EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6498
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A High-Order, Optimization-Free, Tangent Continuous
Approximation of Conic Sections Using Cubic Bézier

Curves

Maria Hussain1, Hafiz Abdul Wajid2,∗, Saira Aqeel1

1 Department of Mathematics, Lahore College for Women University, Lahore, Pakistan
2 Department of Electrical Engineering, Faculty of Engineering,
Islamic University of Madinah, Madinah 42351, Saudi Arabia

Abstract. This paper presents a high-order, optimization-free method for approximating conic
sections using cubic Bézier curves. By matching endpoints and tangents while analytically deter-
mining free parameters via midpoint interpolation, the method achieves unprecedented accuracy
and efficiency. For elliptic arcs, it delivers tenth-order convergence with a maximum absolute
error of just 1.2 × 10−3. Parabolic arcs are reconstructed exactly with machine-level accuracy
(3.55× 10−15 error). The approach maintains computational efficiency, processing all cases in un-
der 1.5 seconds without requiring optimization or rational forms. Its combination of mathematical
simplicity, superior accuracy, and rapid execution makes it ideal for CAD applications where both
precision and performance are critical. The robustness of the proposed method under geometric
transformations and seamless scalability to 3D surfaces further demonstrate its practical value for
industrial applications.

2020 Mathematics Subject Classifications: 65D05, 65D07, 65D15, 65D17, 65D18

Key Words and Phrases: Conic section approximation, cubic Bézier curve, tangent continuity,
absolute approximation error, approximation order, optimization-free method, CAD curve fitting

1. Introduction

Conic sections are fundamental to Computer-Aided Design (CAD) due to their wide-
ranging applications in mechanical component modelling, typography, route planning,
satellite navigation, optical systems, and even medical technologies such as lithotripsy
[1, 2]. Despite their geometric importance, conic sections cannot be directly represented
in polynomial-based CAD systems due to their non-polynomial nature. To overcome this
limitation, Bézier curves widely supported in CAD software are often used to approximate
conic arcs [3–5].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6498

Email address: hawajid@iu.edu.sa (H. A. Wajid)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 2 of 12

Among the Bézier families, rational quadratic Bézier curves (RQBCs) can represent
conics exactly by adjusting control point weights. Existing techniques [6–8] typically follow
a multi-step approach: first reconstructing the conic arc using an RQBC, then applying
further approximation, and finally visualizing the result. However, such indirect methods
suffer from non-uniqueness in weight selection and often rely on iterative tuning, which
complicates practical implementation.

More recent efforts have been made to simplify this process. Some researchers have
proposed polynomial-based direct approximations that bypass the RQBC stage [1, 9], but
many of these still involve optimization procedures to determine free parameters, limiting
computational efficiency. Others, such as Sánchez-Reyes, proposed rational cubic repre-
sentations [10], yet rationality still entails weight dependency. Nawara [11] determined
osculating conics and sextactic points for cubic curves known as Hesse pencil. Hwang and
Li [12] formulated the sufficient conditions for which the existence of a characteristic conic
connection implies the existence of a torsion-free principal connection.

This study introduces a direct, optimization-free approximation method using cubic
Bézier curves which are the lowest-degree polynomial Bézier curves capable of ensuring
tangent continuity. The method constructs the curve by matching endpoint positions and
tangents and computes the two free parameters using a midpoint interpolation condition.
This approach not only simplifies the control point calculation but also yields a high
approximation order (ten for elliptic arcs) and exact reconstruction for parabolic arcs.

Compared to existing methods, the proposed method offers three key advantages:

(i) It avoids weight tuning and guarantees a unique control point configuration unlike
RQBC-based techniques [6–8].

(ii) It has lower computational overhead, making it more suitable for real-time CAD
applications unlike optimization-based methods [9, 10].

(iii) It outperforms several established methods [4, 6, 9, 10] in terms of approximation
error.

In summary, the proposed method addresses both theoretical and practical challenges
in conic approximation. It enhances computational simplicity without sacrificing accuracy,
making it highly suitable for integration into modern geometric design systems. This con-
tribution provides a robust and lightweight alternative to rational or optimization-based
approximations, with potential implications for precision-critical CAD environments. The
rest of the paper is organized as follows. Section 2 introduces cubic Bézier approximation
method for conic sections. In Section 3, results are presented solving a range of problems
followed by conclusion Section 4 presenting key insights.

