Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

Computer Aided Geometric Design 29 (2012) 129–140

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Computer Aided Geometric Design www.elsevier.com/locate/cagd

Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions ✩ Rushan Ziatdinov a,1 , Norimasa Yoshida b , Tae-wan Kim a,c,∗ a b c

Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744, Republic of Korea Department of Industrial Engineering and Management, Nihon University, 1-2-1 Izumi-cho, Narashino Chiba 275-8575, Japan Research Institute of Marine Systems Engineering, Seoul National University, Seoul 151-744, Republic of Korea

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 11 March 2011 Received in revised form 26 August 2011 Accepted 17 November 2011 Available online 3 December 2011 Keywords: Log-aesthetic curve Spiral Linear logarithmic curvature graph Log-aesthetic spline Fair curve

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 an 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. © 2011 Elsevier B.V. All rights reserved.

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 curves with monotonically varying curvature. Such curves are generally assumed to be fair (Burchard et al., 1994). Plane curves with monotone curvature were studied by Mineur et al. (1998), and this research has been extended by Farin (2006), who introduced Class A 3D Bézier curves with monotone curvature and torsion. Meek and Walton (1989) 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 (1997) have been used to construct transition curves of monotone curvature. Frey and Field (2000) analyzed the curvature distributions of segments of conic sections represented as rational quadratic Bézier curves in standard form. The conditions sufficient for planar Bézier and B-spline curves to have monotone curvature have been described by Wang et al. (2004). Sapidis and Frey (1992) described a simple ✩

This paper has been recommended for acceptance by Ramon F. Sarraga. Corresponding author at: Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744, Republic of Korea. Tel.: +82 2 880 1434; fax: +82 2 888 9298. E-mail addresses: [email protected] (R. Ziatdinov), [email protected] (N. Yoshida), [email protected] (T.-w. Kim). URL: http://caditlab.snu.ac.kr/ (T.-w. Kim). 1 Also holds a position of Assistant Professor in the Department of Computer and Instructional Technologies, Fatih University, 34500 Büyükçekmece, Istanbul, Turkey.

*

0167-8396/$ – see front matter doi:10.1016/j.cagd.2011.11.003

© 2011

Elsevier B.V. All rights reserved.

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R. Ziatdinov et al. / Computer Aided Geometric Design 29 (2012) 129–140

Fig. 1. An example of a LAC segment (red line), and its evolute (purple line). Like a quadratic Bézier curve, an LAC segment can be controlled by three control points and specifying α . (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 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 G 0 continuity).

geometric condition that indicates when a quadratic Bézier curve segment has monotone curvature. An interesting idea was presented by Xu and Mould (2009), 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. (1999, 2002) introduced log-aesthetic curves, which exhibit monotonically varying curvature because they have linear logarithmic curvature graphs (LCGs). Harada et al. (1999, 2002) 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 Yoshida et al. (2010). Curves with LCGs which are straight lines were called log-aesthetic curves (Yoshida and Saito, 2006), and they called curves with nearly straight LCGs quasi-log-aesthetic curves (Yoshida and Saito, 2007). 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) (Dankwort and Podehl, 2000), in which designers evaluate 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. Log-aesthetic splines, which consist of many LAC segments connected with tangent or curvature continuity, can be associated with fair curves (Levien and  Séquin, 2009). They are actually non-linear splines, the theory of which arising from a variational criterion of the type κ 2 ds → min has been briefly described in Mehlum (1974). Moreover, one of the LAC cases, Cornu spiral, were used for “staircase” approximation in Mehlum (1974). Figs. 2, 3 exhibit the usage of G 1 logaesthetic spline with shape parameter α = 3/2 in car body and Japanese characters design respectively.

R. Ziatdinov et al. / Computer Aided Geometric Design 29 (2012) 129–140

131

Fig. 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 G 0 continuity).

Table 1 Comparison of the present study with previous work on log-aesthetic curves. Paper

Equations

Features

Harada et al. (1999, 2002)

–

It was shown that many of the aesthetic curves in artificial objects and the natural world have LCGs that can be approximated by straight lines.

Miura (2006)

A general formula for log-aesthetic curves was defined as a function of arc length.

