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

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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),......

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.

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  • 1. Analytic parametric equations of log-aesthetic curves in terms ofincomplete gamma functionsComputer Aided Geometric Design, 29(2)(2012), pp. 129-140.Rushan Ziatdinova,1, Norimasa Yoshidab, Tae-wan Kimc,∗aDepartment of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744,Republic of KoreabDepartment of Industrial Engineering and Management, Nihon University, 1-2-1 Izumi-cho,Narashino Chiba 275-8575, JapancDepartment of Naval Architecture and Ocean Engineering, and Research Institute of Marine SystemsEngineering, Seoul National University, Seoul 151-744, Republic of KoreaAbstractLog-aesthetic curves (LACs) have recently been developed to meet the requirements ofindustrial design for visually pleasing shapes. LACs are defined in terms of definiteintegrals, and adaptive Gaussian quadrature can be used to obtain curve segments. Todate, 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 realvalue of α. The computation time for generating a LAC segment using the incompletegamma functions is up to 13 times faster than using direct numerical integration. Ourequations 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-aestheticspline, fair curve2010 MSC: 65D17, 68U071. IntroductionIn designing shapes, such as the exterior surfaces of automobiles, which are subjectto very significant aesthetic considerations, the quality of the surfaces is often assessed interms of reflections of a linear light source. Convoluted reflection lines usually are takento indicate that the corresponding part of the shape not acceptable. Since variation incurvature 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 9298Email 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)1Also 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 befair [3].Plane curves with monotone curvature were studied by Mineur et al. [17], and thisresearch 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 arcsof circles and Cornu spirals with continuous curvature, which is greatest for one of thecircular arcs. The Pythagorean-hodograph curves introduced by Farouki et al. [7] havebeen used to construct transition curves of monotone curvature. Frey et al. [8] analysedthe curvature distributions of segments of conic sections represented as rational quadraticB´ezier curves in standard form. The conditions sufficient for planar B´ezier and B-splinecurves 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 curvesegment 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 ofa 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 artificialobjects have approximately linear LCGs. LCGs together with logarithmic torsion graphs(LGTs) for analyzing planar and space curves were studied in [31]. Curves with LCGswhich are straight lines were called log-aesthetic curves by Yoshida et al. [32], and theycalled curves with nearly straight LCGs quasi-log-aesthetic curves [33]. Both of thesetypes of curve can be used for aesthetic shape modelling, and are likely to be an importantcomponent of next-generation CAD systems. Log-aesthetic curves can be also consideredin 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´eziercurve, 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 orcurvature continuity, can be associated with fair curves [14]. They are actually non-linearsplines, the theory of which arising from a variational criterion of the type∫κ2ds → minhas been briefly described in [16]. Moreover, one of the LAC cases, Cornu spiral, were usedfor “staircase” approximation in [16]. Fig. 2, 3 exhibit the usage of G1log-aesthetic splinewith 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 G0continuity).3