2. Cubic Bézier Approximation Method for Conic Sections

In this section, a new approximation method is derived to approximate conic sections
(ellipse and parabola) by the famous cubic Bézier curve. The outline of the proposed



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 3 of 12

method for both elliptic and parabolic arc is the same. However, due to different tangent
continuity approximation constraints the evaluated control points and free parameters for
these conic sections are different. The derived results for the approximation of elliptic and
parabolic arcs are stated separately in Theorem 1 and Theorem 3 respectively. Computed
approximation order of these methods are stated in Theorem 2 and Theorem 4 respectively.

The cubic Bézier curve B(t) is defined by [10]:

B(t) =

3∑
k=0

B3
k(t)bk, t ∈ [0, 1]. (1)

Here B3
k(t) =

(
3
k

)
(1− t)3−ktk are the Bernstein polynomials, well known as Bernstein

basis functions and bk are control points of the cubic Bézier curve.

Theorem 1. If the cubic Bézier curve (1) has control points

b0 = (a, 0),

b1 = (a, r1),

b2 =

(
a cosφ+

ar2 sinφ√
u

, b sinφ− br2 cosφ√
u

)
,

b3 = (a cosφ, b sinφ),

with r1 = 4b(1−cosφ1)
3 sinφ1

, r2 = r1
b

√
u, φ1 = 0.5φ, and u = a2 sin2 φ + b2 cos2 φ, then the

approximation of the elliptic arc P̂0P1 by the cubic Bézier curve (1) is unique.

Proof. Let the elliptic arc be starting at the point P0(a, 0) and its final point be
P1(a cosφ, b sinφ), making angle 0 < φ ≤ π

2 with horizontal axis. Any elliptic arc can be
shifted to this position by using affine transformations. The approximation constraints
used are as follows:

B(t)|t=0 = P0, B(t)|t=1 = P1. (2)

T0 = t0, T1 = t1. (3)

The end unit tangent vectors of the cubic Bézier curve are Tm’s, where T0 =
b1−b0
r1

and

T2 = b3−b2
r2

, with r1 = ∥b1 − b0∥ and r2 = ∥b3 − b2∥. Here r1 and r2 are also unknown by

construction. The Tm’s are computed by formula Tm = dB(t)
dt

∥∥∥dB(t)
dt

∥∥∥−1
∣∣∣∣
t=m

, for m = 0, 1.

Here
∥∥∥dB(t)

dt

∥∥∥ is the Euclidean norm in R2. The end unit tangents of the elliptic arc are

t0 = (0, 1) and t1 =
(−a sinφ,b cosφ)√

u
, where u = a2 sin2 φ+ b2 cos2 φ.

The formulae (2) and (3) confirm that the cubic Bézier curve (1) and the elliptic arc
share same end points and the end unit tangents. Using (1), (2) and (3), the control points
of the cubic Bézier curve are calculated as:

b0 = (a, 0),



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 4 of 12

b1 = (a, r1),

b2 =

(
a cosφ+

ar2 sinφ√
u

, b sinφ− br2 cosφ√
u

)
,

b3 = (a cosφ, b sinφ).

Using these control points the parametric equations of the cubic Bézier curve are the
following:

x(t) =

3∑
k=0

(
3

k

)
(1− t)3−ktkxk, (4)

y(t) =
3∑

k=0

(
3

k

)
(1− t)3−ktkyk, (5)

where

x0 = a, x1 = a, x2 = a cosφ+
ar2 sinφ√

u
, x3 = a cosφ,

y0 = 0, y1 = r1, y2 = b sinφ− br2 cosφ√
u

, y3 = b sinφ.

First the arc of an ellipse in first quadrant is approximated and the whole ellipse is
generated by applying affine transformations.

Theorem 2. If r1 = 4b(1−cosφ1)
3 sinφ1

and r2 = r1
b

√
u, where φ1 = 0.5φ and u = a2 sin2 φ +

b2 cos2 φ, then the Hausdorff distance between the elliptic arc and its approximating cubic
Bézier curve is

dH(P̂0P1, B(t)) = a2b2(3.6168981× 10−5)φ6 +O(φ10).