The first step towards a mathematical theory of log-aesthetic curves.

Miura et al. (2005)

A general equation of aesthetic curves was introduced that describes the relationship between radius of curvature and length.

Log-aesthetic curves was shown to exhibit self-affinity.

Yoshida and Saito (2006)

The general equations of a log-aesthetic plane curves were represented by definite integrals. It was noted that exact analytic equations can only be found for values 0, 1 and 2 of the shape parameter.

Numerical integration using adaptive 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 logaesthetic 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 integration (Yoshida and Saito, 2006).

1.1. Main results We show how to derive analytic parametric equations of log-aesthetic curves in terms of tangent angle efficiently and accurately. Yoshida and Saito (2006) used numerical integration based on adaptive Gaussian quadrature (Kronrod, 1964; Laurie, 1997) to evaluate log-aesthetic curves. Representing 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 surfaces 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 numerical integration;

• Results obtained using our equations have been shown to agree with numerical results obtained using CAS Mathematica and Maple. 1.2. Organization The rest of this paper is organized as follows. In Section 2 we briefly review the basic mathematical concepts of logaesthetic curves. In Section 3 we derive the general analytic equations of log-aesthetic curves and discuss particular cases, illustrated with the shapes 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.

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Table 2 Notation.

ρ

Radius of curvature

ρ

Change in radius of curvature

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

κ

Curvature of a curve

c

A constant, log λ

θ, ψ

Tangent angle (angle between a tangent line and the x-axis) (see Fig. 5)

s

Change in arc length

[x(ψ), y (ψ)]

Parametric equation of a log-aesthetic curve in terms of tangent angle

Γ (a, z)

Incomplete gamma function

P n (u )

Polynomial of degree n

E (n)

Integral part of a real number

P n (u )

Derivative of P n (u ) of order k

C (t ) and S (t )

Fresnel integrals

γ

Any natural number except zero

(k)

Fig. 4. (a) A curve subdivided into infinitesimal segments over which ρ /ρ is constant. (b) Geometrical meaning of a linear LCG of a curve with slope

α.

2. Preliminaries 2.1. Nomenclature We are going to use the notation presented in Table 2. 2.2. Fundamentals of log-aesthetic curves Miura (2006) 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 ,

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 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

=

ρ α −1 λ

.

(1)

α (see

(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 P r 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 (Yoshida and Saito, 2006):

• Scaling: ρ = 1 at P r , which means that s = 0 and θ = 0 at the reference point; • Translation: P r is placed at the origin of Cartesian coordinate system; • Rotation: the tangent line to curve at P r is parallel to x-axis. Subsequently, after integrating Eq. (2) with respect to ρ with its upper and lower limits 1 and ρˆ respectively, and then replacing ρˆ with ρ , Yoshida and Saito (2006) found the intrinsic (natural) equation of the log-aesthetic curve, also known as the Cesáro equation (Yates, 1952):

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133

Fig. 5. The geometric meaning of the parameter θ in Eqs. (8) and (9).



ρ (s) =

α = 0,

e λs ,

(λα s

1

+ 1) α

, otherwise,

(3)

where λ = e −c , 0 < λ < ∞. The following relation, which arises in geometric interpretation of the curvature of a regular curve, is well-known in differential geometry (Pogorelov, 1974; Struik, 1988)

κ=

1

=

ρ

dθ ds

(4)

.

If we substitute Eq. (3) into this equation, integrate with respect to s from 0 to sˆ , and afterwards replace sˆ by s, and set θ = 0 when s = 0 we obtain the Whewell equation (Whewell, 1849) that relates the tangent angle θ with the arc length s:

θ(s) =

⎧ 1−e−λs , ⎪ ⎪ ⎨ λ

α = 0,

log(λs+1)

, λ ⎪ ⎪ 1 − ⎩ (λα s+1) α1 −1 λ(α −1)

α = 1,

(5)

, otherwise.

From Eqs. (2) and (4) we can further obtain:

dθ dρ

=

ds

ρ dρ

=

ρ α −2 λ

(6)

.

