Rushan Ziatdinov, Norimasa Yoshida, Tae-wan Kim, 2012. Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions, Computer Aided Geometric Design 29(2), 129-140. Log-aesthetic curves (LACs) have recently been developed to meet the requirements of industrial design for visually pleasing shapes. LACs are defined in terms of definite integrals, and adaptive Gaussian quadrature can be used to obtain curve segments. To date, these integrals have only been evaluated analytically for restricted values (0, 1, 2) of the shape parameter. We present parametric equations expressed in terms of incomplete gamma functions, which allow us to find an exact analytic representation of a curve segment for any real value of alpha. The computation time for generating a LAC segment using the incomplete gamma functions is up to 13 times faster than using direct numerical integration. Our equations are generalizations of the well-known Cornu, Nielsen, and logarithmic spirals, and involutes of a circle.

1. Analytic parametric equations of log-aesthetic curves in terms of
incomplete gamma functions
Computer Aided Geometric Design, 29(2)(2012), pp. 129-140.
Rushan Ziatdinova,1
, Norimasa Yoshidab
, Tae-wan Kimc,∗
a
Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744,
Republic of Korea
b
Department of Industrial Engineering and Management, Nihon University, 1-2-1 Izumi-cho,
Narashino Chiba 275-8575, Japan
c
Department of Naval Architecture and Ocean Engineering, and Research Institute of Marine Systems
Engineering, Seoul National University, Seoul 151-744, Republic of Korea
Abstract
Log-aesthetic curves (LACs) have recently been developed to meet the requirements of
industrial design for visually pleasing shapes. LACs are defined in terms of definite
integrals, and adaptive Gaussian quadrature can be used to obtain curve segments. To
date, these integrals have only been evaluated analytically for restricted values (0, 1, 2)
of the shape parameter α.
We present parametric equations expressed in terms of incomplete gamma functions,
which allow us to find an exact analytic representation of a curve segment for any real
value of α. The computation time for generating a LAC segment using the incomplete
gamma functions is up to 13 times faster than using direct numerical integration. Our
equations are generalizations of the well-known Cornu, Nielsen, and logarithmic spirals,
and involutes of a circle.
Keywords: log-aesthetic curve, spiral, linear logarithmic curvature graph, log-aesthetic
spline, fair curve
2010 MSC: 65D17, 68U07
1. Introduction
In designing shapes, such as the exterior surfaces of automobiles, which are subject
to very significant aesthetic considerations, the quality of the surfaces is often assessed in
terms of reflections of a linear light source. Convoluted reflection lines usually are taken
to indicate that the corresponding part of the shape not acceptable. Since variation in
curvature determines the pattern of the reflections, a lot of work has been done to generate
∗
Corresponding author, Tel.: +82 10 2739 7364; fax: +82 2888 9298
Email addresses: rushanziatdinov@yandex.ru (Rushan Ziatdinov), norimasa@acm.org
(Norimasa Yoshida), taewan@snu.ac.kr (Tae-wan Kim)
URL: http://caditlab.snu.ac.kr/ (Tae-wan Kim)
1
Also holds a position of Assistant Professor in the Department of Computer and Instructional Tech-
nologies, Fatih University, 34500 B¨uy¨uk¸cekmece, Istanbul, Turkey

2. curves with monotonically varying curvature. Such curves are generally assumed to be
fair [3].
Plane curves with monotone curvature were studied by Mineur et al. [17], and this
research has been extended by [6], who introduced Class A 3D B´ezier curves with mono-
tone curvature and torsion. Meek et al. [15] showed how to construct a curve from arcs
of circles and Cornu spirals with continuous curvature, which is greatest for one of the
circular arcs. The Pythagorean-hodograph curves introduced by Farouki et al. [7] have
been used to construct transition curves of monotone curvature. Frey et al. [8] analysed
the curvature distributions of segments of conic sections represented as rational quadratic
B´ezier curves in standard form. The conditions sufficient for planar B´ezier and B-spline
curves to have monotone curvature have been described by Wang et al. [27]. Sapidis et al.
[22] described a simple geometric condition that indicates when a quadratic B´ezier curve
segment has monotone curvature. An interesting idea was presented by Xu et al. [29],
who used a particle-tracing method to create curves that simulate the orienting effect of
a magnetic field on iron filings. These curves are known to be circular or helical.
Recently, Harada et al. [10, 34] introduced log-aesthetic curves, which exhibit mono-
tonically varying curvature because they have linear logarithmic curvature graphs (LCGs).
Harada et al. [10, 34] noted that many attractive curves in both natural and artificial
objects have approximately linear LCGs. LCGs together with logarithmic torsion graphs
(LGTs) for analyzing planar and space curves were studied in [31]. Curves with LCGs
which are straight lines were called log-aesthetic curves by Yoshida et al. [32], and they
called curves with nearly straight LCGs quasi-log-aesthetic curves [33]. Both of these
types of curve can be used for aesthetic shape modelling, and are likely to be an important
component of next-generation CAD systems. Log-aesthetic curves can be also considered
in the context of computer-aided aesthetic design (CAAD) [4], in which designers eval-
uate the quality of a curve by looking at plots of curvature or radius of curvature. Fig.
1 shows an example of a log-aesthetic curve segment together with its smooth evolute,
which means that radius of curvature is changing monotonically.
2

