Computer-Aided Design 44 (2012) 591–596

Contents lists available at SciVerse ScienceDirect

Computer-Aided Design

journal homepage: www.elsevier.com/locate/cad

Fitting G2 multispiral transition curve joining two straight linesRushan Ziatdinov a,1, Norimasa Yoshida b, Tae-wan Kim a,c,∗

a Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744, Republic of Koreab Department of Industrial Engineering and Management, Nihon University, 1-2-1 Izumi-cho, Narashino Chiba 275-8575, Japanc 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

Article history:Received 4 August 2011Accepted 21 January 2012

Keywords:Log-aesthetic curveMultispiralTransition curveFair curveHigh-quality curveSpiral

a b s t r a c t

In this paper, we describe an algorithm for generating a C-shaped G2 multispiral transition curve betweentwo non-parallel straight lines. The G2 multispiral is a curve that consists of two or more log-aestheticcurve segments connected with curvature continuity, and it has inflection endpoints. Compound-rhythmlog-aesthetic curves are not directly applicable to the generation of transition curves between two straightlines, which is important in highway and railroad track design, because both endpoints are required to beinflection points. Thus, a newapproach for generating transition curves is necessary. The two log-aestheticcurve segments with shape parameter α < 0 are connected at the origin, and they form amultispiral. Theproblem is to find a similar triangle, as in the given data. Depending on the parameterα, theG2 multispiraltransition curve may have different shapes; moreover, the shape of the curve approximates a circular arcas α decreases. The obtained curves also find applications in gear design and fillet modeling.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

A transition curve, also known as a spiral easement, is amathematically calculated curve on a section of a highway or rail-road track, where a straight section changes into a curve.

Such curves are used to reduce the effects of centrifugal forceexperienced by users. The transition begins at an infinite radiusand ends with the same radius as the curve itself, thereby formingan extremely broad spiral. If such easements are not applied, thelateral acceleration of a rail vehicle would change abruptly at thetangent point where the straight track meets the curve. Transitioncurves may connect not only a line with a circle but also any twolines with different topology; they find applications in several CADapplications, for example, in font design.

The usage of transition curves in railroad track design hasbeen described in several studies by Kellogg and Nordling [1],Searles [2], Talbot [3], Fox [4], Howard [5], Lee [6], Stephens [7],

∗ Corresponding author at: Department of Naval Architecture and OceanEngineering, Seoul National University, Seoul 151-744, Republic of Korea. Tel.: +822 880 1434; fax: +82 2 888 9298.

E-mail addresses: rushanziatdinov@yandex.ru (R. Ziatdinov),norimasa@acm.org (N. Yoshida), taewan@snu.ac.kr (T.-w. Kim).

URL: http://caditlab.snu.ac.kr (T.-w. Kim).1 Now holds a position of Assistant Professor in the Department of Computer

and Instructional Technologies, Fatih University, 34500 Büyükçekmece, Istanbul,Turkey. During this research until August 2011 was holding a position of AssistantProfessor in the Department of Special Mathematics, Tupolev Kazan State TechnicalUniversity (Kazan University of Aviation), 420111 Kazan, Russia.

0010-4485/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2012.01.007

Allen [8], Shunk [9], Cary [10]. Comprehensive information onthe classification and applications of transition curves in highwaydesign are found in [11].

The use of a fair cubic Bézier curve for G2 circle blendinghas been demonstrated in [12], and the use of a fair PH quinticcurve for G2 blending of two circles has been described in [13].A method for joining two circles with an S-shaped or a brokenback C-shaped transition curve composed of at most two spiralsegments, which is desirable in highway and railway routedesign or car-like robot path planning, is described by Habib andSakai [14]. The composition of G2 curves by joining circular arcsand/or straight line segments with cubic Bézier and Pythagoreanhodograph quintic spiral transitions has been presented in [15].A very short algorithm with a less restrictive ratio of the largerto the smaller radii of the given circular arcs, which is moreconvenient for practical applications than the approach adoptedin [13], is presented in [16]. An algorithm for composing asingle Pythagorean hodograph (PH) quintic Bézier spiral segmentbetween two circles, with one circle inside the other, is discussedin [17]. A method for the family of G2 planar cubic Bézier spiraltransitions from a straight line to a circle is investigated in [18].However, as pointed out in [19], the curvature of a polynomialcurve near an inflection point gets close to be linear in most cases.The use of log-aesthetic curves gives us more freedom near theinflection. Moreover, log-aesthetic curves are known to be highquality curves which appear in real world and the artificial objects.

