Quadratic Log-Aesthetic Curves

Transcription

1 Quadratic Log-Aethetic Curve Norimaa Yohida[ ] and Takafumi Saito[ X] Nihon Univerit, Toko Univerit of Agriculture and Technolog, ABSTRACT Thi paper propoe quadratic log-aethetic curve that are curve whoe logarithmic curvature graph are quadratic. In previou work, generalized log-aethetic curve are derived b hifting either the curvature or the radiu of curvature of log-aethetic curve. Quadratic log-aethetic curve are another generalization of log-aethetic curve b making logarithmic curvature graph quadratic. We derive the equation of quadratic log-aethetic curve and clarif their characteritic. For drawing quadratic log-aethetic curve, we need to compute the invere of the error and imaginar error function. We preent a method for computing thee invere and confirm that the curve can be generated in real time. Keword: Quadratic log-aethetic curve, Logarithmic curvature graph, Error and imaginar error function. INTRODUCTION In the deign of highl aethetic urface, uch a the eterior of automobile and other, the ue of highl aethetic curve i important ince uch urface are contructed baed on aethetic feature curve. If the curvature of a feature curve i not monotonicall varing, the reflected image of the generated urface get eail ditorted, which i not aetheticall pleaing. See Fig. 9 of [7] for eample. In aethetic hape deign, the ue of curve with monotonicall varing curvature (and placing curvature etrema intentionall) i important. Freeform curve, uch a Bézier curve or NURBS curve are widel ued in CAD tem, but controlling the curvature of freeform curve i not ea. Among variou curve with monotonicall varing curvature, Harada et al. ha propoed logaethetic curve baed on their original anali [] of man aethetic curve including feature curve of automobile. Log-aethetic curve [,7,] are curve with linear logarithmic curvature graph whoe lope i α. Special cae of log-aethetic curve are the Clothoid if α =, Nielen piral if α = 0, a logarithmic piral if α =, the circle involute if α =, and a circle if α = ±. Variou work ha been done for logaethetic curve. Some of them are: compound-rhthm log-aethetic curve [3], etenion to pace curve uing the linearit in the logarithmic torion graph [4], proving the uniquene of the curve egment when α < 0 (curve with inflection point) [6], proving the evolute of log-aethetic curve are alo log-aethetic curve [6], fat computation of curve egment uing incomplete Gamma function[], generalization b hifting either the curvature or the radiu of curvature[,0], G Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

2 Hermite interpolation uing three connected egment [9], and application to generating urface [3,4,8]. Thi paper propoe quadratic log-aethetic curve, which are curve whoe logarithmic curvature graph [5] are quadratic. In previou work, generalized log-aethetic curve [,0] have been propoed b Miura and Gobithaaan. Generalized log-aethetic curve are derived b hifting either the curvature or the radiu of curvature of log-aethetic curve. Quadratic log-aethetic curve are another generalization of log-aethetic curve b making logarithmic curvature graph quadratic. It ha been hown that the incomplete Gamma function arie in the tangential angle formulation of logaethetic curve [0]. We how that other pecial function, which are the error and imaginar error function, arie in the formulation of quadratic log-aethetic curve. We derive the equation of quadratic log-aethetic curve and clarif their characteritic. One notable difference from generalized log-aethetic curve i that quadratic log-aethetic curve include curve with finite arc length and their curvature varing from 0 to infinit. We how that uch curve can be obtained if the quadratic coefficient γ in the logarithmic curvature graph i negative. For drawing quadratic log-aethetic curve, we need to compute the invere of the error and imaginar error function. We preent a method for computing thee invere and confirm that the curve can be generated in real time. ERROR AND IMAGINARY ERROR FUNCTIONS Thi ection briefl review the error and imaginar error function [5]. A will be hown later, we have to compute thee function a well a their invere function for drawing quadratic log-aethetic curve. Since we compute thee function uing floating point arithmetic, uch a the double preciion binar floating point arithmetic, we have to be careful about the domain of thee function. erfi( z) The error function and imaginar error function are defined a z t = e d t, 0 (.) erfi ( z) = i erf ( iz) (.) where i i the imaginar unit. Although i erfi( z) appear in the right ide of Eqn. (.), i alwa real if z i real. Figure how the error and imaginar error function. erfi z ( ) 4 4 Fig. The error and imaginar error function. erfi( z ), we have to be careful that we are computing uing the double When we compute or preciion. A z approache, approache ±. In the double preciion, if z i greater than z 5 approimatel 5.9, i become eentiall. Thu we afel aume when erf ( z ) erfi computing. When computing ( z ) erfi, we have to be careful o that ( z) i within the range of 9 z 6 erfi( z ) erfi( 6 the double preciion. We aume when computing, ince )( ) i within Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