3. Figure 1: An example of a LAC segment (red line), and its evolute (purple line). Like a quadratic B´ezier
curve, a LAC segment can be controlled by three control points and specifying α.
Log-aesthetic splines, which consist of many LAC segments connected with tangent or
curvature continuity, can be associated with fair curves [14]. They are actually non-linear
splines, the theory of which arising from a variational criterion of the type
∫
κ2
ds → min
has been briefly described in [16]. Moreover, one of the LAC cases, Cornu spiral, were used
for “staircase” approximation in [16]. Fig. 2, 3 exhibit the usage of G1
log-aesthetic spline
with shape parameter α = 3/2 in car body and Japanese characters design respectively.
(a)
(b)
Figure 2: Aesthetic design of a car body by means of log-aesthetic splines: (a) with control polygon, (b)
without control polygon (some points intentionally satisfy only G0
continuity).
3

4. (a) (b)
Figure 3: Aesthetic design of Japanese word “shape” by means of log-aesthetic splines: (a) with control
polygon, (b) without control polygon, and colored in black. (some points intentionally satisfy only G0
continuity).
Main results
We show how to derive analytic parametric equations of log-aesthetic curves in terms
of tangent angle efficiently and accurately. Yoshida et al. [32] used numerical integration
based on adaptive Gaussian quadrature [12, 13] to evaluate log-aesthetic curves. Repre-
senting them in analytic form in some cases avoids numerical integration and makes them
more suitable for interactive applications, which is specially important if we generate sur-
faces containing many log-aesthetic curve segments. Furthermore, analytic equations will
facilitate research on log-aesthetic curves. Table 1 compares previous results with ours.
Our work makes the following contributions:
• We obtain analytic parametric equations of log-aesthetic curves in terms of tangent
angle, from which we can obtain exact representations of any real value of shape
parameter;
• Because our obtained parametric equations consist of incomplete gamma functions,
for which good approximation methods exist, we can compute log-aesthetic curve
segments accurately;
• Our analytic formulation allows log-aesthetic curve segment to be computed up to
13 times faster than using the Gauss-Kronrod or Newton-Cotes methods of numer-
ical integration;
• Results obtained using our equations have been shown to agree with numerical
results obtained using CAS Mathematica and Maple.
Organization
The rest of this paper is organized as follows. In Section 2 we briefly review the basic
mathematical concepts of log-aesthetic curves. In Section 3 we derive the general analytic
equations of log-aesthetic curves and discuss particular cases, illustrated with the shapes
4

5. Paper Equations Features
Harada et al. [10, 34] - It was shown that many of the aes-
thetic curves in artifical objects and
the natural world have LCGs that
can be approximated by straight
lines.
Miura et al. [19] A general formula for log-
aesthetic curves was de-
fined as a function of arc
length.
The first step towards a mathemat-
ical theory of log-aesthetic curves.
Miura et al. [18] A general equation of
aesthetic curves was in-
troduced that describes
the relationship between
radius of curvature and
length.
Log-aesthetic curves was shown to
exhibit self-affinity.
Yoshida et al. [32] The general equations of a
log-aesthetic plane curves
were represented by defi-
nite integrals. It was noted
that exact analytic equa-
tions can only be found for
values 0, 1 and 2 of the
shape parameter.
Numerical integration using adap-
tive Gaussian quadrature was used
to evaluate log-aesthetic curves. A
new method of using a log-aesthetic
curve segment to perform Hermite
interpolation was proposed.
Present study We have obtained the
parametric equations of
log-aesthetic plane curves
which allow exact analytic
representations for ∀α ∈ R
to be found.
Log-aesthetic curve segments can be
computed accurately, up to 13 times
faster than by direct numerical inte-
gration [32].
Table 1: Comparison of the present study with previous work on log-aesthetic curves.
5

