Journal of Computational and Applied Mathematics

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1 Journal of Computatonal and Appled Mathematcs 236 (212) Contents lsts avalable at ScVerse ScenceDrect Journal of Computatonal and Appled Mathematcs journal homepage: G 3 quntc polynomal approxmaton for Generalsed Cornu Spral segments Benjamn Cross, Robert J. Crpps Geometrc Modellng Group, School of Mechancal Engneerng, Unversty of Brmngham, Brmngham, B15 2TT, UK a r t c l e n f o a b s t r a c t Artcle hstory: Receved 24 September 211 Receved n revsed form 26 January 212 Keywords: Quntc Bezer polynomal Approxmaton Generalsed Cornu Spral Wthn Computer Aded Desgn (CAD) there s a need to construct far curves. The Generalsed Cornu Sprals (GCSs) are a set of curves wth a monotonc curvature profle and are hence consdered far but mplementaton n current CAD systems s not straghtforward, partly due to not beng n the usual polynomal form. A method to approxmate a GCS usng a quntc polynomal curve s presented. The method seeks to nterpolate the GCS to satsfy the G 3 constrants at the end ponts wth a quntc Bézer, leavng two degrees of freedom. An ntal approxmaton s shown to be effectve for the majorty of GCS curves. Moreover, t s possble to determne when an ntal approxmaton s lkely to be poor. If ths approxmaton does not meet the tolerance requred, a search nvolvng two parameters s performed. Characterstcs of the search doman are used to establsh a sutable startng value. 212 Elsever B.V. All rghts reserved. 1. Introducton Wthn CAD there s a need to construct far curves [1]. A planar curve s consdered far f the curvature profle, the curvature plot wth respect to (w.r.t.) arc length, conssts of relatvely few smooth monotonc segments [2]. A common curve form wthn CAD s the Bézer representaton [3]. Usng ths representaton drectly t s dffcult to control the curvature; the polynomal nature of the Bézer curve often produces undulatng oscllatons n curvature [4]. An alternatve approach s to defne a curve drectly from ts curvature profle to ensure farness. A flexble set of curves defned n ths way are the Generalsed Cornu Sprals (GCS) [5]. These curves are defned to have a ratonal lnear monotonc curvature profle. Explctly, the curvature profle of a GCS has the form: κ(s) = κ S + (κ 1 κ + rκ 1 )s s [, S], r ( 1, ), S + rs where the parameter s represents the arc length of the curve, κ and κ 1 represent the start and end curvature values respectvely, S corresponds to the length of the GCS and r s referred to as the shape factor. GCS curves are useful n span generaton such as transton curves between two data ponts [5]. Ths s partcularly true for the Cornu spral, whch s tself a member of the GCS famly, havng applcatons n hghway desgn, robot trajectores and roller coaster desgn [6,7]. Rewrtng the GCS as κ(s) = p + qs S + rs wth p = κ S and q = κ 1 κ + rκ 1, a famly of degenerate curves can be represented. Correspondng author. Tel.: ; fax: E-mal address: r.crpps@bham.ac.uk (R.J. Crpps) /$ see front matter 212 Elsever B.V. All rghts reserved. do:1.116/j.cam

2 3112 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) If p = and q = then κ(s) = and the curve s a straght lne. If q = and r = then κ(s) = p = κ S and the curve s a crcular arc. If q and r = then κ(s) = (p+q)s and the curve s a Cornu spral. If q = and r then κ(s) = p S (S+rs) and the curve s a logarthmc Spral. The curvature profle however only descrbes the shape of the curve. Translatons or rotatons do not affect the curvature. Therefore, n order to completely defne the curve n R 2, ts ntal locaton, (x, y ), and orentaton, θ, must be defned [8]. The orentaton s decded by the angle, θ, that the ntal tangent vector makes wth the postve x-axs. Scalng of the curve can also be accounted for. Applyng a scalng factor λ, new values for the parameters can be calculated as: S = λs, κ = κ λ, κ = κ 1 1 λ and r = r [8]. These propertes lead to a defnton of a normalsed GCS. Gven a GCS, t s always possble to express t n normal form. Ths s acheved by frstly translatng the GCS to the the orgn (therefore settng x =, y = ). The curve s then rotated so that the ntal tangent ponts along the postve x-axs (θ = ). Fnally the curve s scaled to make the arc length equal to 1. The normalsed GCS curve can then be synthessed parametrcally from [5]: x(t) = y(t) whch gves x(t) = y(t) x(t) = y(t) t σ x + cos θ + κ(s)ds dσ t σ y + sn θ + κ(s)ds dσ t σ p + qs cos = 1 + rs ds dσ t σ p + qs sn 1 + rs ds dσ t t t t cos pσ + 12 qσ 2 dσ sn pσ + 12 qσ 2 dσ f r =, rqσ + (pr q)ln(1 + rσ ) cos dσ r 2 rqσ + (pr q)ln(1 + rσ ) otherwse. sn dσ r 2 In ths ntegral form t s not always possble to represent ths curve as a composton of fundamental functons. Apart from trval exceptons, for example the crcle, the solutons are thus usually found by numercal ntegraton [5]. Ths representaton makes them undesrable for drect use wthn CAD. Ths paper outlnes a method whch helps ncorporate the GCS nto CAD systems. The dea s to produce an approxmaton that accurately reflects the curvature profle therefore mmckng the behavour of a GCS. Specfcally a quntc polynomal n Bézer form wll be used. The polynomal s of quntc order so as to accommodate geometrc arguments wthout leavng too many degrees of freedom and the Bézer form s used to explot ts geometrcal propertes. Exstng curve approxmaton research focuses on degenerate curve forms of the GCS such as the crcle [9], Cornu spral [1] and logarthmc spral [11]. These approxmaton methods measure the error wth respect to the poston of the curve. Consequently, the curvature profle of the approxmaton s not controlled. These approxmaton methods are also not flexble enough to accommodate an approxmaton to a general GCS curve. In [9] symmetrcal propertes of the crcle are exploted, whereas [11] uses an equangular property; both propertes of whch do not extend to a general GCS. A lnear curvature profle of a Cornu spral s studed n [1] wheren an approxmaton to the Fresnel ntegrals s presented. However, a general GCS s curvature profle s not lnear but ratonal lnear and so s not expressed usng Fresnel ntegrals. Another way to approxmate a GCS curve could be to use a Hermte nterpolant [6]. Ths method nterpolates the start and end dervatve data, of order k, from the GCS wth the unque polynomal of order (2k + 1) whch share these values. If the approxmaton s not satsfactory, a hgher order nterpolant can be used. However, n order to obtan a satsfactory approxmaton, k may be too large for practcal use. Smlarly, a transton curve across G 2 pont data of the GCS could be used as an approxmaton. A method n whch a Pythagorean Hodograph (PH) quntc curve s used to transton between two crcles s outlned n [12]. However, ths method only outlnes how to blend between crcles and not G 2 pont data. The curvature profle may also contan oscllatons snce monotoncty s not guaranteed. (1)

3 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) A G 2 method s outlned n [13] whch uses a par of PH quntc spral segments. However, the requrement that the two PH curves have zero curvature at ther ntersecton ponts would result n an unnecessary nflecton pont. Thus for non-nflectng GCS curves t would be a poor approxmaton. Research has been carred out to create an approxmaton method for a general GCS [4]. A method exsts whch uses a quntc Bézer; matchng the G 2 condtons at the end ponts leavng four degrees of freedom. These four degrees of freedom are then vared n a search untl an error wthn a tolerance s acheved. Ths error s measured wth respect to the relatve curvature error between the GCS and the quntc Bézer approxmatng t. Although ths method produces approxmatons to a GCS, ts relance on a numerc search makes t computatonally expensve. Each confguraton of the four free varables requre an error functon to be calculated. Each tme ths functon s calculated the quntc must be reparametersed w.r.t. arc length and the curvature profles compared. Thus ths method s mpractcal for CAD mplementaton. The proposed method seeks to nterpolate the GCS to satsfy the G 3 constrants at the end ponts wth a quntc Bézer, leavng two degrees of freedom. The G 3 constrants are shown to have the propertes of a G 2 nterpolaton that also matches the frst dervatve of the curvature profle at the start and end ponts. The two free varables hold geometrc sgnfcance and an ntal value for these varables can be argued for. Ths provdes an ntal approxmaton for the GCS. However, for a small subset of GCS curves ths ntal approxmaton s unacceptable. In ths case a search s employed on the two ndependent varables. Characterstcs of the searchng doman are utlsed to manpulate a searchng algorthm to generate approxmatons more effcently by selectng an approprate startng value. The new method ams to reduce the computatonal expense requred to create an approxmaton to the GCS. Ths s done by reducng the necessty for a search. However, f a search s stll requred the degrees of freedom are reduced from the four n [4] to two. The remander of the paper s organsed as follows. Secton 2 ntroduces the background theory. Secton 3 descrbes the approxmaton method n detal. Analyss of the performance of the