Integrating this equation with respect to θ from 0 to θˆ , and then replacing θˆ with θ yields a formulation of a log-aesthetic curve that relates the radius of curvature ρ to the tangent angle θ :



ρ (θ) =

α = 1,

e λθ , 1

((α − 1)λ θ + 1) α−1 , otherwise.

(7)

Using the quadratures by which a plane curve given by its natural equation can be represented (Pogorelov, 1974; Struik, 1988), we can obtain the parametric equations of a log-aesthetic curve:

ψ x(ψ) =

ρ (θ) cos θ dθ,

(8)

ρ (θ) sin θ dθ,

(9)

0

ψ y (ψ) = 0

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 )), there are no upper or lower bounds on the tangent angle ψ (Yoshida and Saito, 2006). Some of the characteristics of log-aesthetic curves are described in details by Yoshida and Saito (2006):

α > 1. If α = 1

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• • • • •

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; 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 (Gradshtein and Ryzhik, 1962; Abramowitz and Stegun, 1965; Temme, 1996b)

∞ Γ (a, z) =

ua−1 e −u du ,

(10)

z

we can derive the general equations of log-aesthetic curves in terms of the tangent angle ψ :

       1 α i (1 + (α − 1)θλ) 1 1 λi (α − 1) α−1 Γ − i cos ,− sin 2 α−1 (α − 1)λ λ(1 − α ) λ(1 − α )     ψ    1 α i (1 + (α − 1)θλ) 1 1  , + (−1) α−1 Γ + i cos , sin  α−1 (α − 1)λ λ(1 − α ) λ(1 − α ) 0        α−1 1 1 1 α i (1 + (α − 1)θλ) 1 1 λi (α − 1) − i sin , cos (−1) α−1 Γ y (ψ) = 2 α−1 (α − 1)λ λ(1 − α ) λ(1 − α )      ψ   α i (1 + (α − 1)θλ) 1 1  . +Γ + i sin ,− cos  α−1 (α − 1)λ λ(1 − α ) λ(1 − α ) 0 x(ψ) =

1

(11)

(12)

According to Abramowitz and Stegun (1965) an incomplete gamma function can be represented by following series:

Γ (a, z) = Γ (a) − za

∞  (− z)k , (a + k)k! k =0

and gamma function’s product representation is (Abramowitz and Stegun, 1965)

Γ ( z) =

∞ e −γ z 

z

 1+

k =1

z

 −1

k

e z/k ,

where γ ∼ = 0.577 is the Euler–Mascheroni constant. An asymptotic expansion can be also useful when | z| → ∞ and | arg z| < 3 π (Amore, 2005): 2

Γ (a, z) ∼ za−1 e −z

∞  k =0

Γ (a) −k z . Γ (a − k)

Now we consider some particular cases of the above equations, using the following well-known formulas (Gradshtein and Ryzhik, 1962; Abramowitz and Stegun, 1965):



E( n )

P n (u ) cos mu du =

2 sin mu 



m

(2k)

(−1)k

Pn

m2k

k =0 E( n )

P n (u ) sin mu du = −

2 cos mu 

m

(u )

(2k)

k

(−1)

k =0

Pn

E ( n +1 )

+

(u )

m2k

2 cos mu 

m

(−1)k−1

k =1 E ( n +1 )

+

2 sin mu 

m

k =1

(k)

(2k−1)

Pn

(−1)k−1

(u )

m2k−1 (2k−1)

Pn

(u )

m2k−1

(13)

,

,

(14)

where P n (u ) is a polynomial of degree n, P n (u ) is a derivative of P n (u ) of order k, and E (n) is the integral part of a real 1 =γ number (smallest integer greater than or equal to a number). We can now reduce Eqs. (8) and (9) for α = 1 and α − 1 γ +1 3 4 ∗ 2 (α = 2, 2 , 3 , . . . , γ ), where γ ∈ N (the set of all natural numbers except zero) to a pair of integrals :

2

We will now and subsequently use (k) to signify the kth derivative with respect to θ .