3 3 the range of the double preciion. Thee aumption are ued a the bound for computing invere of thee function, which are necear for drawing quadratic log-aethetic curve. 3 THE FUNDAMENTAL EQUATION OF QUADRATIC LOG-AESTHETIC CURVES The equation of log-aethetic curve were derived from the linearit in logarithmic curvature graph. Similarl, the equation of quadratic log-aethetic curve are derived from quadratic curve in logarithmic curvature graph. Logarithmic curvature graph are graph whoe horizontal ai i log ( ρ d / dρ ) and vertical ai i. Here, ρ i the radiu of curvature and i the arc length. Fig. how the linear logarithmic curvature graph with it lope α =. See [,7,,4] for more detail of log-aethetic curve and logarithmic curvature graph. Quadratic log-aethetic curve are curve whoe logarithmic curvature graph are quadratic a hown in Fig. or. Quadratic curve include ellipe (including circle), parabola, and hperbola. Among them, onl parabola in the logarithmic curvature graph guarantee the monotonicit of the curvature ince other tpe of quadratic curve ma have a turn in the horizontal ai ( ). A parabola in the logarithmic curvature graph i log d ρ = γ ( ) + αlog ρ + c dρ (3.) where γ, α are quadratic and linear coefficient repectivel and c i a contant. Eqn. (3.) i the fundamental equation for quadratic log-aethetic curve. Note that Eqn. (3.) include a line if γ = 0, which mean that Eqn. (3.) i the generalization of the fundamental equation of log-aethetic curve. In comparion with log-aethetic curve, quadratic log-aethetic curve have additional parameter γ. If we modif γ near 0, we can generate curve cloe to log-aethetic curve guaranteeing the monotonicit of the curvature. If γ = 0, quadratic log-aethetic curve become eactl log-aethetic curve. d d d α =, γ = 0 α =, γ = 0.0 α =, γ = 0.0 Fig. Linear and quadratic logarithmic curvature graph 4 DERIVING THE EQUATIONS FOR QUADRATIC LOG-AESTHETIC CURVES Thi ection derive the curvature of quadratic log-aethetic curve a a function of arc length baed on Eqn. (3.). Once the curvature function i derived the curve can be drawn uing Frenet Serret formula. Uing d curvature, Eqn. (3.) can be written a log = γ ( log) αlog + d d. (4.) Eqn. (3.) and Eqn. (4.) are eentiall the ame equation. Taking the eponential of both ide of Eqn. (4.), d we get α+ γ log d = e d. (4.) Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

4 4 Let Λ= e d. The reaon that Λ i et to like thi i that the curve become a circular arc if Λ= 0 α []. Then Eqn. α + γ log (4.) become e erfi α+ γ log d γ = if γ 0 d Λ γ > Λ. (4.3) α α γ log Integrating Eqn. e (4.3) erf with repect to, we get ( ) = γ if γ < 0 γ Λ α if α 0 and γ = 0 αλ log if α = 0 and γ = 0. Λ (4.4) α α + γ log α e erfi erfi γ γ if γ > 0 γ Λ α Similarl a in [], we conider the tandard form. In the tandard form, we et = at = 0 α γ log α e erf erf. Then Eqn. (4.4) ( ) = become γ γ if γ < 0 γ Λ α if α 0 and γ = 0 αλ log if α = 0 and γ = 0. Λ (4.5 α α γe Λ+ erfi γ α+ γerfi γ e if γ > 0 ( ) Solving for, we get α α γe Λ+ erf ( ) = γ α γerf γ e if γ < 0 ( + αλ) α if α 0 and γ = 0 Λ e if α = 0 and γ = 0. (4.6) In Eqn. (4.6) and (4.7), the cae of γ = 0 are eactl the ame a the one in log-aethetic curve. If γ > 0 or γ < 0 erfi ( ) erf ( ), invere imaginar function or invere error function arie. For computing the invere of thee function, we ue a hbrid method combining the biection method and Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

5 5 the Newton method. In the region where z in Eqn. (.) or Eqn. (.) i near 0 where the Newton method i reliable, we ue the Newton method. Otherwie, the biection method i ued. In the biection method, the lower and upper bound of are -5 and 5 repectivel, and the lower and erfi( z) the upper bound of are -6 and 6 repectivel. Thee bound are derived a wa decribed in Section. if γ > 0 Since the curvature of a quadratic log-aethetic curve varie from 0 to α, we have to be careful about the bound of arc length. ma have upper and lower bound depending on γ, α α, and Λ erf. Putting = e + 0 and = into Eqn. γ (4.5), we get if 0 ( 0 γ ) = < γ Λ if α 0 and γ = 0 if α < 0 and γ = 0 αλ (4.7) if γ > 0 α α e + erf γ if γ 0 ( < ) = γ Λ if α > 0 and γ = 0 αλ if α 0 and γ = 0. (4.8) ( 0) ( ) and are the upper and lower bound of arc length, repectivel. Thee bound are necear for drawing quadratic log-aethetic curve. If ( 0) =, it mean that the arc length to the point at = 0 i infinit. Thu the inflection point doe not eit. In other word, the inflection point ( ) = i at infinit. Similarl, if, the arc length to the point at = i infinit. In ummar, the following propertie can be derived for quadratic log-aethetic curve. () The curve include an inflection point if α < 0 and γ = 0 or if γ < 0. () The curve include a point at = if α > 0 and γ = 0 or if γ < 0. (3) If γ < 0, the arc length of the curve i finite and it curvature varie from 0 to without depending on the value of α. (4) For curve with γ < 0, the arc length get longer a Λ get maller. ( ( ( ))) Propert (4) can be verified from Eqn. (4.7) and (4.8). + erf α / γ in the numerator of γ < 0 in ( 0,). Thu ( 0) Eqn. (4.7) i alwa poitive ince get larger a Λ get maller, ince the numerator of γ < 0 in Eqn. (4.7) are poitive. Similarl, ince the numerator of γ < 0 in Eqn. (4.8) are ( ) negative, get maller a Λ get maller. Thu the arc length of the curve with γ < 0, which i ( 0) ( ), get longer a Λ get maller. In the limit of Λ= 0, the arc length of the curve become infinite with it curvature being a contant. Once Eqn. (4.6) and the bound of arc length are derived, the curve can be drawn b integrating the Frenet Serret formula with an initial condition. Similarl a in [5], we et the tangent and the curve point at = 0 a [ 0 ]T and the origin, repectivel. Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