6. of different spirals. In Section 4 we compare the computation time of the curve segment
using the analytical equations with the computation time using numerical integration. In
Section 5, we conclude our paper and suggest future work.
2. Preliminaries
2.1. Nomenclature
We are going to use the notation presented in Table 2.
2.2. Fundamentals of log-aesthetic curves
Miura et al. [19] defined log-aesthetic curves as having a radius of curvature which is
a function of their arc length s as below:
log
(
ρ
ds
dρ
)
= α log ρ + c, (1)
where the constant c = − log λ, and (0, c) are the coordinates of the intersection of the y-
axis with a line of a slope α (see Fig.4(b)), which is the shape parameter that determines
the type of a log-aesthetic curve.
After simply manipulating Eq. (1), and recollecting that c is a constant we obtain
ds
dρ
=
ρα−1
λ
. (2)
When α = −1, 0, 1, 2 or ∞, we obtain a clothoid, a Nielsen’s spiral, a logarithmic
spiral, the involute of a circle, and a circle respectively.
To derive a formula of a log-aesthetic curve we need to consider a reference point Pr
on the curve. The reference point can be any point on the curve except for the point
whose radius of curvature is either 0 or ∞. The following constraints are placed at the
reference point [32]:
• Scaling: ρ = 1 at Pr, which means that s = 0 and θ = 0 at the reference point;
• Translation: Pr is placed at the origin of Cartesian coordinate system;
• Rotation: the tangent line to curve at Pr is parallel to x-axis.
6

7. (a) (b)
Figure 4: (a) A curve subdivided into infinitesimal segments over which ∆ρ/ρ is constant. (b) Geomet-
rical meaning of a linear LCG of a curve with slope α.
Subsequently, after integrating Eq. (2) with respect to ρ with its upper and lower
limits 1 and ˆρ respectively, and then replacing ˆρ with ρ, Yoshida et al. [32] found the
intrinsic (natural) equation of the log-aesthetic curve, also known as the Ces´aro equation
[30]:
ρ(s) =
{
eλs
, α = 0
(λαs + 1)
1
α , otherwise
, (3)
where λ = e−c
, 0 < λ < ∞. The following relation, which arises in geometric interpreta-
tion of the curvature of a regular curve, is well-known in differential geometry [21, 23]
κ =
1
ρ
=
dθ
ds
. (4)
If we substitute Eq. (3) into this equation, integrate with respect to s from 0 to ˆs, and
afterwards replace ˆs by s, and set θ = 0 when s = 0 we obtain the Whewell equation [28]
that relates the tangent angle θ with the arc length s:
θ(s) =



1−e−λs
λ
, α = 0
log(λs+1)
λ
, α = 1
(λαs+1)1− 1
α −1
λ(α−1)
, otherwise
. (5)
From Eqs. (2) and (4) we can further obtain:
dθ
dρ
=
ds
ρdρ
=
ρα−2
λ
. (6)
Integrating this equation with respect to θ from 0 to ˆθ, and then replacing ˆθ with θ yields
a formulation of a log-aesthetic curve that relates the radius of curvature ρ to the tangent
angle θ:
7

8. ρ(θ) =
{
eλθ
, α = 1
((α − 1) λ θ + 1)
1
α−1 , otherwise
. (7)
Using the quadratures by which a plane curve given by its natural equation can be
represented [21, 23], we can obtain the parametric equations of a log-aesthetic curve:
x(ψ) =
ψ∫
0
ρ(θ) cos θdθ, (8)
y(ψ) =
ψ∫
0
ρ(θ) sin θdθ, (9)
Figure 5: The geometric meaning of the parameter θ in Eqs. (8) and (9).
where the upper bound on the tangent angle ψ is 1/(λ(1 − α)), α < 1, and its lower
bound is 1/(λ(1−α)), α > 1. If α = 1 there are no upper or lower bounds on the tangent
angle ψ [32].
Some of the characteristics of log-aesthetic curves are described in details by Yoshida
et al. [32]:
• The radius of curvature ρ of log-aesthetic curves can grow from 0 to ∞;
• When α < 0, a log-aesthetic curves have an inflection points, and the curve is a
spiral until the point at which ρ = 0;
• When α = 0, the curve is also a spiral until ρ = 0. The point at which ρ = ∞ is at
infinity;
• When 0 < α < 1, the distance to the point at which ρ = 0 is finite, and there is an
inflection point at infinity;
8