R. Ziatdinov et al. / Computer Aided Geometric Design 29 (2012) 129–140

ψ x(ψ) =



(α − 1)λθ + 1

α−1 1

135

cos θ dθ

0



E ( 2(α1−1) )



= sin θ

 1 (2k) (−1)k (α − 1)λ θ + 1 α−1

k =0 E ( 2(αα−1) )



+ cos θ

k −1

(−1)

k =1

ψ y (ψ) =



ψ  α−1 1 (2k−1)  (α − 1)λ θ + 1  , 

(15)

0

(α − 1)λθ + 1

α−1 1

sin θ dθ

0



E ( 2(α1−1) )



= − cos θ

 1 (2k) (−1)k (α − 1)λθ + 1 α−1

k =0 E ( 2(αα−1) )



+ sin θ

k −1

(−1)



k =1

ψ α−1 1 (2k−1)  (α − 1)λ θ + 1  . 

(16)

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

ψ x(ψ) =

1 2

2 λθ + 1

cos θ dθ

0



 2 (2k)  2 (2k−1) ψ 1 2   1 1  k k −1 = sin θ (−1) λθ + 1 + cos θ (−1) λθ + 1   2 2 k =0 k =1 0   

ψ 2    λ2 1 1 = λθ + 1 − sin θ + λ λθ + 1 cos θ   =

2

1 2

ψ  y (ψ) =

λψ + 1

1 2

2

2 −

λ

2

2

2

0

   1 sin ψ + λ λψ + 1 cos ψ − λ, 2

2 λθ + 1

sin θ dθ

0



 2 (2k)  2 (2k−1) ψ 1 2   1 1  k k −1 = − cos θ (−1) λθ + 1 + sin θ (−1) λθ + 1   2 2 k =0 k =1 0    

 2   ψ 2  λ 1 1 = − λθ + 1 − cos θ + λ λθ + 1 sin θ  2 2 2 0    2   λ2 λ2 1 1 =− λψ + 1 − cos ψ + λ λψ + 1 sin ψ − + 1. 2

2

2

2

The family of LACs with α = 3/2 is shown in Fig. 6. • 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.

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R. Ziatdinov et al. / Computer Aided Geometric Design 29 (2012) 129–140

Fig. 6. Log-aesthetic curves with

α = 3/2. The value of θ is changing from its lower bound to 10 radians.

Fig. 7. Log-aesthetic curves with

α = 2. The value of θ is changing from its lower bound to 10 radians.

R. Ziatdinov et al. / Computer Aided Geometric Design 29 (2012) 129–140

Fig. 8. Log-aesthetic curves (clothoids) with

137

α = −1. The value of θ is changing from −10 radians to its upper bound.

• Setting α = −1 and integrating Eqs. (8) and (9) yields the following equations:



x(ψ) =

π



 cos

λ

1

 

C √

2λ  √





1

+ sin

 



1

S √

2λ

πλ  √  1 − 2λψ − cos C S − sin , √ √ 2λ 2λ πλ πλ           π 1 1 1 1 y (ψ) = − cos S √ C √ − sin λ 2λ 2λ πλ πλ   √   √   1 1 − 2λψ 1 1 − 2λψ − cos S C + sin , √ √ 2λ 2λ πλ πλ 

πλ

1

1 − 2λψ

1





1

which refer to extended Cornu spiral, the graphs of which are shown in Fig. 8.

• When α = 3 we obtain (Fig. 9):

 x(ψ) = λ

−

√



π cos

√



π cos



1 2λ 1 2λ

 



1

S √

πλ

 √

2λψ + 1

√

S



−

πλ  



√

π sin

 +

1 2λ

√

 

π sin



√

C



1



πλ  √

1 2λ



C

+



 2λψ + 1 sin(ψ)

2λψ + 1

√

 

πλ



1

λ

,



√ √ 1 1 1 1 1 C √ S √ − π sin √ − π cos 2λ 2λ πλ πλ λ λ    √   √    √ 1 √ 1 1 2λψ + 1 2λψ + 1 + π cos C S − 2λψ + 1 sin(ψ) + π sin , √ √ λ 2λ 2λ πλ πλ

1 y (ψ) = √

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 representations (Gradshtein and Ryzhik, 1962; Abramowitz and Stegun, 1965; Olver, 1997; Temme, 1996a):

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R. Ziatdinov et al. / Computer Aided Geometric Design 29 (2012) 129–140

Fig. 9. Log-aesthetic curves with

t S (t ) =



sin u

2



t du ,

0

α = 3. The value of θ is changing from its lower bound to 10 radians.





cos u 2 du .

