0,(P4; 1) with P0−1
2; 0,
−→
P1−1
4;−1
4,−→
P2(0; 0),−→
P3−1
4;1
4and P4=P0ap-
plying the translation of vector −1
2−→
ı.
On the same Figure, the Descartes folium loop
γQis modeled by a rational Bézier cubic with the
following mass control points : Q0(0; 2),ωQ0= 1,
Figure 5: G2between the half-circle γ0and the
segment [BC]by a rational quartic Bézier curve γ.
Figure 6: G2blends between two parallel segments
by a quartic rational Bézier curve: four control
points and the control null vector.
−→
Q1(2; 0),ωQ1= 0,−→
Q2(0; 2),ωQ2= 0, and Q3=
Q0,ωQ3= 1, applying the translation of vector
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Figure 7: G2blends between two parallel segments
by a quartic rational Bézier curve: two control
points, two control non-null vectors and the con-
trol null vector.
1
2−→
ı+ 2−→
ȷ.
The Bézier representation of the two loops is
used to compute their osculating circles at the
given points that are P4for γPand Q0for γQ.
As −→
P2=−→
0, the curvature at P4equals zero. The
curvature at Q0equals ρ1=1
3.
The connecting curve denoted γRis a Bézier
quintic curve with mass control points (Ri, ωi),
i∈[[0,5]], dened by R0=Q0,ω0= 2,−→
R12
5,2
5,
ω1= 0,R23
2;−2,ω2= 1,R3−1; 7
3,ω3= 1,
−→
R4−6
5,0,ω4= 0,R5=P5,ω5= 1. The Fig-
ure 11 shows a G2and C1connection between the
loop of lemniscate γPand the loop of the folium γQ
by a quintic Bézier curve γRand both osculating
circles.
4.6.2 Finding C2connection for two loops
The quintic rational Bézier curve doing the G2
connection between the loop of Descartes folium
and one loop of Bernouilli lemniscate is built from
the six mass control points (Si, ωi)with i∈[[0,5]].
In case of a C2connection is necessary, the method
can also be used to compute it. Freeing S2, S3
provides a G1or C1at most. Then, S01
2,2=
Figure 8: Two G2blends using a lathe from the
curve in the Figure 6.
Q3,ω0= 2,S11
2,−2
5,ω1= 1,S217
10 ,14
10 ,
ω2= 1,S33
10 ,4
5,ω3= 1,S4−2
5,1
10 ,ω4= 2,
S5−1
2,0=P0and ω5= 1. The Figure 12 shows
aG2joints to C2connection between both loops
and also osculating circles.
Figure 13 shows a future application of G2-
continuity to handwriting modeling.
5 Conclusion and perspectives
The paper has shown a method that computes a
Bézier curve that G2connects two sets of data.
Each set of data is composed by a given tangent
line and a given curvature at the given point. The
connection curve is at least a quintic rational curve
dened by mass control points that are weighted
points with any type of weight. In the case of a
null weight a vector is obtained. The article shows
that the G2connection oers two degree of free-
dom. For a C1connection, there is one degree of
freedom left. There is no more degree of freedom
for a C2connection implying only a unique solu-
tion. The results perform a handwriting modelling
by the use of that type of Bézier curve. The thick
and thin strokes of handwriting is an application.
Further applications are on the way of the 3d do-
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Figure 9: Two G2blends using a lathe from the
curve in the Figure 7.
Figure 10: Two loops for G2connection
main as the torsion of space curves based on the
Frenet frame.