9. • When α = 1, the curve is a spiral that converges to the point at which ρ = 0 with
a finite arc length. In the other direction the curve is a spiral that diverges to the
point at which ρ = ∞;
• When α > 1, the point at ρ = 0 has a fixed tangent direction;
• The curve is a spiral that diverges to the point at which ρ = ∞.
3. General equations and overall shapes of log-aesthetic curves
After integrating in Eqs. (8) and (9), and applying the incomplete gamma function
[9, 1, 26]
Γ(a, z) =
∞∫
z
ua−1
e−u
du, (10)
we can derive the general equations of log-aesthetic curves in terms of the tangent angle
ψ:
x(ψ) =
1
2
(λi(α − 1))
1
α−1
{
Γ
(
α
α − 1
, −
i(1 + (α − 1)θλ)
(α − 1)λ
)
×
(
sin
(
1
λ(1 − α)
)
− i cos
(
1
λ(1 − α)
))
+ (−1)
1
α−1 Γ
(
α
α − 1
,
i(1 + (α − 1)θλ)
(α − 1)λ
)
×
(
sin
(
1
λ(1 − α)
)
+ i cos
(
1
λ(1 − α)
))} ψ
0
, (11)
y(ψ) =
1
2
(λi(α − 1))
1
α−1
{
(−1)
1
α−1 Γ
(
α
α − 1
,
i(1 + (α − 1)θλ)
(α − 1)λ
)
×
(
cos
(
1
λ(1 − α)
)
− i sin
(
1
λ(1 − α)
))
+ Γ
(
α
α − 1
, −
i(1 + (α − 1)θλ)
(α − 1)λ
)
×
(
cos
(
1
λ(1 − α)
)
+ i sin
(
1
λ(1 − α)
))} ψ
0
. (12)
According to [1] an incomplete gamma function can be represented by following series:
Γ(a, z) = Γ(a) − za
∞∑
k=0
(−z)k
(a + k)k!
,
and gamma function’s product representation is [1]
Γ(z) =
e−γz
z
∞∏
k=1
(
1 +
z
k
)−1
ez/k
,
9

10. where γ ≈= 0.577 is the Euler-Mascheroni constant. An asymptotic expansion can be
also useful when |z| → ∞ and |arg z| < 3
2
π [2]:
Γ(a, z) ∼ za−1
e−z
∞∑
k=0
Γ(a)
Γ(a − k)
z−k
.
Now we consider some particular cases of the above equations, using the following
well-known formulas [9, 1]:
∫
Pn(u) cos mu du =
sin mu
m
E(n
2 )∑
k=0
(−1)k P
(2k)
n (u)
m2k
+ (13)
+
cos mu
m
E(n+1
2 )∑
k=1
(−1)k−1 P
(2k−1)
n (u)
m2k−1
,
∫
Pn(u) sin mu du = −
cos mu
m
E(n
2 )∑
k=0
(−1)k P
(2k)
n (u)
m2k
+ (14)
+
sin mu
m
E(n+1
2 )∑
k=1
(−1)k−1 P
(2k−1)
n (u)
m2k−1
,
where Pn(u) is a polynomial of degree n, P
(k)
n (u) is a derivative of Pn(u) of order k,
and E(n) is the integral part of a real number (smallest integer greater than or equal
to a number). We can now reduce Eqs. (8) and (9) for α ̸= 1 and 1
α−1
= γ (α =
2, 3
2
, 4
3
, . . . , γ+1
γ
), where γ ∈ N∗
(the set of all natural numbers except zero) to a pair of
integrals2
:
x(ψ) =
ψ∫
0
((α − 1) λ θ + 1)
1
α−1 cos θdθ =



sin θ
E( 1
2(α−1) )
∑
k=0
(−1)k
[
((α − 1) λ θ + 1)
1
α−1
](2k)
+ (15)
cos θ
E( α
2(α−1) )
∑
k=1
(−1)k−1
[
((α − 1) λ θ + 1)
1
α−1
](2k−1)



ψ
0
,
y(ψ) =
ψ∫
0
((α − 1) λ θ + 1)
1
α−1 sin θdθ =
2
We will now and subsequently use (k) to signify the kth
derivative with respect to θ.
10

11. 