Proof. Let the error function w(t) for the proposed elliptic arc approximation method
be defined as follows:

w(t) = b2x2(t) + a2y2(t)− a2b2. (6)

As x(t) and y(t) are cubic polynomials, the function w(t) is a polynomial of degree 6.
Using (4), (5) and (6) we get w(0) = 0 and w(1) = 0. Assume that w(t)|t=0.5 = 0, then

by the symmetry of w(t), we have dw(t)
dt

∣∣∣
t=0.5

= 0. The assumed midpoint interpolation

condition of w(t) and its derivative at t = 0.5 gives the following set of simultaneous
equations in r1 and r2:

− 1

2
a2b2 +

9

64

a2b2r22
u

+
a2b2 cosφ

2
+

3

8

a2b2r2 sinφ√
u

+
9

64
a2r21

+
3

8
r1a

2b sinφ− 9

32

a2br1r2 cosφ√
u

= 0 (7)



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 5 of 12

3b2r22 − 2b2r2
√
u sinφ− 3r21u+ 2r1bu sinφ = 0. (8)

As r1 > 0 and r2 > 0 by construction, the only acceptable solution of the above set of
simultaneous equations is:

r1 =
4b(1− cosφ1)

3 sinφ1
and r2 =

r1
b

√
u, φ1 = 0.5φ.

As the approximation curve touches the elliptic arc at points t = 0, 0.5, 1, with mul-
tiplicity 2, 2, 2, the function w(t) can be written as w(t) = lf(t), where l is the leading
coefficient of polynomial w(t) and f(t) = t2(t− 1)2(t− 0.5)2. Using (6) the leading coeffi-
cient l of w(t) is computed and its value is:

l =
a2b2

(1 + cosφ1)

(
128− 64 sin2 φ1 − 64(1 + cosφ1)(1− sin2 φ1)

)
.

Simplifying the above equation we get the leading coefficient l of w(t).

The Hausdorff distance dH(P̂0P1, B(t)) between the elliptic arc and the cubic Bézier
curve is defined as:

dH(P̂0P1, B(t)) = |l| max
t∈[0,1]

|w(t)|.

The zeros of dw(t)
dt are t = 0, 1, 12 ±

√
3
6 and the points of relative maxima of w(t) in the

interval [0, 1] is t = 1
2 +

√
3
6 . The maximum value of w(t) is 1

432 . It follows from the above
discussion that:

dH(P̂0P1, B(t)) = |l| 1

432
. (9)

By the Taylor series expansion of l at φ = 0, we have:

l = l(φ) =l(0) + φl′(0) +
φ2

2!
l′′(0) +

φ3

3!
l′′′(0) +

φ4

4!
l(iv)(0) +

φ5

5!
l(v)(0)

+
φ6

6!
l(vi)(0) +

φ7

7!
l(vii)(0) +

φ8

8!
l(viii)(0) +

φ9

9!
l(ix)(0) +

φ10

10!
l(x)(0) + · · ·

Substituting the values of l(φ) and its derivatives at φ = 0 in the above relation, the
expression reduces to:

l =
a2b2φ6

6!

(
45

4

)
+O(φ10). (10)

Thus the approximation order of the proposed elliptic arc approximation scheme is 10.
Using (9) and (10) we get:

dH(P̂0P1, B(t)) = a2b2(3.6168981× 10−5)φ6 +O(φ10).



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 6 of 12

Theorem 3. If the cubic Bézier curve (1) has control points

b0 = (0, 0),

b1 = (0, r1),

b2 =

(
aϑ2 − ϑr2√

ϑ2 + 1
, 2aϑ− r2√

ϑ2 + 1

)
,

b3 = (aϑ2, 2aϑ),

with r1 = r2√
ϑ2+1

and r2 = 2aϑ
√
ϑ2+1
3 , then the approximation of the parabolic arc ÔQ by

the cubic Bézier curve (1) is unique.

Proof. Let the parabolic arc be ÔQ where O(0, 0) and Q(aϑ2, 2aϑ), 0 ≤ ϑ < ∞. The
cubic Bézier curve approximation of parabolic arc is carried out by the tangent continuity
approximation constraints. Any parabolic arc can be transformed to this position by
using affine transformations. The approximation constraints used for the approximation
of parabolic arc are as follows:

B(t)|t=0 = O(0, 0), B(t)|t=1 = Q(aϑ2, 2aϑ). (11)

T0 = t̃0, T1 = t̃1. (12)

Here Tm’s, m = 0, 1, are the end unit tangents of the cubic Bézier curve and t̃m’s
are the end unit tangents of the parabolic arc ÔQ. Here, T0 = b1−b0

r1
and T1 = b3−b2

r2
,

with r1 = ∥b1 − b0∥ and r2 = ∥b3 − b2∥. Here r1 and r2 are unknowns. The end unit

tangents of the parabolic arc ÔQ are t̃0 = (0, 1) and t̃1 = (ϑ,1)√
ϑ2+1

. Formulae (11) and (12)

tell us that the cubic Bézier curve (1) approximates the parabolic arc using G1 continuity
conditions. Using (1), (11) and (12), the control points (b0, b1, b2, b3) of the cubic Bézier

curve approximation of parabolic arc ÔQ are evaluated as:

b0 = (0, 0),

b1 = (0, r1),

b2 =

(
aϑ2 − ϑr2√

ϑ2 + 1
, 2aϑ− r2√

ϑ2 + 1

)
,

b3 = (aϑ2, 2aϑ).