  • 4. (a) (b)Figure 3: Aesthetic design of Japanese word “shape” by means of log-aesthetic splines: (a) with controlpolygon, (b) without control polygon, and colored in black. (some points intentionally satisfy only G0continuity).Main resultsWe show how to derive analytic parametric equations of log-aesthetic curves in termsof tangent angle efficiently and accurately. Yoshida et al. [32] used numerical integrationbased 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 themmore suitable for interactive applications, which is specially important if we generate sur-faces containing many log-aesthetic curve segments. Furthermore, analytic equations willfacilitate 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 tangentangle, from which we can obtain exact representations of any real value of shapeparameter;• Because our obtained parametric equations consist of incomplete gamma functions,for which good approximation methods exist, we can compute log-aesthetic curvesegments accurately;• Our analytic formulation allows log-aesthetic curve segment to be computed up to13 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 numericalresults obtained using CAS Mathematica and Maple.OrganizationThe rest of this paper is organized as follows. In Section 2 we briefly review the basicmathematical concepts of log-aesthetic curves. In Section 3 we derive the general analyticequations of log-aesthetic curves and discuss particular cases, illustrated with the shapes4
  • 5. Paper Equations FeaturesHarada et al. [10, 34] - It was shown that many of the aes-thetic curves in artifical objects andthe natural world have LCGs thatcan be approximated by straightlines.Miura et al. [19] A general formula for log-aesthetic curves was de-fined as a function of arclength.The first step towards a mathemat-ical theory of log-aesthetic curves.Miura et al. [18] A general equation ofaesthetic curves was in-troduced that describesthe relationship betweenradius of curvature andlength.Log-aesthetic curves was shown toexhibit self-affinity.Yoshida et al. [32] The general equations of alog-aesthetic plane curveswere represented by defi-nite integrals. It was notedthat exact analytic equa-tions can only be found forvalues 0, 1 and 2 of theshape parameter.Numerical integration using adap-tive Gaussian quadrature was usedto evaluate log-aesthetic curves. Anew method of using a log-aestheticcurve segment to perform Hermiteinterpolation was proposed.Present study We have obtained theparametric equations oflog-aesthetic plane curveswhich allow exact analyticrepresentations for ∀α ∈ Rto be found.Log-aesthetic curve segments can becomputed accurately, up to 13 timesfaster 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 segmentusing the analytical equations with the computation time using numerical integration. InSection 5, we conclude our paper and suggest future work.2. Preliminaries2.1. NomenclatureWe are going to use the notation presented in Table 2.2.2. Fundamentals of log-aesthetic curvesMiura et al. [19] defined log-aesthetic curves as having a radius of curvature which isa function of their arc length s as below:log(ρdsdρ)= 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 determinesthe type of a log-aesthetic curve.After simply manipulating Eq. (1), and recollecting that c is a constant we obtaindsdρ=ρα−1λ. (2)When α = −1, 0, 1, 2 or ∞, we obtain a clothoid, a Nielsen’s spiral, a logarithmicspiral, 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 Pron the curve. The reference point can be any point on the curve except for the pointwhose radius of curvature is either 0 or ∞. The following constraints are placed at thereference 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 lowerlimits 1 and ˆρ respectively, and then replacing ˆρ with ρ, Yoshida et al. [32] found theintrinsic (natural) equation of the log-aesthetic curve, also known as the Ces´aro equation[30]:ρ(s) ={eλs, α = 0(λαs + 1)1α , 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 =1ρ=dθds. (4)If we substitute Eq. (3) into this equation, integrate with respect to s from 0 to ˆs, andafterwards 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−λsλ, α = 0log(λs+1)λ, α = 1(λαs+1)1− 1α −1λ(α−1), otherwise. (5)From Eqs. (2) and (4) we can further obtain:dθdρ=dsρdρ=ρα−2λ. (6)Integrating this equation with respect to θ from 0 to ˆθ, and then replacing ˆθ with θ yieldsa formulation of a log-aesthetic curve that relates the radius of curvature ρ to the tangentangle θ:7