The spiral condition for a single Bézier cubic transition curveof G2 contact between two circles, with one circle inside theother, is derived by Habib and Sakai [20]; the use of Pythagorean

592 R. Ziatdinov et al. / Computer-Aided Design 44 (2012) 591–596

hodograph quintic transition curves for solving the same problemis described in [21]. A method for joining a circle to another circlewith an S-shaped or a broken back C-shaped spiral transitionusing a Pythagorean hodograph quintic curve is introduced in [22].In [23], it has been shown that a single cubic curve can beused for blending or as a transition curve between two circlespreserving G2 continuity, regardless of the distance between theircenters and their radii. The lateral change in acceleration of avehicle moving along a new transition curve that is used in thedesign of the horizontal geometry of routes has been investigatedin [24]; discontinuities in the composite transition curve thatconsists of two spiral segments and a circular arc are observedas jumps in the diagrams of lateral change in acceleration, andthese discontinuities reduce travel comfort and cause wear andtear of vehicle wheels and rails. The more general cubic Bézierspiral, which has seven degrees of freedom, is presented in [25];it can be used in an interactive graphics environment to drawcomposite cubic Bézier curves that match G2 Hermite data, andto adjust the control points of the curves so that each segmentis a spiral. The application of an S-shaped transition curve in thedesign and generation of a spur gear has been introduced in [26].Furthermore, transition spirals and transition curves that consistof several spiral segments can be used for designing spiral bevelgears, which are widely used in the transmissions of helicopters,cars, etc. [27]. A new hysteretic rule that facilitates the modelingof gradual stiffness transitions that occur in real systems has beenpresented in [28]. A method to construct S-shape and C-shapetransition curves for G2 contact between two separated circlesusing a family of the quartic Bézier spiral is described in [29], lateron [30] proposed a G3 transition curve based on quartic Bézierspiral to be used in the design of horizontal geometry of routes. Theconstruction of a single C-Bézier curve with a shape parameter forjoining two circular arcs with G2 continuity is described by Cai andWang [31].

Because so-called log-aesthetic curves (LACs) [32,33,19] areaesthetically pleasing, they, as well as compound-rhythm LACs[34], can be used as transition curves in industrial design or, forexample, in computer aided aesthetic design [35]. Log-aestheticcurves as well as class A Bézier curves [36] have monotone cur-vature, and the LAC family can be considered as a generalization ofmany well-known spirals such as Cornu, Nielsen, and logarithmicspirals, and involutes of a circle. In this study, a transition curveconsists of two LACs, and logically, it can be called amultispiral. Forinstance, well known biarcs [37,38] can be considered as a limitingcases of multispirals.

Fig. 1 shows an application of aG2 multispiral transition curve ingear [26] and highway and stable motion of a vehicle, respectively.As it was noted by Yahaya et al. [26],

. . . the newdesign2 will give another alternative to the designersor manufacturers use, and the important is that the C-shapedcurve application can be applied in gear tooth shape especially,in spur gear.

Main resultsWe describe the construction of a G2 multispiral curve that can

be used as a transition between two non-parallel straight lines, andwe present an algorithm for drawing the curve. It can find severalapplications in highway or railroad track horizontal alignment,gear design, and font design.

Our work has the following features:

• The G2 multispiral transition is a curvature continuous curve (ithas only one curvature extremum);

2 By using curves with G2 continuity.

• The shape of a G2 multispiral transition curve can be easilychanged by varying α1 ∈ (−∞; 0);

• By decreasing α1, one may obtain a G2 multispiral transitioncurve, the half-part shape of which approximates a circular arc.