− cos θ
E( 1
2(α−1) )
∑
k=0
(−1)k
[
((α − 1) λ θ + 1)
1
α−1
](2k)
+ (16)
sin θ
E( α
2(α−1) )
∑
k=1
(−1)k−1
[
((α − 1) λ θ + 1)
1
α−1
](2k−1)



ψ
0
,
We can use Eqs. (15) and (16) to derive exact analytic equations of log-aesthetic curves
in terms of trigonometric functions for some special values of α. These and further curves
in the present work are drawn using general parametric equations.
• For the case of α = 3/2 we have
x(ψ) =
ψ∫
0
(
1
2
λθ + 1
)2
cos θdθ =



sin θ
1∑
k=0
(−1)k
[(
1
2
λθ + 1
)2
](2k)
+
cos θ
2∑
k=1
(−1)k−1
[(
1
2
λθ + 1
)2
](2k−1)



ψ
0
=
{[(
1
2
λθ + 1
)2
−
λ2
2
]
sin θ+
[
λ
(
1
2
λθ + 1
)]
cos θ
} ψ
0
=
[(
1
2
λψ + 1
)2
−
λ2
2
]
sin ψ+
[
λ
(
1
2
λψ + 1
)]
cos ψ − λ.
y(ψ) =
ψ∫
0
(
1
2
λθ + 1
)2
sin θdθ =



− cos θ
1∑
k=0
(−1)k
[(
1
2
λθ + 1
)2
](2k)
+
sin θ
2∑
k=1
(−1)k−1
[(
1
2
λθ + 1
)2
](2k−1)



ψ
0
=
{
−
[(
1
2
λθ + 1
)2
−
λ2
2
]
cos θ+
[
λ
(
1
2
λθ + 1
)]
sin θ
} ψ
0
= −
[(
1
2
λψ + 1
)2
−
λ2
2
]
cos ψ+
[
λ
(
1
2
λψ + 1
)]
sin ψ −
λ2
2
+ 1.
The family of LACs with α = 3/2 is shown in Fig. 6.
11

12. λ = 0.01 λ = 0.05
λ = 0.1 λ = 1
Figure 6: Log-aesthetic curves with α = 3/2. The value of θ is changing from its lower bound to 10
radians.
• Applying the same approach for α = 2 yields:
x(ψ) = sin ψ − λ + λ(cos ψ + ψ sin ψ),
y(ψ) = 1 − cos ψ + λ(sin ψ − ψ cos ψ),
which are the parametric equations of involutes of a circle shown in Fig . 7.
12

13. λ = 0.01 λ = 0.05
λ = 0.1 λ = 1
Figure 7: Log-aesthetic curves with α = 2. The value of θ is changing from its lower bound to 10 radians.
• Setting α = −1 and integrating Eqs. (8) and (9) yields the following equations:
x(ψ) =
√
π
λ
{
cos
(
1
2λ
)
C
(
1
√
πλ
)
+ sin
(
1
2λ
)
S
(
1
√
πλ
)
− cos
(
1
2λ
)
C
(√
1 − 2λψ
√
πλ
)
− sin
(
1
2λ
)
S
(√
1 − 2λψ
√
πλ
)}
,
y(ψ) = −
√
π
λ
{
cos
(
1
2λ
)
S
(
1
√
πλ
)
− sin
(
1
2λ
)
C
(
1
√
πλ
)
− cos
(
1
2λ
)
S
(√
1 − 2λψ
√
πλ
)
+ sin
(
1
2λ
)
C
(√
1 − 2λψ
√
πλ
)}
,
which refer to extended Cornu spiral, the graphs of which are shown on Fig. 8.
13

14. λ = 0.01 λ = 0.05
λ = 0.1 λ = 1
Figure 8: Log-aesthetic curves curves (chlotoids) with α = −1. The value of θ is changing from -10
radians to its upper bound.
• When α = 3 we obtain (Fig. 9):
x(ψ) = λ
{
√
π cos
(
1
2λ
)
S
(
1
√
πλ
)
−
√
π sin
(
1
2λ
)
C
(
1
√
πλ
)
+
√
2λψ + 1 sin(ψ)
√
1
λ
−
√
π cos
(
1
2λ
)
S
(√
2λψ + 1
√
πλ
)
+
√
π sin
(
1
2λ
)
C
(√
2λψ + 1
√
πλ
)}
,
y(ψ) =
1
√
λ
{
1
√
λ
−
√
π cos
(
1
2λ
)
C
(
1
√
πλ
)
−
√
π sin
(
1
2λ
)
S
(
1
√
πλ
)
−
√
2λψ + 1 sin(ψ)
√
1
λ
+
√
π cos
(
1
2λ
)
C
(√
2λψ + 1
√
πλ
)
+
√
π sin
(
1
2λ
)
S
(√
2λψ + 1
√
πλ
)}
,
where S(x) and C(x) are Fresnel integrals. These are two transcendental functions which
commonly occur in the physics of diffraction, and have the following integral representa-
tions [9, 1, 20, 25]:
14