method s consdered n Secton 4. Fnally, some concludng remarks on the method s effectveness and on possble mprovements are gven n Secton Prelmnares 2.1. GCS dervatve nformaton Gven a normalsed GCS segment, t s possble to calculate the dervatve nformaton at the start and end ponts. Recall that the parametersaton for a GCS, F(t) for t [, 1], s gven by (1). The dervatve data at the endponts can thus be calculated as: F() = (, ) F(1) = (x, y) F () = (1, ) F (1) = (cos θ, sn θ) F () = (, κ ) F (1) = κ 1 ( sn θ, cos θ) F () = F () κ() κ () F ()κ() 2 = ( κ 2, (κ 1 κ )(1 + r)) F (1) = F (1) κ(1) κ (1) F (1)κ(1) 2 = F (1) (κ 1 κ ) F (1)κ 2 1 (1 + r) where represents dfferentaton w.r.t. the parameter t, (x, y) s the end pont and θ s the wndng angle calculated from: θ = 1 = κ + κ 1 2 κ(s)ds = 1 f r =, (κ 1 κ + rκ 1 )s + κ ds rs + 1 = r(κ 1(1 + r) κ ) + (1 + r)(κ κ 1 )ln(1 + r) r 2 otherwse. The end ponts can be calculated by numercal ntegraton of (1), such as Romberg ntegraton [14]. Let the th dervatve w.r.t. t evaluated at s = a be denoted by D a so that F() (a) = D a = D a,x, D a,y. For example F () = D 2 = (, κ ). The dervatve data can therefore be represented by {D j }=,1,2,3 j=, Quntc Bézer dervatve nformaton The proposed method uses a quntc polynomal n Bézer form as the approxmatng functon. The quntc Bézer s a quntc polynomal governed by the control ponts V and s defned by:

4 3114 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) V(u) = = 5 B 5 (u)v = 5 5 (1 u) 5 u V. = Dervatves of the quntc Bézer can be calculated from [3]: d k V(u) = 12 5 k k k ( 1) k j V du k +j B 5 k (u). (5 k)! j = Start and end pont dervatve data are therefore gven by: j= V() = V V(1) = V 5 V () = 5(V 1 V ) V (1) = 5(V 5 V 4 ) V () = 2(V 2 2V 1 + V ) V (1) = 2(V 5 2V 4 + V 3 ) V () = 6(V 3 3V 2 + 3V 1 V ) V (1) = 6(V 5 3V 4 + 3V 3 V 2 ) Geometrc contnuty The approxmaton method proposed n ths paper seeks an nterpolatng quntc Bézer to satsfy the G 3 condtons wth the GCS at the end ponts. In order to understand the geometrc sgnfcance of the G 3 condtons, frst ther dervaton must be explaned. The G 3 condtons are derved from the defnton of thrd order geometrc contnuty [15]. Geometrc contnuty s a specal case comparable to standard parametrc contnuty. For nth order parametrc contnuty, C n, between two vector functons f(t) and g(u) at a then t must be that: d dt f(a) = d du g(a) =... n.e. ther dervatves match exactly. Geometrc contnuty extends from ths defnton but uses the arc-length parametersaton. The defnton states that f(t) and g(u) are G n contnuous at a f and only f ther arc length parametersatons meet wth C n contnuty at a [15]. That s: d ds f(a) = d ds g(a) =... n (2) where the parameter s represents arc length. Another mportant equvalent defnton for geometrc contnuty allows for the functons f(t) and g(u) to be reparametersed. Ths reparametersaton however must produce geometrcally equvalent curves and so must concde wth each other at the end ponts and le n the same range [15]. For nth order geometrc contnuty, G n, between two functons f(t) and g(u) at a t must be that; there exsts a parametersaton g(r(u)) equvalent to g(u) and that [15]: d dt f(a) = d g(a) =... n. du Ths defnton s used to create a coeffcent matrx wth accompanyng shape factors α = 1... n. The α s are real, free varables whch relate to the reparametersaton functon [15]. The coeffcent matrx correspondng to G 3 contnuty between f(t) and g(u) at a can be calculated as: f(a) 1 g(a) f (a) α 1 g (a) f (a) = α 2 α 2 f 1 g (a). (3) (a) α 3 3α 1 α 2 α 3 g (a) 1 Notce C 3 contnuty can be acheved by settng α 1 = 1 and α 2 = α 3 =. In general parametrc contnuty can be acheved from geometrc contnuty by settng the shape factors α 1 = 1 and α = for = 1... n Measurng the accuracy of the approxmaton Part of the motvaton behnd usng a GCS curve s that the curvature s a smooth monotonc functon. Ths property, whch ensures farness, s what the approxmaton should be mmckng. Therefore when questonng how good an approxmaton s, a measure of error should reflect on ts curvature.