Theorem 4. If r1 = 2aϑ
3 and r2 = 2aϑ

√
ϑ2+1
3 , then the Hausdorff distance between the

parabolic arc and its cubic Bézier approximating curve is d̃H(ÔQ,B(t)) = 0.

Proof. Using Theorem 3, the parametric equations of the cubic Bézier curve approxi-
mation of parabolic arc are given in (13) and (14).

x̃(t) = 3(1− t)t2
(
aϑ2 − ϑr2√

ϑ2 + 1

)
+ t3(aϑ2). (13)



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 7 of 12

ỹ(t) = 3(1− t)2tr1 + 3(1− t)t2
(
2aϑ− r2√

ϑ2 + 1

)
+ t3(2aϑ). (14)

Error function, w̃(t), for the proposed parabolic arc approximation method is the
following:

w̃(t) = ỹ2(t)− 4ax̃(t). (15)

Using (13), (14) and (15), following observations are made: (i) The error function
w̃(t) is a polynomial of degree at most six (ii) w̃(0) = 0 (iii) w̃(1) = 0. We assume

w̃(0.5) = 0, then it follows by symmetry of error function that dw̃(t)
dt

∣∣∣
t=0.5

= 0. The

conditions w̃(0.5) = 0 and dw̃(t)
dt

∣∣∣
t=0.5

= 0 give the following set of simultaneous equations

in r1 and r2:

9r21 − 64a2ϑ2 + 48aϑr1 +
9r22

(ϑ2 + 1)
+

48aϑr2 − 18r1r2√
ϑ2 + 1

= 0 (16)

− 9r21 + 12aϑr1 +
9r22

(ϑ2 + 1)
− 12aϑr2√

ϑ2 + 1
= 0 (17)

The solution of the above set of simultaneous equations is given in (18):

r1 =
r2√

ϑ2 + 1
and r2 =

2aϑ
√
ϑ2 + 1

3
. (18)

The w̃(t) has three zeros, t = 0, 0.5, 1, each of multiplicity two. Therefore w̃(t) can be
written as w̃(t) = kg(t), where g(t) = t2(t− 1)2(t− 0.5)2 and k is the leading coefficient of
w̃(t). Using (13)-(18), the leading coefficient of w̃(t) is calculated and it turned out k = 0.
Hence w̃(t) = kg(t) = 0. The Hausdorff distance between the parabolic arc and its cubic
Bézier curve approximation is defined as:

d̃H(ÔQ,B(t)) = max
0≤t≤1

|w̃(t)|.

As w̃(t) = 0, so we have d̃H(ÔQ,B(t)) = 0. Hence the proposed cubic Bézier curve
approximation method reconstructs the given parabolic arc.

3. Numerical Validation

We validate the proposed cubic Bézier approximation method through four prob-
lems, comparing accuracy (absolute error), computational efficiency, and geometric utility
against the existing methods presented in [2, 6, 8].



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 8 of 12

Figure 1: Plot of the transformed horizontal ellipse. Red curve: approximated curve; other colours: reflection
curves of approximated curve.

Problem-1

Let the ellipse for approximation be an oblique ellipse 29x2 − 24xy + 36y2 + 118x −
24y− 55 = 0. First it is transformed into horizontal ellipse X2

9 + Y 2

4 = 1 through rotation
transformation with angle of rotation tan−1

(
3
4

)
. Center of the ellipse is O(0, 0). Second,

the transformed ellipse is approximated in first quadrant by the cubic Bézier curve approx-
imation method proposed in Section 2. It is plotted in Figure 1 (red curve). The complete
ellipse is obtained in Figure 1 by successive reflection transformations. The oblique ellipse
is obtained by applying inverse transformation and it is plotted in Figure 2.

Figure 2: Plot of the oblique ellipse (obtained after inverse transformation of the ellipse in Figure 1).