OrganizationThe remainder of this paper is organized as follows. In Section 2,

we briefly review previous studies on log-aesthetic curves andcompound-rhythm log-aesthetic curves, and their applicationsin industrial design. In Section 3, we provide the definition ofa multispiral, and we present an algorithm for G2 multispiraltransition curve generation. In Section 4, we present severalapplication examples of the proposed curve in road and spurdesign. Finally, in Section 6, we conclude the paper and discuss thescope for future work.

2. Previous work

2.1. Nomenclature

We are going to use the notation listed in Table 1.

2.2. Family of a log-aesthetic curves

Log-aesthetic curves (LACs) have been recently proposed tomeet the requirements of industrial design, in which aestheticallypleasing shapes are very significant. The family of log-aestheticcurves, which exhibit monotonically varying curvature and havelinear logarithmic curvature graphs (LCGs), was introduced byHarada et al. [39], Yoshimoto and Harada [40], who first notedthat many attractive curves in both natural and artificial objectshave approximately linear LCGs. Such objects include birds’ eggs,butterflies’wings, Japanese swords [39,40], horns, seashells, bones,leaves, flowers, and tree trunks [41–43]. In [19,44], Harada’s ideashave been extended, and LCGs and logarithmic torsion graphs(LTGs) have been studied for analyzing planar and space LACs; inaddition, the following definition has been given.

Definition 1. A planar log-aesthetic curve is a curve for which thelogarithmic curvature graph is linear.

A quasi-log-aesthetic curve has been introduced in [45]; thiscurve has a nearly straight logarithmic curvature graph. Boththese types of curves can be used for aesthetic shape modelingin computer aided design [46], and they are potentially importantcomponents of next-generation CAD systems or computer-aidedaesthetic design (CAAD) [35], in which designers evaluate thequality of a curve by examining its curvature plots. The generalformula of log-aesthetic curves was defined as the arc lengthfunction, and itwas presented byMiura [32],Miura et al. [33]. LAC’sparametric equation was obtained in terms of the tangent angle instandard form Yoshida and Saito [19] as

S(ψ) =

x(ψ)y(ψ)

=

ψ

0ρ(θ) cos θdθ ψ

0ρ(θ) sin θdθ

. (1)

The Gauss–Kronrod numeric integrationmethod [47] was used forcomputing the curve segment. The radius of curvature of a log-aesthetic curve in Eq. (1) is defined as

ρ(θ) =

eλθ , α = 1((α − 1) λ θ + 1)

1α−1 , otherwise,

(2)

where α ∈ R denotes the LAC’s shape parameter, θ is tangentangle, λ > 0, and λ ∈ R is the second parameter of a LAC.

R. Ziatdinov et al. / Computer-Aided Design 44 (2012) 591–596 593

a b

Fig. 1. Applications of G2 multispiral transition curve in (a) gear and (b) highway design. The blue points denote the junctions of spiral segments with straight linesegments. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 1Notation.

γ , ϕ, ω The angles of a triangle that is fitted by a multispiral transition curve (Fig. 3(b)).θ1, θ2 The tangent angles of the first and second LAC, respectively.α1, α2 The shape parameters of the first and second LAC, respectively.λ1, λ2 The second parameters of the first and second LAC, respectively.ρ(θ) Radius of curvature.[x(ψ), y(ψ)] Parametric equation of a log-aesthetic curve in terms of tangent angle.

2.3. Compound-rhythm curves

According to Yoshida and Saito [34],

Definition 2. A compound-rhythm log-aesthetic curve consists oftwo log-aesthetic curve segments, which are connected with G3

continuity.

In other words, compound-rhythm log-aesthetic curves arecurves whose logarithmic curvature graphs are represented bytwo connected segments, whose slopes are α1 and α2. A methodfor generating a log-aesthetic curve segment by specifying thetwo endpoints and their tangents has been proposed in [34]. Theidea of generating compound-rhythm log-aesthetic curves is notdirectly applicable to the generation of transition curves becauseboth endpoints are required to be inflection points. Thus, we needa new method for generating transition curves.