  • 8. ρ(θ) ={eλθ, α = 1((α − 1) λ θ + 1)1α−1 , otherwise. (7)Using the quadratures by which a plane curve given by its natural equation can berepresented [21, 23], we can obtain the parametric equations of a log-aesthetic curve:x(ψ) =ψ∫0ρ(θ) cos θdθ, (8)y(ψ) =ψ∫0ρ(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 lowerbound is 1/(λ(1−α)), α > 1. If α = 1 there are no upper or lower bounds on the tangentangle ψ [32].Some of the characteristics of log-aesthetic curves are described in details by Yoshidaet 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 aspiral 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 atinfinity;• When 0 < α < 1, the distance to the point at which ρ = 0 is finite, and there is aninflection point at infinity;8
  • 9. • When α = 1, the curve is a spiral that converges to the point at which ρ = 0 witha finite arc length. In the other direction the curve is a spiral that diverges to thepoint 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 curvesAfter integrating in Eqs. (8) and (9), and applying the incomplete gamma function[9, 1, 26]Γ(a, z) =∞∫zua−1e−udu, (10)we can derive the general equations of log-aesthetic curves in terms of the tangent angleψ:x(ψ) =12(λi(α − 1))1α−1{Γ(αα − 1, −i(1 + (α − 1)θλ)(α − 1)λ)×(sin(1λ(1 − α))− i cos(1λ(1 − α)))+ (−1)1α−1 Γ(αα − 1,i(1 + (α − 1)θλ)(α − 1)λ)×(sin(1λ(1 − α))+ i cos(1λ(1 − α)))} ψ0, (11)y(ψ) =12(λi(α − 1))1α−1{(−1)1α−1 Γ(αα − 1,i(1 + (α − 1)θλ)(α − 1)λ)×(cos(1λ(1 − α))− i sin(1λ(1 − α)))+ Γ(αα − 1, −i(1 + (α − 1)θλ)(α − 1)λ)×(cos(1λ(1 − α))+ i sin(1λ(1 − α)))} ψ0. (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−γzz∞∏k=1(1 +zk)−1ez/k,9
  • 10. where γ ≈= 0.577 is the Euler-Mascheroni constant. An asymptotic expansion can bealso useful when |z| → ∞ and |arg z| < 32π [2]:Γ(a, z) ∼ za−1e−z∞∑k=0Γ(a)Γ(a − k)z−k.Now we consider some particular cases of the above equations, using the followingwell-known formulas [9, 1]:∫Pn(u) cos mu du =sin mumE(n2 )∑k=0(−1)k P(2k)n (u)m2k+ (13)+cos mumE(n+12 )∑k=1(−1)k−1 P(2k−1)n (u)m2k−1,∫Pn(u) sin mu du = −cos mumE(n2 )∑k=0(−1)k P(2k)n (u)m2k+ (14)+sin mumE(n+12 )∑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 equalto a number). We can now reduce Eqs. (8) and (9) for α ̸= 1 and 1α−1= γ (α =2, 32, 43, . . . , γ+1γ), where γ ∈ N∗(the set of all natural numbers except zero) to a pair ofintegrals2:x(ψ) =ψ∫0((α − 1) λ θ + 1)1α−1 cos θdθ =sin θE( 12(α−1) )∑k=0(−1)k[((α − 1) λ θ + 1)1α−1](2k)+ (15)cos θE( α2(α−1) )∑k=1(−1)k−1[((α − 1) λ θ + 1)1α−1](2k−1)ψ0,y(ψ) =ψ∫0((α − 1) λ θ + 1)1α−1 sin θdθ =2We will now and subsequently use (k) to signify the kthderivative with respect to θ.10
  • 11. − cos θE( 12(α−1) )∑k=0(−1)k[((α − 1) λ θ + 1)1α−1](2k)+ (16)sin θE( α2(α−1) )∑k=1(−1)k−1[((α − 1) λ θ + 1)1α−1](2k−1)ψ0,We can use Eqs. (15) and (16) to derive exact analytic equations of log-aesthetic curvesin terms of trigonometric functions for some special values of α. These and further curvesin the present work are drawn using general parametric equations.• For the case of α = 3/2 we havex(ψ) =ψ∫0(12λθ + 1)2cos θdθ =sin θ1∑k=0(−1)k[(12λθ + 1)2](2k)+cos θ2∑k=1(−1)k−1[(12λθ + 1)2](2k−1)ψ0={[(12λθ + 1)2−λ22]sin θ+[λ(12λθ + 1)]cos θ} ψ0=[(12λψ + 1)2−λ22]sin ψ+[λ(12λψ + 1)]cos ψ − λ.y(ψ) =ψ∫0(12λθ + 1)2sin θdθ =− cos θ1∑k=0(−1)k[(12λθ + 1)2](2k)+sin θ2∑k=1(−1)k−1[(12λθ + 1)2](2k−1)ψ0={−[(12λθ + 1)2−λ22]cos θ+[λ(12λθ + 1)]sin θ} ψ0= −[(12λψ + 1)2−λ22]cos ψ+[λ(12λψ + 1)]sin ψ −λ22+ 1.The family of LACs with α = 3/2 is shown in Fig. 6.11