6 6 α =, γ = 0. α = 0, γ = 0.5 α =, γ = 0 α = 0, γ = 0 α =, γ = 0. α = 0, γ = 0.5 Fig. 3 α = with different γ Fig. 4 α = 0 with different γ Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

7 7 α =, γ = 0.5 α =, γ = α =, γ = 0 α = 0, γ = α =, γ = Fig. 5 α = with different γ Fig. 6 α = 3, γ = γ = with different α Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

8 8 γ = γ = 0.5 γ = 0. (d) γ = 0 (e) γ = (f) γ = 0.5 Fig. 7 α = with variou γ ( Λ= ). (g) γ = 0. Λ= 0. Λ= 0.5 Λ= (d) Λ=.5 Λ= Fig. 8 α = 0., γ =0.5 Fig. 9 α =, γ =30, Λ= 0.5 Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

9 9 5 RESULTS We have implemented the code for drawing a quadratic log-aethetic curve including it logarithmic curvature graph and curvature plot uing C++ on an Intel Core i7.ghz proceor. In our eperimental tem, the curve and it related graph are hown in real time b interactivel modifing the value of γα, and Λ. Fig. 3, 4, 5 and 6 how quadratic log-aethetic curve with variou αγ, and their logarithmic curvature graph and curvature plot. In all of Fig. 3, 4, 5 and 6, Λ i et to. In each of to of thee figure, the upper left figure i a quadratic log-aethetic curve, the upper right i the logarithmic curvature graph, and the bottom i the curvature plot. Fig 3, 4 and 5 are the curve of α =, 0,, repectivel, with variou γ. A hown in Eqn. (4.4) and (4.5), the arc length of the quadratic logaethetic curve i finite if γ < 0 with it curvature varing from 0 to infinit. Thee cae are hown in Fig. 3, 4, 5 and 6. If γ = 0, quadratic log-aethetic curve become eactl log-aethetic curve. If γ > 0, the arc length of the curve become infinite (Fig. 3, for eample). In other word, the point at = 0 and = are at infinit. Fig. 7 how how a log-aethetic curve ( α =, the Clothoid) change b modifing the quadratic coefficient γ. Fig. 8 how the curve of α = 0., γ =0.5 but with different Λ. A Λ i decreaed, the arc length of the curve get longer if γ < 0. The hape of the curve alo change if the value of Λ i changed ecept when α = []. Fig. 9 how an intereting eample of α =, γ = 30, Λ= 0.5. From the curve hape and it curvature plot, the curve get cloer to the circular arc (contant ) a get cloe to ±. Thi fact mean that G interpolation algorithm of [5] doe not work properl ince more than one curve that fit the pecified control triangle ma be found. Thu a different method i necear and we are working with thi problem. 6 CONCLUSIONS: Thi paper propoed quadratic log-aethetic curve b etending log-aethetic curve o that the logarithmic curvature graph become quadratic. We derived the curvature function in term of arc length for drawing the curve and clarified the characteritic. We have implemented the propoed curve in C++ and confirmed that the curve can be generated full in real time. Quadratic logaethetic curve have additional degree of freedom γ and can repreent a curve egment with finite arc length and the curvature varing from 0 to if γ < 0. Future work include more detail anali of the characteritic of the curve, and the application to G and G Hermite interpolation. ACKNOWLEDGEMENT: Thi work wa upported b JSPS KAKENHI Grant Number Norimaa Yohida, Takafumi Saito, RERFERENCES: [] Gobithaaan, R.U.; Miura, K.T.: Aethetic Spiral for Deign, Sain Malaiana, 40(), 0, [] Harada, T.: Stud of quantitative anali of the characteritic of a curve, FORMA, (), 997, [3] Inoue, J.; Harada T.; Hagihara, T.: An Algorithm for Generating Log-Aethetic Curved Surface and the Development of a Curved Surface Generation Stem uing VR, IASDR009 proceeding, 009. [4] Kineri, Y.; Endo, S.; Maekawa, T.: Surface deign baed on direct curvature editing, Computer-Aided Deign, 55(0), 04, -. DOI: 0.06/j.cad Computer-Aided Deign & Application, 4(), 07, CAD Solution, LLC,

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