3. Multispirals

For multispiral transition curve construction, we are going touse two log-aesthetic curve segments; the first one is given by Eq.(1) and the second is a reflected LAC with respect to the y-axis,which parametric equations are

Sref(ψ) =

−x(ψ)y(ψ)

=

−

ψ

0ρ(θ) cos θdθ ψ

0ρ(θ) sin θdθ

. (3)

Note that reflection does not change the characteristics of an LAC,such as curvature and arclength. Two log-aesthetic curves areconnected at the origin (Fig. 3(a)), where they have equal radiusof curvature from the definition of the standard form. Thus, theobtained composite curve is curvature continuous.

In this study, we present two new definitions.

Definition 3. AMultispiral is a curve that consists ofm ≥ 2,m ∈ Zspiral segments connected with at least G0 continuity.

Definition 4. A Gn multispiral is a multispiral with Gn continuity.

According to Definition 4, compound-rhythm log-aestheticcurves areG3 multispirals, even though theyhave some restrictionson the values of curvature at the endpoints; in particular, theycannot have zero curvature at the two endpoints, and they cannotbe used as a transition curve between two straight lines. In thiscase, it is very important to determine a method by which a curveconsisting of two spiral segments and having zero curvatures at theendpoints can be constructed. If the upper bound for the tangentangle of a LAC with α < 0 is

θ =1

λ(1 − α), (4)

the corresponding point is an inflection point [19]. Only LACs withshape parameter α < 0 have inflection points together with finitearclength; hence, we are going to use only this subfamily of log-aesthetic curves. It includes the Cornu spiral, which is the mostpopular curve in railroad track and highway design.

In this section, we present an algorithm for drawing a G2

multispiral transition curve with inflection endpoints when thetwo endpoints, their tangents, and the shape parameter α1 of thefirst log-aesthetic curve segment are given (Algorithm 1). The keyidea of this algorithm is to compute a curve segment that fitsa similar triangle defined by three control points, as shown inFig. 2. The similarity of two triangles △P∗

1R∗P∗

2 (Fig. 2) and △P1RP2(Fig. 3(b)) can be determined by comparing two of their angles. Thecurvature graph of the generatedmultispiral is shown in Fig. 4, andthe LCG is described in Fig. 5.

Like a compound-rhythm curve [34], a G2-multispiral withinflection endpoints is generated by searching for the similar

594 R. Ziatdinov et al. / Computer-Aided Design 44 (2012) 591–596

Fig. 2. Hermite interpolation problem.

a

b

Fig. 3. (a) Configuration for generating a G2 multispiral transition curve. (b)Generated G2 multispiral transition curve.

Fig. 4. Curvature graph of multispiral generated in Fig. 3(b). α1 and α2 denote theshape parameters of the corresponding log-aesthetic curve segments.

triangle formed by three vertices (two inflection points and thepoint of intersection of the two tangent lines).

The method for generating the multispiral transition curve (seeFig. 3(a)) is described by the following algorithm.

Fig. 5. Logarithmic curvature graph of multispiral generated in Fig. 3(b). α1 and α2denote the slopes of the corresponding lines. At the origin of configuration proposedin Fig. 3(b), ρ = 1; hence, here, log(ρ) = 0.

Algorithm 1 MultispiralTransitionCurveGeneration(γ , ϕ, α1)1: Input: The two control points (endpoints) with unit tangent

vectors, zero curvatures at endpoints; shape parameter of thefirst spiral, α1 < 0, α1 ∈ R

2: Find the coordinates of the third control point R, which isformed by the intersection of the two tangent lines at theendpoints

3: Find the angles γ , ϕ, ω of the triangle, which are formed by thethree control points

4: Set θ1 = ϕ and θ2 = ω such that λ1, λ2 are related byθ1 + θ2 + γ = π , which is obvious from △Q1Q2R

5: Compute λ1 and λ2 = λ2(α2) using Eq. (4)6: Compute the Y -coordinates y(θ1), y(θ2) of inflection points P1

and P2 of both spirals7: Find α2 numerically by Newton’s method such that y(θ1) =

y(θ2)8: if α2 < 0 then9: Print(Multispiral transition curve can be constructed)

10: else11: Print(Multispiral transition curve cannot be constructed)12: end if13: Output: A set of two log-aesthetic curve parameters (α, λ),

which are necessary to construct a multispiral for G2 Hermiteinterpolation

The scaling and rotation of the obtained multispiral transitioncurve does not change the zero curvatures at the endpoints.