5 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) Ths suggests that rather than usng tradtonal measures of error, such as Hausdorff dstance, the curvature profles of the GCS and the approxmaton should be compared; as proposed n [4]. However, before any comparson can be made a curvature profle of the approxmaton must be calculated. Ths nvolves reparametersng the curve so that the parameter reflects arc-length (as s the case for the GCS). The arc-length at parameter value t can be calculated as a(t) = t V (τ) dτ. A possble approach to calculate the functon a(t) could be to use a numercal ntegraton method, for example Romberg ntegraton [14]. The arc-length parametersaton of V(t) has the form V(a 1 (s)) for s [, S b ] where S b s the total length of the curve. Before the two curvature profles are compared, a 1 (s) s normalsed by a factor 1 so that the new curvature functon S b κ b (s) and the normalsed GCS curvature functon κ g (s) both range over the values [, 1] as proposed n [4]. Two ways to compare these curvature functons are the absolute curvature dfference (ϵ a ) or the relatve curvature dfference (ϵ r ); ϵ a (κ b, κ g ; s) = κ b (s) κ g (s) and ϵ r (κ b, κ g ; s) = κ b(s) κ g (s). κ g (s) The absolute curvature dfference s not a good measure when dealng wth large curvature values. On the other hand, relatve curvature dfference s not a good measure when dealng wth curvature values close to. A compromse was formed n [4] by only usng the absolute value when suffcently close to zero curvature. However, the value on whch ths change occurs was arbtrary and not well defned and so a new measure s consdered. Choosng the error functon, ϵ, as the mnmum of the absolute and relatve dfference not only s ϵ well defned t s also contnuous: ϵ(κ b, κ g ; s) = mn{ϵ a (κ b, κ g ; s), ϵ r (κ b, κ g ; s)} = κ b(s) κ g (s) max{ κ g (s), 1}. An approxmaton can thus be assgned an error value equal to the maxmum error experenced throughout the doman of functon. That s: κ b (s) κ g (s) ϵ = ϵ(κ b, κ g ) = max s [,1] max{ κ g (s), 1}. Approxmatons are then sought such that the error s wthn some tolerance.e. ϵ µ. If µ s too large the level of accuracy dmnshes. Conversely, f µ s too small then the tolerance cannot always be met. Although n theory µ can be set to any value, a reasonable suggeston that ensures hgh qualty defnton s µ =.5 [16]. For the remander of the paper an approxmaton s classfed as acceptable f ϵ The method Before the G 3 approxmaton s presented another well known approxmaton method, a quntc C 2 Hermte splne [6], s appled to a GCS n order to gan some nsght nto the shape parameters α. Ths method nterpolates the parametrc dervatve data, up to second order, of the functon to be approxmated. An example of a C 2 Hermte approxmaton to a GCS, along wth the curvature plot, s gven n Fg. 1. At the start and end ponts the approxmaton method matches the second dervatve data and thus the curvature values agree. However, the tangents of the curvature profle do not agree and so an approxmaton could be mproved by ensurng ths property. Mathematcally, the followng condtons need to be satsfed: d ds κ V () = d ds κ F () d ds κ V (1) = d ds κ F (1) where κ V (u) and κ F (t) are the curvature functons for V(u) and F(t) respectvely. Snce the curvature profle s just the second dervatve of a functon w.r.t. arc length, (4) s equvalent to the G 3 contnuty condtons (as s shown by the followng equaton usng (3)). d 3 ds 3 V(a) = d ds d 2 ds 2 V(a) = d ds κ V (a) = d ds κ F (a) = d3 ds 3 F(a). (4)

6 3116 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) Fg. 1. [a] A C 2 Hermte approxmaton (black) to the normalsed GCS (κ = 1, κ 1 = 3, r =.75) (grey). [b] The correspondng curvature plot. [c] A close up of the ntal curvature. [d] The relatve curvature error (ϵ =.45). Therefore n order to satsfy property (4) usng a quntc Bézer to approxmate a GCS, the followng sets of equatons can be formed va (2). Shape factors β 1, β 2, β 3 are used for the ntal pont condtons and γ 1, γ 2, γ 3 are used for the end pont condtons, note that the β, γ are real parameters ndependent of each other. The geometrcal constrants at the start pont are: V 5(V 1 V ) 2(V 2 2V 1 + V ) 6(V 3 3V 2 + 3V 1 V ) = 1 β 1 β 2 β 2 1 β 3 3β 1 β 2 β 3 1 For the end pont, the geometrcal constrants are: V 5 1 5(V 5 V 4 ) γ 1 2(V 5 2V 4 + V 3 ) = γ 2 γ 2 1 6(V 5 3V 4 + 3V 3 V 2 ) γ 3 3γ 1 γ 2 γ 3 1 After some manpulaton the constrants can be seen to be: V 6 V 1 V = β β β 1 3β 2 1 V 3 6 β 3 + 9β β 1 9β 2 + 3β 1 1β 2 β 3 1 V γ 1 V 4 V 3 V 2 = 1 6 D D 1 D 2 D 3 D 1 D 1 1 D 2 1 D γ 2 24γ 1 3γ γ 3 + 9γ 2 36γ 1 9γ 2 1 3γ 1γ 2 γ 3 1 Then V, V 1, V 4, V 5 are completely defned by the dervatve data and the shape factors β 1 and γ 1. However, V 2 and V 3 need to smultaneously satsfy the constrants at the start and end ponts. That s, the equatons nvolvng V 2 and V 3 :.. D D 1 D 2 D 3 6D + (3β β 1 )D 1 + 3β2 1 D2 = 6D 1 + ( γ 3 + 9γ 2 36γ 1 )D (9γ 2 1 3γ 1γ 2 )D 2 1 γ 3 1 D3 1, 6D + (β 3 + 9β β 1 )D 1 + (9β β 1β 2 )D 2 + β3 1 D3 = 6D 1 + (3γ 2 24γ 1 )D γ 2 1 D2 1, need to be satsfed., D 1 D 1 1 D 2 1 D 3 1.