Figure 11: A quintic Bézier rational curve con-
necting G2and C1beween the loop of Descartes
folium and a loop a Bernouilli lemniscate
Figure 12: A quintic Bézier rational curve con-
necting G2and C2beween the loop of Descartes
folium and a loop a Bernouilli lemniscate
Figure 12: A quintic Bézier rational curve con-
necting G2and C2beween the loop of Descartes
folium and a loop a Bernouilli lemniscate
Figure 12: A quintic Bézier rational curve con-
necting G2and C2beween the loop of Descartes
folium and a loop a Bernouilli lemniscate
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Appendix
A Proof of Formula (3)
From the relation
−−−−→
P0γ(t) = χ(ω1)n t (1 −t)n−1f
P1
+χ(ω2)n(n−1)
2t2(1 −t)n−2f
P2
+
n
X
k=3
χ(ωi)Bk,n (t)f
Pk
the velocity vector is equal to
d
dt−−−−→
P0γ(t)
=χ(ω1)n(1 −t)n−1−t(n−1) (1 −t)n−2f
P1
+χ(ω1)n(n−1)
22t(1 −t)n−2f
P1
+χ(ω1)n(n−1)
2−t2(n−2) (1 −t)n−3f
P1
+
n
X
k=3
χ(ωi)d
dtBk,n (t)f
Pk
The acceleration vector is computed as follows
d2−−−−→
P0γ(t)
dt2
=−d00 (t)d(t)−2 (d0(t))2
(d(t))3−−−−→
P0γ(t)
−2d0(t)
(d(t))2
d
dt−−−−→
P0γ(t) + 1
d(t)
d2
dt2−−−−→
P0γ(t)
(4)
and the acceleration vector is simplied to
d2
dt2−−−−→
P0γ(t)
=χ(ω1)n(n−1) −(1 −t)n−2f
P1
−χ(ω1)n(n−1) (1 −t)n−2+t(1 −t)n−3f
P1
+χ(ω2)n(n−1)
22 (1 −t)n−2f
P2
+χ(ω2)n(n−1)
2−2 (n−2) t(1 −t)n−2f
P2
+χ(ω2)n(n−1)
2−2t(n−2) (1 −t)n−3f
P2
+χ(ω2)n(n−1)
2t2(n−2) (n−3) (1 −t)n−4f
P2
+
n
X
k=3
χ(ωi)d2
dt2Bk,n (t)f
Pk
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which, at t= 0 leads to
−−−−→
P0γ(0) = −→
0
d
dt−−−−→
P0γ(0) = nχ (ω1)f
P1
d2
dt2−−−−→
P0γ(t) = n(n−1) −2χ(ω1)f
P1+χ(ω2)f
P2
Let d(t)be dened by
d(t) = ω0(1 −t)n+ω1n t (1 −t)n−1
+ω2
n(n−1)
2t2(1 −t)n−2+
n
X
k=3
ωiBk,n (t)
and the expression of the rst derivative of dis
equal to
d0(t)
=−ω0n(1 −t)n−1
+ω1n(1 −t)n−1−(n−1) t(1 −t)n−2
+ω2
n(n−1)
22t(1 −t)n−2
−ω2
n(n−1)
2(n−2) t2(1 −t)n−3
+
n
X
k=2
ωi
d
dtBk,n (t)
and the expression of the second derivative of d
equals
d00 (t)
=ω0n(n−1) (1 −t)n−2
−2ω1n(n−1) (1 −t)n−2
+ω1n(n−1) (n−2) t(1 −t)n−2
+ω2n(n−1) (1 −t)n−2−t(1 −t)n−3
−ω2
n(n−1)
2(n−2) 2t(1 −t)n−3
−ω2
n(n−1)
2(n−3) t2(1 −t)n−3
+
n
X
k=2
ωi
d2
dt2Bk,n (t)
that leads to
d(0) = ω0
d0(0) = n(ω1−ω0)
d00 (0) = n(n−1) (ω0−2ω1+ω2)
and
d00 (0) d(0) −2 (d0(0))2
(d(0))3
=n(n−1) ω0(ω0−2ω1+ω2)−2n2(ω1−ω0)2
ω3
0
=n(n−1) (ω0−2ω1+ω2)−2n(ω1−ω0)2
ω2
0
and d0(0)
(d(0))2=nω1−ω0
ω2
0
Based on Formula (4), the acceleration vector
equals
d2
dt2−−−−→
P0γ(0)
=−2nω1−ω0
ω2
0
nχ (ω1)f
P1
+1
ω0
n(n−1) −2χ(ω1)f
P1+χ(ω2)f
P2
=−2nω1−ω0
ω2
0
nχ ω1f
P1
+1
ω0
n(n−1) −2χ(ω1)f
P1+χ(ω2)f
P2
=−2nnω1−ω0
ω2
0
+n−1
ω0χ(ω1)f
P1
+1
ω0
n(n−1) χ(ω2)f
P2
= 2nω0−nω1
ω2
0χ(ω1)f
P1
+1
ω0
n(n−1) χ(ω2)f
P2
and four cases must be distinguished for the ac-
celeration vector at P0:
• if ω1=ω2= 0,
d2
dt2−−−−→
P0γ(0) = 2n
ω0
−→
P1+nn−1
ω0
−→
P2
• if ω16= 0 and ω2= 0,
d2
dt2−−−−→
P0γ(0)
= 2nω0−nω1
ω2
0ω1−−→
P0P1
+1
ω0
n(n−1) −→
P2
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• if ω1= 0 and ω26= 0,
d2
dt2−−−−→
P0γ(0) = 2n
ω0
−→
P1+ω2
ω0
n(n−1) −−→
P0P2
• if ω1ω26= 0,
d2
dt2−−−−→
P0γ(0)
= 2nω0−nω1
ω2
0ω1−−→
P0P1
+ω2
ω0
n(n−1) −−→
P0P2
■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
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