4. Examples

Fig. 6 shows an example of fitting a triangle by theG2 multispiraltransition curve;α1 is given by the user, andα2 is computed so thatthemultispiral transition curve can be generated. A decrease in theshape parameter α1 results in a quasi-circular shape, which meansthat the shape approximates a circular arc. The quasi-circularshapes of multispiral transition curves are likely to be obtainedfrom isosceles triangles when α1 = α2, for example, see Fig. 7.However, in comparison with a circular arc, transition multispiralshave zero curvatures at their end points, i.e., they have radius ofcurvature continuous connection.

5. Computation cost

Yoshida and Saito [19] noted that the computation timerequired to evaluate a log-aesthetic curve segment depends on theshape parameter α, as well as integration range, and number ofpoints for curve drawing. Our computations were coded in CAS

R. Ziatdinov et al. / Computer-Aided Design 44 (2012) 591–596 595

Fig. 6. Fitting triangle (γ = 70°, ϕ = 60°, ω = 50°) with G2 multispiral transitioncurve. The junction of the two log-aesthetic curve segments is shown by a redpoint. (For interpretation of the references to colour in this figure legend, the readeris referred to the web version of this article.)

Fig. 7. Fitting isosceles triangle (γ = 90°, ϕ = ω = 45°) with G2 multispiraltransition curve, for every case α1 = α2 , and −100 ≤ α1 ≤ −0.01. The junction ofthe two log-aesthetic curve segments is shown by a red point. (For interpretationof the references to colour in this figure legend, the reader is referred to the webversion of this article.)

Table 2The multispiral segment (Fig. 6) computation time (in seconds). Number of pointsis denoted as powers of 10.

α1 α2 105 104 103 102

−0.01 −0.213 15.632 3.136 1.888 1.794−0.1 −0.491 15.474 2.855 1.856 1.716−10 −2.279 32.713 7.815 5.460 5.211−100 −3.268 38.579 13.307 11.060 10.640

Table 3The multispiral segment (Fig. 7) computation time (in seconds). Number of pointsis denoted as powers of 10.

α1 = α2 105 104 103 102

−0.01 14.212 1.981 0.889 0.811−0.1 14.290 1.950 0.905 0.718−10 28.065 3.541 1.295 0.967−100 29.281 4.960 2.465 2.215

Maple Version 15 [48], and performed on a Intel(R) Core(TM)2 DuoCPU 2.00 GHz computer. On every LAC segmentwe computed from102 to 105 points. It can be simply seen from Tables 2 and 3 thatmultispiral segment computation time increases with increase of|α|.

Our very recent work on log-aesthetic curves [49], whereanalytic parametric equations of LACs in terms of incompletegamma functionswere obtained, allow to generate an LAC segmentup to 13 times faster than using the Gauss–Kronrod adaptiveintegration used in present research.

6. Conclusions and future work

We introduced a G2 multispiral curve that can be used asa transition between two non-parallel straight lines, and wepresented an algorithm for its construction. This curve can finddifferent applications in highway or railroad track design, geardesign, and font design. Depending on parameter α ∈ (−∞; 0),the G2 multispiral transition curve may have different shapes;moreover, one may obtain quasi-circular shapes by decreasing α.

There is considerable scope for future work. It is importantto find an algorithm using which arbitrary curvatures at theendpoints of a multispiral can be obtained; this would be veryuseful for generating transition curve between two circles (both C-type transition curve, and S-type transition curve with inflectionpoint) or between a circle and a straight line (J-shaped) as well asmultispiral splines. In addition, we believe that the application ofour G2 multispiral transition curve generation algorithm to areassuch as the planning of robot trajectories and font design requiresfurther investigation. A logical way related with planning robottrajectories is to investigate the motion stability of differentialwheeled robot, or even construct a kinematical model of adifferential drive mobile robot for such a family of transitioncurves.