  • 12. λ = 0.01 λ = 0.05λ = 0.1 λ = 1Figure 6: Log-aesthetic curves with α = 3/2. The value of θ is changing from its lower bound to 10radians.• 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 = 0.05λ = 0.1 λ = 1Figure 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) =√πλ{cos(12λ)C(1√πλ)+ sin(12λ)S(1√πλ)− cos(12λ)C(√1 − 2λψ√πλ)− sin(12λ)S(√1 − 2λψ√πλ)},y(ψ) = −√πλ{cos(12λ)S(1√πλ)− sin(12λ)C(1√πλ)− cos(12λ)S(√1 − 2λψ√πλ)+ sin(12λ)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 = 0.05λ = 0.1 λ = 1Figure 8: Log-aesthetic curves curves (chlotoids) with α = −1. The value of θ is changing from -10radians to its upper bound.• When α = 3 we obtain (Fig. 9):x(ψ) = λ{√π cos(12λ)S(1√πλ)−√π sin(12λ)C(1√πλ)+√2λψ + 1 sin(ψ)√1λ−√π cos(12λ)S(√2λψ + 1√πλ)+√π sin(12λ)C(√2λψ + 1√πλ)},y(ψ) =1√λ{1√λ−√π cos(12λ)C(1√πλ)−√π sin(12λ)S(1√πλ)−√2λψ + 1 sin(ψ)√1λ+√π cos(12λ)C(√2λψ + 1√πλ)+√π sin(12λ)S(√2λψ + 1√πλ)},where S(x) and C(x) are Fresnel integrals. These are two transcendental functions whichcommonly occur in the physics of diffraction, and have the following integral representa-tions [9, 1, 20, 25]:14
  • 15. S(t) =t∫0sin(u2)du, C(t) =t∫0cos(u2)du.The simultaneous parametric plot of S(t) and C(t) is the Cornu spiral or clothoid.λ = 0.01 λ = 0.05λ = 0.1 λ = 1Figure 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 estimationYoshida 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 numberof points needed. Our computations were coded in CAS Mathematica Version 7 [5], andperformed on a Pentium Core i7 3.07GHz computer. On every segment we computed100 points. The tangent angle of every curve segment varies from 0 to 1. We set λ suchthat the curve segment is defined in interval θ ∈ [0, 1]. If λ takes value greater than1, we set λ = 1. Our analytic approach is compared with different numerical methodsin Table 3. It can be seen that from analytic equations log-aesthetic curve segmentscan 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. Thisis because these curves are formulated as incomplete gamma functions which have goodapproximation methods [1, 24] and an exact series representation [2]. Other numerical15
  • 16. methods are slower; and moreover the Monte-Carlo method may fail for values of αaround 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 forcomputation 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 thatthe Gauss-Kronrod and the Newton-Cotes methods may have significant errors in theneighbourhood of α = 1; in other cases the maximum errors are negligible.5. Conclusions and future workWe have introduced analytic parametric equations for log-aesthetic curves consistingof trigonometric and incomplete gamma functions. Whereas previous authors [32] formu-lated parametric equations for particular cases (α = 0, 1, 2), our general equations allowan 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, 32, 43, . . . , γ+1γ, γ ∈ N∗, and in terms of Fresnel integralswhen α = −1, 3. Depending on the parameter α, the availability of general parametricequations (11), (12) allows log-aesthetic curve segments to be obtained up to 13 timesfaster than the Gauss-Kronrod numeric integration used previously. This will be espe-cially significant in the construction of log-aesthetic surfaces [11] containing many curvesegments.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 atinflection points. We are going to examine the possibility of deriving such an equation.6. AcknowledgementThe 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 byNational Research Foundation of Korea (NRF) grant No. 2010-0014404, funded by theKorean government (MEST).References[1] Abramowitz, M., Stegun, I. A., 1965. Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables. Dover, New York.[2] Amore, P., 2005. Asymptotic and exact series representations for the incompletegamma function. Europhysics Letters 71 (1), 1 – 7.[3] Burchard, H., Ayers, J., Frey, W., Sapidis, N., 1994. Designing Fair Curves andSurfaces. SIAM, Philadelphia, USA, pp. 3 – 28.[4] Dankwort, C. W., Podehl, G., 2000. A new aesthetic design workflow: results fromthe european project FIORES. In: CAD Tools and Algorithms for Product Design.Springer-Verlag, Berlin, Germany, pp. 16 – 30.16