15. S(t) =
t∫
0
sin(u2
)du, C(t) =
t∫
0
cos(u2
)du.
The simultaneous parametric plot of S(t) and C(t) is the Cornu spiral or clothoid.
λ = 0.01 λ = 0.05
λ = 0.1 λ = 1
Figure 9: Log-aesthetic curves with α = 3. The value of θ is changing from its lower bound to 10 radians.
4. Computation cost and maximum error estimation
Yoshida et al. [32] observed that the computation time required to evaluate a log-
aesthetic curve segment depends on the parameter α, the range of integration and number
of points needed. Our computations were coded in CAS Mathematica Version 7 [5], and
performed on a Pentium Core i7 3.07GHz computer. On every segment we computed
100 points. The tangent angle of every curve segment varies from 0 to 1. We set λ such
that the curve segment is defined in interval θ ∈ [0, 1]. If λ takes value greater than
1, we set λ = 1. Our analytic approach is compared with different numerical methods
in Table 3. It can be seen that from analytic equations log-aesthetic curve segments
can be obtained up to 13 times faster than by means of the Gauss-Kronrod method
[12, 13] used by Yoshida et al. [32]; the precise ratio depends on the value of α. This
is because these curves are formulated as incomplete gamma functions which have good
approximation methods [1, 24] and an exact series representation [2]. Other numerical
15

16. methods are slower; and moreover the Monte-Carlo method may fail for values of α
around 1, and we did not consider it to be worth close examination. Since Eqs. (15)
and (16) are represented by simple and exact analytic functions, they can be useful for
computation of the maximum errors of numerical methods used in previous work [32].
Table 4 includes such a comparison for several values of α and λ, and it can be seen that
the Gauss-Kronrod and the Newton-Cotes methods may have significant errors in the
neighbourhood of α = 1; in other cases the maximum errors are negligible.
5. Conclusions and future work
We have introduced analytic parametric equations for log-aesthetic curves consisting
of trigonometric and incomplete gamma functions. Whereas previous authors [32] formu-
lated parametric equations for particular cases (α = 0, 1, 2), our general equations allow
an accurate evaluation of log-aesthetic curve segments for ∀α ∈ R.
We have simplified the general equations and represented them in terms of trigono-
metric functions when α = 2, 3
2
, 4
3
, . . . , γ+1
γ
, γ ∈ N∗
, and in terms of Fresnel integrals
when α = −1, 3. Depending on the parameter α, the availability of general parametric
equations (11), (12) allows log-aesthetic curve segments to be obtained up to 13 times
faster than the Gauss-Kronrod numeric integration used previously. This will be espe-
cially significant in the construction of log-aesthetic surfaces [11] containing many curve
segments.
An analytic equation of a log-aesthetic curve in terms of arc length is also required,
since the equation in terms of tangent angle is unstable when ρ → ∞, which occurs at
inflection points. We are going to examine the possibility of deriving such an equation.
6. Acknowledgement
The authors appreciate the issues and remarks of the anonimous reviewers and as-
sociate editor, as well as suggestions of Prof. Kenjiro T. Miura (Shizuoka University,
Japan) which helped to improve the quality of this paper. This work was supported by
National Research Foundation of Korea (NRF) grant No. 2010-0014404, funded by the
Korean government (MEST).
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17. ρ radius of curvature ∆ρ change in radius of cur-
vature
α first parameter of an LAC
is the slope of a line in the
LCG (shape parameter)
s arc length of a curve
λ second parameter of an
LAC
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Parameters Our approach Approximate methods Comp. cost
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17

18. Parameters Maximum error
α λ Gauss-Kronrod N-C
2
1 2.0 × 10−15
1.5 × 10−10
10 7.6 × 10−15
3.7 × 10−10
100 7.0 × 10−14
3.5 × 10−9
4
3
1 2.1 × 10−15
7.5 × 10−10
10 1.5 × 10−13
3.0 × 10−9
100 6.2 × 10−11
9.0 × 10−7
10
9
1 3.3 × 10−15
5.0 × 10−10
10 4.9 × 10−10
5.8 × 10−8
100 0.3 × 100
1.0 × 100
Table 4: Error estimations for LAC segment computation.
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