Acknowledgments

The authors appreciate the issues and remarks of the anony-mous reviewers and associate editor which helped to improve thequality of this paper. This work was supported by grant No. 2011-0018023 and 2011-0000325 from the National Research Founda-tion of Korea (NRF), funded by the Korean government (MEST).

References

[1] Kellogg NB, NordlingW. The transition curve or curve of adjustment as appliedto the alignment of railroads by the method of rectangular co-ordinates andby deflection angles (or polar coordinates). New York: McGraw-Hill; 1907.

[2] Searles WH. The railroad spiral: the theory of compound transition curvereduced to practical formulae and rules for application in field work. NewYork: J. Wiley & Sons; 1882.

[3] Talbot AN. The railway transition spiral. NewYork: EngineeringNews Pub. Co.;1901.

[4] Fox WG. Transition curves: a field book for engineers, containing rules andtables for laying out transition curves. New York: D. Van Nostrand Company;1893.

[5] Howard CR. The transition-curve fieldbook. New York: J. Wiley & Sons; 1891.[6] Lee P. The railroad taper: the theory and application of a compound transition

curve based upon thirty-foot chords. New York: J. Wiley & Sons; 1915.[7] Stephens JR. The six-chord spiral. NewYork: Engineering News Publishing Co.;

1907.[8] Allen CF. Railroad curves and earthwork. New York: McGraw-Hill Book

Company; 1920.[9] Shunk WF. A practical treatise on railway curves and location. Philadelphia:

E.H. Butler & Co.; 1854.[10] Cary ER. Solution of railroad problems by the slide rule. New York: D. Van

Nostrand Co.; 1913.[11] Lamm R, Psarianos B, Mailaender T. Highway design and traffic safety

engineering handbook. New York: McGraw-Hill; 1999.[12] Walton D, Meek D. Planar G2 transition between two circles with a fair cubic

Bézier curve. Computer-Aided Design 1999;31(14):857–66.[13] Walton DJ, Meek DS. Planar G2 transition with a fair Pythagorean hodograph

quintic curve. Journal of Computational and Applied Mathematics 2002;138(1):109–26.

[14] Habib Z, Sakai M. G2 cubic transition between two circles with shape control.Journal of Computational and Applied Mathematics 2009;223(1):133–44.

[15] Habib Z, Sakai M. Simplified and flexible spiral transitions for use in computergraphics and geometric modelling. In: Proceedings. Third internationalconference on image and graphics. 2004. p. 426–9.

[16] Habib Z, Sakai M. G2 cubic transition between two circles with shape control.International Journal of Computer Mathematics 2003;80(8):959–67.

[17] Habib Z, Sakai M. G2 PH quintic spiral transition curves and their applications.Scientiae Mathematicae Japonicae 2005;61(2):207–17.

[18] Habib Z, Sakai M. Spiral transition curves and their applications. ScientiaeMathematicae Japonicae 2005;61(2):195–206.

[19] Yoshida N, Saito T. Interactive aesthetic curve segments. The Visual Computer2006;22(9):896–905.

596 R. Ziatdinov et al. / Computer-Aided Design 44 (2012) 591–596

[20] Habib Z, Sakai M. An answer to an open problem on cubic spiral transitionbetween two circles. In: Computer algebra—design of algorithms. 2006. p.46–52.

[21] Habib Z, SakaiM.OnPHquintic spirals joining two circleswith one circle insidethe other. Computer-Aided Design 2007;39(2):125–32.

[22] Habib Z, Sakai M. G2 Pythagorean hodograph quintic transition between twocircles with shape control. Computer Aided Geometric Design 2007;24(5):252–66.

[23] Dimulyo S, Habib Z, SakaiM. Fair cubic transition between two circleswith onecircle inside or tangent to the other. Numerical Algorithms 2009;51:461–76.

[24] Baykal O, Tari E, Coskun Z, Sahin M. New transition curve joining two straightlines. Journal of Transportation Engineering 1997;123(5):337–47.

[25] Walton DJ, Meek DS, Ali JM. Planar G2 transition curves composed of cubicBézier spiral segments. Journal of Computational and Applied Mathematics2003;157(2):453–76.