7 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) There are sx degrees of freedom, namely the sx shape parameters, and four equatons to be satsfed, two each for the x and y values of V 2 and V 3. The four parameters β 2, β 3, γ 2, γ 3 behave lnearly wthn these four equatons hence a soluton can be determned for these parameters gven nformaton about β 1 and γ 1 (snce the dervatve data D j s known). The followng soluton was calculated usng Maple software [17]: where β 2 = B(β 1, γ 1 ) D(β 1, γ 1 ) and γ 2 = G(β 1, γ 1 ) D(β 1, γ 1 ), (5) B(β 1, γ 1 ) = 1 3 6D 1 1,y D1 1,x D 1,y 3γ 3 1 (D2 1,y )2 D 1 1,x (D1 1,y )2 γ 3 1 D3 1,x + 3γ 3 1 D2 1,y D2 1,x D1 + 1,y D1 1,y D1 γ 3 1,x 1 D3 1,y 6D 1,y D1 1,x D2 γ 1,y 1 24β 1 (D 1 1,y )2 + 6(D 1 1,y )2 D 1,x β3 1 D3,y D2 γ 1,x 1D 1 1,y + β 3 1 D3,y D1 1,x D2 γ 1,y 1 + 9D 2,y β2 1 D1 1,x D2 γ 1,y 1 + 6D 1,y D2 γ 1,x 1D 1 1,y 9D 2,y β2 1 D2 γ 1,x 1D 1 + 1,y 3D2,y D1 1,y D1 1,x β D1 1,y D1 γ 2 1,x 1 D2 1,y 15(D 1 1,y )2 γ 2 1 D2 1,x. G(β 1, γ 1 ) = 1 3 3D 1 1,x β3 1 (D2,y )2 6D 1 1,x D 1,y β 1D 2,y 3D1 1,y γ 2 1 D2 1,y + D 1 γ 3 1,x 1 D3 β 1,y 1D 2,y 15D1 1,y β2 1 D2 +,y 6D 1,x D1 β 1,y 1D 2,y + 9γ 2 1 D2 1,x D1 β 1,y 1D 2 γ 3,y 1 D3 1,x D1 β 1,y 1D 2 +,y D1 1,y β3 1 D3,y 6D 1 1,y D 1,y + 24(D1 1,y )2 γ 1 9D 1 1,x γ 2 1 D2 1,y β 1D 2,y D(β 1, γ 1 ) = β 1 γ 1 (D 1 1,x D2 1,y D2,y D2 1,x D1 1,y D2,y ) (D1 1,y )2. Moreover, t suffces to only calculate the shape factors β 2 and γ 2 as β 3 and γ 3 are not requred to defne the control vertces. Ths s because the values for the control ponts V are: V = D, V 1 = β 1 5 D1 + D, V 2 = β2 1 2 D2 + β β 1 D D, V 3 = γ D2 + γ γ 1 D D, 1 V 4 = γ 1 5 D1 1 + D 1, V 5 = D 1. Although β 3 and γ 3 are not requred they can be used as a check to valdate the G 3 condtons at V 2 and V 3. Ther dervaton s shown n the Appendx. Thus an approxmaton s completely defned by the dervatve data of the GCS along wth values for β 1 and γ 1. The two free parameters β 1 and γ 1 have geometrc sgnfcance. Notce from the start and end pont constrants that: 5(V 1 V ) = β 1 D 1, 5(V 5 V 4 ) = γ 1 D 1 1. Therefore β 1 and γ 1 relate to the magntude of the vectors V 1 V and V 5 V 4 respectvely and hence the magntude of the ntal and end tangents. Recall from Secton 2.3 that C 2 contnuty s equvalent to G 2 contnuty when β 1 = γ 1 = 1 and β 2 = γ 2 =. Drawng nsght from the Hermte C 2 nterpolant a sensble ntal value for the shape factors could therefore be β 1 = γ 1 = 1. In Fg. 2 examples of ths ntal approxmaton are gven along wth the curvature profles and relatve curvature errors. From Fg. 2 t s apparent that ths approxmaton method wll not always suffce. Ths s generally caused by unreasonable values for β 2 and γ 2. If these values are too large the vertces V 2 and V 3 are a relatvely large dstance away from V 1 and V 4 respectvely..

8 3118 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) Fg. 2. [a, d] An ntal G 3 approxmaton (black) to the normalsed GCS (grey) ([a] - κ = 1, κ 1 = 3, r =.75; [d] - κ =.36, κ 1 = 2.7, r =.1). [b, e] The correspondng curvature plots. [c, f] The relatve curvature errors ([c] - ϵ =.16; [f] - ϵ =.23). Ths problem s caused when the denomnator, D(β 1, γ 1 ), n (5) approaches. As the denomnator, D(β 1, γ 1 ), tends to the approxmaton becomes more unreasonable, caused by a dvergence of the values β 2 (β 1, γ 1 ), γ 2 (β 1, γ 1 ) towards. Ths behavour occurs when the denomnator s suffcently close to. Numercal testng suggests unreasonable approxmatons occur when D(β 1, γ 1 ).1. Recallng the dervatve data from Secton 2.1, the denomnator of an approxmaton can be calculated usng: D(β 1, γ 1 ) = β 1 γ 1 (D 1 1,x D2 1,y D2,y D2 1,x D1 1,y D2,y ) (D1 1,y )2, = β 1 γ 1 (cos(θ)κ 1 cos(θ)κ ( κ 1 sn(θ) sn(θ)κ )) sn 2 (θ), = β 1 γ 1 κ κ 1 sn 2 (θ). (6) 3.1. A search method If an ntal approxmaton s unacceptable (.e. ϵ > µ), a numercal search nvolvng the free parameters (β 1, γ 1 ) s employed. The search s measured wth respect to the error functon ϵ whch s to be mnmsed and s performed untl wthn a desred tolerance ϵ µ. When employng a search algorthm an ntal startng pont s often requred [14]. Establshng an approprate startng pont s essental to ncrease the effcency of the rate of convergence to a soluton and the lkelhood of yeldng one. Ths mples a search should begn not only close to a soluton but also n a stable regon. A crtera, P, for a possble startng pont can be defned wth the am of ncreasng the effcency of the search. Propertes that the startng pont should possess are frst establshed and then used to restrct values to defne P. Recall that β 1 and γ 1 reflect the sze of the ntal and end tangent vectors. In Bézer form ths translates to fve tmes the dstance between V to V 1 and V 5 to V 4. Obvously these values should not be negatve, otherwse the tangents wll pont n the opposte drecton. Thus a lower bound of β 1, γ 1 > s chosen. Intutvely, β 1 and γ 1 should also not be too large as ths would result n V (V 5 ) beng too far from V 1 (V 4 ). Therefore an upper bound for these values should also be defned.