C (t ) = 0

The simultaneous parametric plot of S (t ) and C (t ) is the Cornu spiral or clothoid. 4. Computation cost and maximum error estimation Yoshida and Saito (2006) 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 (Dyakonov, 2008), and performed on a Pentium Core i7 3.07 GHz 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 (Kronrod, 1964; Laurie, 1997) used by Yoshida and Saito (2006); 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 (Abramowitz and Stegun, 1965; Temme, 1975) and an exact series representation (Amore, 2005). Other numerical 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 (Yoshida and Saito, 2006). 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 neighborhood 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 (Yoshida and Saito, 2006) formulated parametric equations for particular cases (α = 0, 1, 2), our general equations allow an accurate evaluation of log-aesthetic curve segments ∀α ∈ R.

R. Ziatdinov et al. / Computer Aided Geometric Design 29 (2012) 129–140

139

Table 3 The log-aesthetic curve segment computation time (in seconds). N–C is a Newton–Cotes numeric integration method. The last column shows how much faster the analytic equations in comparison with Gauss–Kronrod method. For the cases when α = { 32 , 43 , 54 , 65 } Eqs. (15), (16) has been used. Parameters

Our approach

Approximate methods

α

λ

Analytic eq.

Gauss–Kronrod

N–C

Comparison

−100 −10 −1 − 0.1 −0.01

0.00825 0.0758 0.417 0.758 0.825 0.833 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.109 0.094 0.094 0.110 0.093 0.094 0.093 0.109 0.671 0.327 0.109 0.094 0.078 0.063 0.047 0.140 0.127

1.232 1.217 1.217 1.295 1.185 1.232 1.248 1.154 1.217 1.139 1.310 1.248 1.311 1.310 1.310 1.264 1.231

1.373 1.357 1.341 1.388 1.357 1.357 1.295 1.404 1.404 1.279 1.373 1.326 1.357 1.358 1.388 1.357 1.357

11.3 12.9 12.9 11.7 12.7 13.1 13.4 10.5 1 .8 3 .4 12.0 13.2 16.8 20.7 27.8 9 .0 9 .8

0 0.01 0.1 0.9 0.99 1.1 6/5 5/4 4/3 3/2 10 100

Comp. cost

Table 4 Error estimations for LAC segment computation. Parameters

Maximum error

α

λ

Gauss–Kronrod

N–C

2

1 10 100

2.0 × 10−15 7.6 × 10−15 7.0 × 10−14

1.5 × 10−10 3.7 × 10−10 3.5 × 10−9

4 3

1 10 100

2.1 × 10−15 1.5 × 10−13 6.2 × 10−11

7.5 × 10−10 3.0 × 10−9 9.0 × 10−7

10 9

1 10 100

3.3 × 10−15 4.9 × 10−10 0.3 × 100

5.0 × 10−10 5.8 × 10−8 1.0 × 100

We have simplified the general equations and represented them in terms of trigonometric functions when α = γ +1 2, 32 , 43 , . . . , 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 especially significant in the construction of log-aesthetic surfaces (Inoue et al., 2009) 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. Acknowledgements The authors appreciate the issues and remarks of the anonymous reviewers and Associate 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 partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0018023). In addition, this work was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0000325). References Abramowitz, M., Stegun, I.A., 1965. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York. Amore, P., 2005. Asymptotic and exact series representations for the incomplete gamma function. Europhysics Letters 71 (1), 1–7. Burchard, H., Ayers, J., Frey, W., Sapidis, N., 1994. Designing Fair Curves and Surfaces. SIAM, Philadelphia, USA, pp. 3–28. Dankwort, C.W., Podehl, G., 2000. A new aesthetic design workflow: results from the European project FIORES. In: CAD Tools and Algorithms for Product Design. Springer-Verlag, Berlin, Germany, pp. 16–30. Dyakonov, V., 2008. Mathematica 5: Programming and Mathematical Computations. DMK Press, Moscow. Farin, G., 2006. Class A Bézier curves. Computer Aided Geometric Design 23 (7), 573–581.

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