[26] Yahaya SH, Salleh MS, Ali JM. Spur gear design with an S-shaped transitioncurve application using mathematica and CAD tools. Dubai (United ArabEmirates): IEEE Computer Society; 2009. p. 273–276.

[27] Sheveleva GI, Volkov AE, Medvedev VI. Algorithms for analysis of meshingand contact of spiral bevel gears. Mechanism andMachine Theory 2007;42(2):198–215.

[28] Wiebe L, Christopoulos C. Using Bézier curves to model gradual stiffnesstransitions in nonlinear elements: application to self-centering systems.Earthquake Engineering & Structural Dynamics 2011;40(14):1535–52.

[29] Ahmad A, Gobithasan R, Ali J.G2 transition curve using quartic Bézier curve. In:Computer graphics, imaging and visualisation. 2007. CGIV’07. 2007. p. 223–8.

[30] Ahmad A, Md Ali J. G3 transition curve between two straight lines. In: Fifthinternational conference on computer graphics, imaging and visualisation.2008. p. 154–9.

[31] Cai H,Wang G. A newmethod in highway route design: joining circular arcs bya single C-Bézier curve with shape parameter. Journal of Zhejiang University—Science A 2009;10:562–9.

[32] Miura KT. A general equation of aesthetic curves and its self-affinity.Computer-Aided Design and Applications 2006;3(1–4):457–64.

[33] Miura K, Sone J, Yamashita A, Kaneko T. Derivation of a general formula ofaesthetic curves. In: 8th international conference on humans and computers.2005. p. 166–71.

[34] Yoshida N, Saito T. Compound-rhythm log-aesthetic curves. Computer-AidedDesign and Applications 2009;6(1–4):243–52.

[35] Dankwort CW, Podehl G. A new aesthetic design workflow: results from theEuropean project FIORES. In: CAD tools and algorithms for product design.Berlin (Germany): Springer-Verlag; 2000. p. 16–30.

[36] Farin G. Class a Bézier curves. Computer Aided Geometric Design 2006;23(7):573–81.

[37] Schonherr J. Smooth biarc curves. Computer-Aided Design 1993;25(6):365–70.

[38] Bolton K. Biarc curves. Computer-Aided Design 1975;7(2):89–92.[39] Harada T, Yoshimoto F, Moriyama M. An aesthetic curve in the field of

industrial design. In: IEEE symposium on visual languages. Institute ofelectrical electronics engineering. 1999. p. 38–47.

[40] Yoshimoto F, Harada T. Analysis of the characteristics of curves in natural andfactory products. In: 2nd IASTED international conference on visualization,imaging and image processing. Malaga (Spain): ACTA Press; 2002. p. 276–81.

[41] Cook T. Spirals in nature and art. Nature 1903;68(1761):296.[42] Cook T. The curves of life. New York: Dover; 1979.[43] Harary G, Tal A. The natural 3D spiral. Computer Graphics Forum 2011;30(2):

237–46.[44] Yoshida N, Fukuda R, Saito T. Logarithmic curvature and torsion graphs.

In: Dahlen M, Floater MS, Lyche T, Merrien J-L, Mørken K, Schumaker LL,editors. Mathematical methods for curves and surfaces. Lecture notes incomputer science, vol. 5862. Berlin (Heidelberg): Springer; 2010. p. 434–43.

[45] Yoshida N, Saito T. Quasi-aesthetic curves in rational cubic Bézier forms.Computer-Aided Design and Applications 2007;4(9–10):477–86.

[46] Farin G. Curves and surfaces for computer aided geometric design: a practicalguide. 4th ed. New York: Academic Press; 1997.

[47] Kronrod A. Doklady Akademii Nauk SSSR 1964;154:283–6.[48] Dyakonov V. Maple 9.5/10 in mathematics, physics and education. Moscow:

Solon Press; 2006.[49] Ziatdinov R, Yoshida N, Kim T. Analytic parametric equations of log-aesthetic

curves in terms of incomplete gamma functions. Computer Aided GeometricDesign 2012;29(2):129–40.