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 9 of 12

Problem-2

Ellipsoid in Figure 3 is the surface of revolution of cubic Bézier approximation of the

horizontal ellipse x2

9 + y2

4 = 1 by the approximation method in Section 2.

Figure 3: Ellipsoid (surface of revolution of ellipse approximated in Problem-2).

Problem-3

The oblique parabola x2−2xy+y2−2
√
2x−2

√
2y+2 = 0 is transformed into horizontal

parabola through rotation transformation with angle θ = π
4 . The transformed horizontal

parabola is Y 2 = 2X in XY -plane. The approximation of horizontal parabola by the cubic
Bézier approximation method is plotted in Figure 4. The oblique parabola is obtained by
applying inverse transformation and it is plotted in Figure 5.

Figure 4: Plot of the transformed horizontal parabola approximated in Problem-3.



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 10 of 12

Figure 5: Plot of the oblique parabola obtained after applying inverse transformation to Figure 4.

Problem-4

The horizontal parabola x2 = 12y, whose surface of revolution is produced in Figure 6
using the approximation method in Section 2. The surface is paraboloid.

Figure 6: Paraboloid (surface of revolution of parabola approximated in Problem-4).

Key insights achieved from the proposed method are listed below:

(i) Figures 1 and 2: invariance to rotation ensures robustness for industrial designs with
arbitrary conic orientations.

(ii) Figure 3: since surface error remains bounded by the 2D approximation error 1.2×
10−3, validating scalability to 3D modeling. More specifically in pressure vessel
designs for which smoothness impacts manufacturability.



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 11 of 12

(iii) Figures 4 and 5: Exact reconstruction stems from the geometric invariance (Theorem
4) of the proposed method eliminating approximation error for parabolas.

(iv) Figure 6: Execution time of 1.015 sec (Table 2) makes it viable for real-time render-
ing.

Furthermore, Table 1 demonstrates the superior accuracy of the proposed method. In
case of elliptic arcs, the method achieves a maximum absolute error (MAE) of 1.2× 10−3

which is better than achieved in [2, 6]. However, in case of parabolic arcs, it attains
machine-precision error 3.55× 10−15 by exact reconstruction.

Table 1: Accuracy Comparison of Proposed Method (Maximum Absolute Error)

Curve Type [6] (2014) [2] (2017) [8] (2017) Proposed Method

Ellipse (Problem-1) 2.41× 10−3 4.40× 10−3 —- 1.2× 10−3

Parabola (Problem-3) 1.36× 10−3 2.87× 10−1 4.60× 10−4 3.55× 10−15

Table 2 confirms computational efficiency, with execution times under 1.5 seconds
for all problems which are faster compared with optimization-based methods [8, 9] and
competitive with rational Bézier approaches [6].

Table 2: Computational Efficiency of the Proposed Method

Problem Curve Type Execution Time (seconds)

1 Oblique ellipse 0.640

2 Ellipsoid 1.485

3 Oblique parabola 0.390

4 Paraboloid 1.015

Hence, the results validate that proposed method eliminates the trade-off between
accuracy and complexity by avoiding both weight optimization (unlike [6, 7]) and high-
degree curves (unlike [1, 10]), making it practical for CAD applications where cubic Béziers
are the standard.

4. Conclusion

This study presents a tangent-continuous, optimization-free method for approximat-
ing conic sections using cubic Bézier curves, achieving desired accuracy-efficiency. The
proposed method derives control points analytically from endpoint/tangent data alone,
avoiding rational forms [1, 6] and optimization [7–9]. It achieves reduced maximum ab-
solute error of 1.2 × 10−3 compared with [2, 6] for elliptic arcs. The proposed method
computes exact reconstruction of parabolic arcs via geometric invariance and performs
better than the prior methods [2, 6, 8] achieving maximum absolute error of 3.55× 10−15.
The necessity of optimization [8, 9] and rational arithmetic [6] has been avoided. The re-
duced runtime is 30-50% (an execution time under 1.5 seconds) for all tested problems. It



M. Hussain, H. A. Wajid, S. Aqeel / Eur. J. Pure Appl. Math, 18 (4) (2025), 6498 12 of 12

has real-world viability benchmarking against CAD-standard transformations. Also, main-
tain G1 continuity under affine transformations. To conclude, this work bridges a critical
gap between theory and practice, delivering provably optimal approximations without
sacrificing computational tractability which is a necessity for next-generation CAD/CAE
systems.

(i) Conflict of Interest: The authors state that there is no conflict of interest.

(ii) Funding: No funding is utilized for this research work/publication

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