  • 17. ρ radius of curvature ∆ρ change in radius of cur-vatureα first parameter of an LACis the slope of a line in theLCG (shape parameter)s arc length of a curveλ second parameter of anLACκ curvature of a curvec a constant, log λ θ, ψ tangent angle (angle be-tween a tangent line andthe x-axis∆s change in arc length [x(ψ), y(ψ)] parametric equation ofa log-aesthetic curve interms of tangent angleΓ(a, z) incomplete gamma function Pn(u) polynomial of degree nE(n) integral part of a real num-berP(k)n (u) derivative of Pn(u) of or-der kC(t)andS(t)Fresnel integrals γ any natural number ex-cept zeroTable 2: Notation.Parameters Our approach Approximate methods Comp. costα λ Analytic eq. Gauss-Kronrod N-C comparison-100 0.00825 0.109 1.232 1.373 11.3-10 0.0758 0.094 1.217 1.357 12.9-1 0.417 0.094 1.217 1.341 12.9-0.1 0.758 0.110 1.295 1.388 11.7-0.01 0.825 0.093 1.185 1.357 12.70 0.833 0.094 1.232 1.357 13.10.01 1.0 0.093 1.248 1.295 13.40.1 1.0 0.109 1.154 1.404 10.50.9 1.0 0.671 1.217 1.404 1.80.99 1.0 0.327 1.139 1.279 3.41.1 1.0 0.109 1.310 1.373 12.06/5 1.0 0.094 1.248 1.326 13.25/4 1.0 0.078 1.311 1.357 16.84/3 1.0 0.063 1.310 1.358 20.73/2 1.0 0.047 1.310 1.388 27.810 1.0 0.140 1.264 1.357 9.0100 1.0 0.127 1.231 1.357 9.8Table 3: The log-aesthetic curve segment computation time (in seconds). N-C is a Newton-Cotes numericintegration method. The last column shows how much faster the analytic equations in comparison withGauss-Kronrod method. For the cases when α = {32 , 43 , 54 , 65 } Eq. (15), (16) has been used.17
  • 18. Parameters Maximum errorα λ Gauss-Kronrod N-C21 2.0 × 10−151.5 × 10−1010 7.6 × 10−153.7 × 10−10100 7.0 × 10−143.5 × 10−9431 2.1 × 10−157.5 × 10−1010 1.5 × 10−133.0 × 10−9100 6.2 × 10−119.0 × 10−71091 3.3 × 10−155.0 × 10−1010 4.9 × 10−105.8 × 10−8100 0.3 × 1001.0 × 100Table 4: Error estimations for LAC segment computation.[5] Dyakonov, V., 2008. Mathematica 5: programming and mathematical computations.DMK Press, Moscow.[6] Farin, G., 2006. Class A B´ezier curves. Computer Aided Geometric Design 23 (7),573 – 581.[7] Farouki, R. T., 1997. Pythagorean-hodograph quintic transition curves of monotonecurvature. Computer-Aided Design 29 (9), 601 – 606.[8] Frey, W. H., Field, D. A., 2000. Designing B´ezier conic segments with monotonecurvature. Computer Aided Geometric Design 17 (6), 457 – 483.[9] Gradshtein, I., Ryzhik, I., 1962. Tables of Integrals, Summations, Series and Deriva-tives, 4th Edition. Vol. 1. GIFML, Moscow.[10] Harada, T., Yoshimoto, F., Moriyama, M., 1999. An aesthetic curve in the field ofindustrial design. In: IEEE Symposium on Visual Languages. Institute of ElectricalElectronics Engineering, Tokyo, Japan, pp. 38 – 47.[11] Inoue, J., Harada, T., Hagihara, T., 2009. An algorithm for generating log-aestheticcurved surfaces and the development of a curved surfaces generation system usingVR. In: IASDR. The International Association of Societies of Design Research,Seoul, Korea, pp. 2513 – 2522.[12] Kronrod, A., 1964. Doklady Akademii Nauk SSSR 154, 283 – 286.[13] Laurie, D. P., 1997. Calculation of Gauss-Kronrod quadrature rules. Mathematics ofComputation 66 (219), 1133 – 1145.[14] Levien, R., S´equin, C., 2009. Interpolating splines: which is the fairest of them all?Computer-Aided Design and Applications 4, 91 – 102.[15] Meek, D., Walton, D., 1989. The use of cornu spirals in drawing planar curves ofcontrolled curvature. Journal of Computational and Applied Mathematics 25 (1), 69– 78.18
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