9 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) Recall that the arc-length of the curve s S b = 1 V(τ) dτ 1. Consderng the smplest Romberg approxmaton (equvalent to the trapezod rule) [14] then S V() V V(1) = 1 4 β V γ 1 > 1 4 (β 1 + γ 1 ). Therefore, a sutable upper bound s β 1, γ 1 < 4. To avod the functon tendng to nfnty locally, the ntal value (β 1, γ 1 ) s restrcted such that the denomnator, D(β 1, γ 1 ) >.1. Therefore the crtera, P, can be defned as: P = {(β 1, γ 1 ) [, 4] [, 4] : D(β 1, γ 1 ) >.1}. Any value n P could be used to start a search, however better startng ponts can be argued for. Prevously, t was argued that large values for β 2 (β 1, γ 1 ) and γ 2 (β 1, γ 1 ) contrbuted to a poor approxmaton. Therefore an dea mght be to control these values. Agan drawng nspraton from the C 2 Hermte model consder β 2 = γ 2 =. Ths mples that the frst and second parametrc dervatves are orthogonal, a property that agrees wth the GCS. Therefore fndng values whch yeld β 2 (β 1, γ 1 ) = γ 2 (β 1, γ 1 ) = would be a good suggeston for a startng value. To calculate whch values of β 1 and γ 1 gve β 2 = γ 2 =, a set of equatons must be solved, equvalent to B(β 1, γ 1 ) = G(β 1, γ 1 ) = from (5). These bvarate polynomals have at most 16 solutons by Bezout s Theorem [18]. These solutons must also satsfy P n order to be a sutable ntal pont. If there are multple sutable solutons then the one wth the smallest error should be used. In the rare event no such solutons exst, a random value n P can be chosen as the startng pont for the search. A search may not be necessary f ths ntal value already meets the tolerance. If ths s not the case a search n 2-dmensons can now be mplemented. Dervatve data of the ϵ functon s dffcult to calculate and so a sutable numercal mnmsaton routne s Powell s method [14]. The method begns from an ntal pont and mnmses along a 1-dmensonal lne of the 2-dmensonal doman. A sutable ntal 1-dmensonal lne of the 2-dmensonal search doman s (β 1, γ 1 ) = (1, 1) whch maxmses the dfference between β 1 and γ 1. The drecton of the lne s then altered after every teraton [14] and a new mnmum sought. Ths mnmum s guaranteed to be at most the sze of the prevous mnmum. The process s repeated untl ether a desred tolerance, ϵ µ, s acheved or the search fals. Ths can occur because ether the number of teratons exceeds a preset number or a local mnma has been dscovered such that ϵ > µ. If ths scenaro were to occur then two possble courses of acton are to ether ncrease the tolerance, µ, or splt the curve nto smaller segments The algorthm A short summary of the constructon algorthm follows. As soon as an acceptable approxmaton s found the algorthm fnshes. Step 1. Apply the ntal G 3 approxmaton wth β 1 = γ 1 = 1. If ϵ µ then the approxmaton s acceptable. Otherwse a search s requred. Step 2. Fnd an ntal pont for the search. Begn by solvng β 2 = γ 2 =. Step 3. Pck the soluton wth smallest ϵ that also satsfes P. If none exst then pck a random value n P. Step 4. Apply Powell s method from the startng value, re-teratng untl ether an acceptable value s found or search fals. Step 5. Ether splt the curve or ncrease the tolerance, µ. Then start the approxmaton agan. 4. Examples and analyss In ths secton examples of approxmatons to several GCSs are gven. To begn an approxmaton to the quarter crcle s gven followed by an nflectng GCS curve. Then a GCS whose ntal approxmaton produces a denomnator less than.1 s consdered. Analyss of the curve and the curvature error s presented and dscussed. Fnally a graph whch shows the accuracy for the ntal β 1 = γ 1 = 1 approxmaton across a range of Cornu sprals s presented. Ths ndcates how often a search may be requred. A tolerance of µ =.5 s used throughout.

10 312 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) Fg. 3. [a] An approxmaton (black) to the quarter-crcle (grey). [b] The relatve curvature error (ϵ = ). Fg. 4. [a] An approxmaton (black) to the normalsed GCS (grey) (κ = 1.5, κ 1 = 1.6, r =.5). [b] The relatve curvature error (ϵ =.14). Fg. 5. [a] An approxmaton (black) to the normalsed GCS (grey) (κ =.36, κ 1 = 2.7, r =.1). [b] The relatve curvature error (ϵ =.79). An approxmaton to the quarter crcle can be found by consderng the normalsed GCS wth κ = κ 1 = π 2, r =, S = 1. The ntal approxmaton consders β 1 = γ 1 = 1. Ths gves β 2 =.78, γ 2 =.78 and ths yelds an error of ϵ = An llustraton of ths approxmaton s gven n Fg. 3. By means of comparson a C 2 Hermte approxmaton has an error of ϵ =.3. Next, a normalsed nflectng GCS wth κ = 1.5, κ 1 = 1.6, r =.5 and S = 1 s consdered. The ntal approxmaton wth β 1 = γ 1 = 1 gves β 2 =.46 and γ 2 =.12. The error s ϵ =.14 and an llustraton of ths approxmaton s gven n Fg. 4. The Hermte approxmaton yelds an error of ϵ =.19. Fnally, the normalsed GCS wth κ =.36, κ 1 = 2.7, r =.1 from Fg. 2 s reconsdered. The denomnator s calculated as D =.28. Snce D s close to zero ths suggests that the ntal β 1 = γ 1 = 1 approxmaton method s unsutable. Ths s confrmed when (β 2, γ 2 ) s calculated as ( 5.82, 2.) whch produces an error ϵ =.23. Thus a search on ϵ(β 1, γ 1 ) should be mplemented. An ntal value for the search must be decded. Solvng β 2 = γ 2 = usng Maple software 14 solutons were found [17] all of whch gve orthogonal frst and second dervatves. Sx solutons were complex, three were negatve, and three had a denomnator less than.1. Recallng β 1, γ 1 must be real, postve and gve a denomnator greater than.1 to le n P, these solutons were dscarded. The remanng soluton wth least error had a startng value of (β 1, γ 1 ) = (.73, 1.11). Ths gave an error of ϵ =.6. A search was then mplemented from ths ntal pont. After a sngle teraton of Powell s method a better approxmaton wth (β 1, γ 1 ) = (.7, 1.15) and an error of ϵ =.79 was found. Ths approxmaton s gven n Fg. 5. An llustraton of the searchng doman s gven n Fg. 6. As a comparson the Hermte approxmaton yelds ϵ =.25. For completeness the other soluton of (β 1, γ 1 ) = (1.87,.6) s also consdered. The ntal pont has an error of ϵ =.86. After 1 teraton (β 1, γ 1 ) = (1.33,.55). Ths gves an error of ϵ =.1 so after 1 teraton the approxmaton s acceptable.

11 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) Fg. 6. [a] The error (ϵ) as a functon of the shape parameters β 1, γ 1 defne the search doman. [b] Ths mage hghlghts the regon wth a zero denomnator (grey) and the sequental approxmatons of [β 1, γ 1 ] = [1, 1], [.73, 1.11] and [.7, 1.15] (whte). Fg. 7. [a] The error (ϵ) of the ntal β 1 = γ 1 = 1 approxmaton for a varety of Cornu sprals wth κ, κ 1 = π... π. [b] Ths mage hghlghts when the denomnator functon s zero (grey). A drect comparson between the effcency of the proposed method and the method n [4] would be napproprate. Ths s because a search routne s not defned n [4]. Furthermore, the choce of parameters and ntal values means that Powell s method s not sutable. However, snce the proposed method only has two degrees of freedom t s ntutve to expect t to converge faster than a search wth four parameters as n [4]. Another mportant property of the proposed method s that a search s not always requred makng t a much more effcent approach than [4]. The fnal fgure, Fg. 7, llustrates the effectveness of the β 1 = γ 1 = 1 ntal approxmaton for a range of cornu sprals and hence how often a search s requred. The values for ntal and end curvatures are vared from π to π From the fgure t s clear that most approxmatons result n an mmedate satsfactory approxmaton (99% of the regon). In fact the only tme that the error exceeds the tolereance, ϵ > µ =.5, s when the denomnator s close to zero. Ths s shown by the proxmty to the grey regon where the denomnator equals zero. 5. Concludng remarks A method to approxmate GCS curve segments usng quntc polynomals has been presented. Geometrc arguments, the G 3 condtons, were used to defne all but two free shape parameters. An ntal approxmaton was shown to be acceptable across a wde range of GCSs. Moreover, t was possble to detect when ths ntal approxmaton was unlkely to be acceptable. Ths method mproves on the prevously best technque from [4] by reducng the necessty of a computatonally expensve search routne. If a search s requred, t s reduced from four free varables to two. Informaton about the characterstcs of the doman for the search functon helped desgn a more effcent search routne. Ths was acheved by usng Powell s method and startng n an area where the denomnator was not suffcently close to zero. The method also produced better approxmatons than the establshed C 2 Hermte method. Therefore ths technque s worthy of further consderaton. A search routne however can stll be computatonally expensve. Also there s a possblty that a search mght not fnd an acceptable approxmaton. It s therefore the ntenton to mprove the method so as not to requre a search. A possble

12 3122 B. Cross, R.J. Crpps / Journal of Computatonal and Appled Mathematcs 236 (212) way to acheve ths s to control the values for the two shape parameters (β 1, γ 1 ) whch would deally be derved wth a geometrcal justfcaton. Appendx. Calculaton of β 3 and γ 3 The followng soluton was calculated usng Maple software [17] where: β 3 = B(β 1, γ 1 ) D(β 1, γ 1 ) and γ 3 = G(β 1, γ 1 ) D(β 1, γ 1 ) B(β 1, γ 1 ) = 36(D 1 1,y )2 β (D 1 1,y )2 D 1,x + D3,x β4 1 D1 1,y D2 1,x γ 1D 2,y D 3,x β4 1 D1 1,x D2 1,y γ 1D 2,y + D1 1,x D2,y β 1D 1 1,y D3 1,x γ D 2 1,x γ 3 1 D1 1,x D2 1,y D2,y β D 1 1,x D2,y β 1D 1 1,y D2 1,x γ 2 1 6D 1,x D1 1,y D2 1,x γ 1D 2,y β 1 + 6D 1,x D1 1,x D2 1,y γ 1D 2,y β 1 D 2,y β 1(D 1 1,x )2 D 3 1,y γ 3 1 3D1 1,y D2 1,x γ 1D 3,y β D 1 1,y D2 1,x γ 1D 2,y β D1 1,x D2 1,y γ 1D 3,y β3 1 9D 1 1,x D2 1,y γ 1D 2,y β2 1 3(D2 1,x )2 γ 3 1 D1 1,y D2,y β 1 15D 2,y β 1(D 1 1,x )2 D 2 1,y γ 2 1 6D1 1,x D2,y β 1D 1 1,y D 1,x + 12D 1 1,y D1 1,x D 1,y 3(D2,y )2 β 3 1 (D1 1,x )2 D 3,x β3 1 (D1 1,y )2 3(D 1 1,y )2 D 3 1,x γ (D1 1,y )2 D 2 1,x γ D1 1,x (D2 1,y )2 γ 3 1 3D 1 1,x D1 1,y D3 1,y γ D1 1,x D2 1,y γ 1D 1,y D 1 1,y D1 1,x D3,y β D2,y β 1(D 1 1,x )2 D 1,y + 9D 1 1,y D2 1,x γ 3 1 D2 1,y + 18D1 1,y D2 1,x γ 1D 1,y + 24D 1 1,y D1 1,x D2,y β D1 1,y D1 1,x D2 1,y γ 2 1, G(β 1, γ 1 ) = D 2,y β 1D 2 1,y γ 4 1 D3 1,x + 6D2,y β 1D 2 1,y γ 1D 1,x 6D2,y β 1D 2 1,x γ 1D 1,y 9D2,y β 1D 1 1,x D2 1,y γ 2 1 3D 2,y β 1D 1 1,x D3 1,y γ D3 1,y γ 4 1 D2 1,x D2,y β 1 + D 1 1,y D3 1,y γ D1 1,y D2 1,y γ 2 1 3(D2 1,y )2 γ 3 1 3D1 1,y D3,y β3 1 9(D 2,y )2 β 3 1 D1 1,x 18D2,y β 1D 1 1,x D 1,y 18D2,y β 1D 1 1,y D 1,x D2 1,y γ 1D 3,y β3 1 36(D1 1,y )2 γ 1 + 3D 2,y β 1D 1 1,y D3 1,x γ D2 1,y γ 1D 2,y β D2,y β 1D 1 1,y D2 1,x γ (D2,y )2 β 3 1 D2 1,x γ 1 + 6D 2 1,y γ 1D 1,y + 12D1 1,y D 1,y + 48D1 1,y D2,y β2 1, D(β 1, γ 1 ) = β 1 γ 1 (D 1 1,x D2 1,y D2,y D2 1,x D1 1,y D2,y ) (D1 1,y )2. References [1] T. Rando, J.A. Rouler, Measures of farness for curves and surfaces, n: Desgnng Far Curves and Surfaces, Socety for Industral and Appled Mathematcs, Phladelpha, 1994, pp [2] G. Farn, G. Ren, N.S. Sapds, A.J. Worsey, Farng cubc B-splnes, Comput. Aded Geom. Desgn 4 (1986) [3] G. Farn, Curves and Surfaces for Computer Aded Graphcal Desgn, Academc Press, 22. [4] R.J. Crpps, M.Z. Hussan, S. Zhu, Smooth polynomal approxmaton of spral arcs, J. Comput. Appl. Math. 233 (21) [5] J.M. Al, R.M. Tookey, J.V. Ball, A.A. Ball, The generalsed Cornu spral and ts applcatons to span generaton, J. Comput. Appl. Math. 12 (1999) [6] J. Sánchez-Reyes, J.M. Chacón, Polynomal approxmaton to clothods va s-power seres, Comput. Aded Des. 35 (23) [7] J. McCrae, K. Sngh, Sketchng pecewse clothod curves, Comput. Graph. 33 (29) [8] H.W. Guggenhemer, Dfferental Geometry, Dover Publcatons, [9] L. Fang, Crcular arc approxmaton by quntc polynomal curves, Comput. Aded Geom. Desgn 15 (1998) [1] L.Z. Wang, K.T. Mura, E. Nakamae, T. Yamamoto, T.J. Wang, An approxmaton approach of the clothod curve defned n the nterval [, π/2] and ts offset by free-form curves, Comput. Aded Des. 33 (21) [11] C. Baumgarten, G. Farn, Approxmaton of logarthmc sprals, Comput. Aded Geom. Desgn (1997) [12] Z. Habb, M. Saka, Transton between concentrc or tangent crcles wth a sngle segment of G 2 PH quntc curve, Comput. Aded Geom. Desgn 25 (28) [13] D.J. Walton, D.S. Meek, G 2 curve desgn wth a par of pythagorean hodograph quntc spral segments, Comput. Aded Geom. Desgn 24 (27) [14] W.H. Press, S.A. Teukolsky, W.T. Vetterlng, B.P. Flannery, Numercal Recpes n C, The Art of Scentfc Computng, second ed., Cambrdge Unversty Press, [15] B.A. Barsky, T.D. DeRose, Geometrc contnuty of parametrc curves: three equvalent characterzatons, IEEE Comput. Graph. Appl. (1989) [16] R.J. Crpps, A.A. Ball, Orthogonal C 2 Cubc Splne Curves Geometrcal Modelng and Computng: Seattle 23, Nashboro Press, 24. [17] Maple 12, Copyrght , Waterloo Maple Inc.. [18] E. Breskorn, H. Knörrer, Plane Algebrac Curves, Brkäuser, Boston, 1981.