Discrete Curvature and Torsion-based Parameterization Scheme for Data Points

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1 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Xongbng Fang Chna Shp Development and Desgn Center Wuhan, 4364, Chna E-mal: Sheng Wu Chna Shp Development and Desgn Center Wuhan, 4364, Chna E-mal: Jangyuan Lu Chna Shp Development and Desgn Center Wuhan, 4364, Chna E-mal: Parameterzaton for data ponts s a fundamental problem n the feld of Computer Aded Geometrc Desgn. Recently, an mproved centrpetal parameterzaton technque s proposed by Fang et al and ts superorty to other common methods, such as unform method, chord length method, centrpetal method, Foley method and unversal method, etc., s valdated by a lot of numercal examples. In ths artcle, a refned formula s frstly gven to resolve the problem that the parameter values obtaned by Fang s method are usually a lttle bgger than the optmum ones. Furthermore, as enlghtened by the local dfferental geometrc propertes of parametrc curves, a novel parameterzaton scheme s put forward by ntroducng the dscrete curvature and torson nformaton at each pont. As ndcated by numercal experments, the devaton measured by a curvature and Euler dstance-based crteron between the nterpolaton B-splne curve obtaned by our method and the polylne constructed by the ponts are smaller than the ones between the curves obtaned by Fang s method, the chord length method, the standard centrpetal method and the polylne. The proposed algorthm s applcable to both 2D and 3D ponts and has a dstnct advantage for 3D ponts as compared wth the three aforementoned methods. PoS(CENet27)93 CENet July 27 Shangha, Chna Speaker and Correspondng author Copyrght owned by the author(s) under the terms of the Creatve Commons Attrbuton-NonCommercal-NoDervatves 4. Internatonal Lcense (CC BY-NC-ND 4.).

2 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts. Introducton Xongbng Fang Parameterzaton for a set of data ponts s one of the fundamental problems n curve and surface nterpolaton applcatons[-3]. Approprate parameters of data ponts can be further used to construct a knot vector and compute the control ponts of the nterpolaton curve. The common parameterzaton methods are unform[], chord length[4], centrpetal[5,], Foley[6] and unversal methods[7,8], among whch the chord length and centrpetal methods are the most popular parameterzaton algorthm n the CAGD communty. Recently, a refned centrpetal parameterzaton method s presented by Fang et al[9]. All the aforementoned methods are obtaned by manly takng planar ponts as research focus and utlzng the nformaton of dstance between any two successve ponts, the angle between two successve lnes and the dscrete curvature at each pont, and they are usually used n 3D ponts parameterzaton and curve/surface nterpolaton problems drectly. The parameterzaton s also one of the key steps for 3D data ponts and surface nterpolaton[2,8]. At present, the famlar way s drectly applyng the above methods n 3D pont cases. For spacng ponts, there s usually an angle φ between the trangles constructed by the three consecutve ponts P -, P, P + and P, P +, P +2 (see Fg. ). It s evdent that the chord length and centrpetal methods only utlze the dstance nformaton. Furthermore, Fang used the nformaton of the external angle θ rather than the angle φ. When the value of φ s not equal to zero, the nterpolaton curve s not only crooked but also torsonal at the pont P +. From the dfferental geometry pont of vew, the local confguraton of a space curves at a pont s confrmed by both of the curvature and torson[]. In ths paper, we concentrate the problem of 3D ponts parameterzaton and a new parameterzaton scheme fully dstlled the hgh-order dfferental geometrc quanttes, such as dscrete curvature and torson, mpled by the ponts s put forward, whch s also approprate for 2D ponts. P -2 P - P ϕ V H P +2 θ P + Ω PoS(CENet27)93 Fgure : Spatal Structure of 3D Ponts 2. Data ponts parameterzaton 2.. B-splne curves B-splne has been proven to be the most popular representaton method for free-form curves and surfaces. In the paper, we adopt B-splne curve nterpolaton to llumnate our parameterzaton scheme. A k th -order B-splne curve s represented as n P() t = B jn jk, () t j= where {B j } s the control ponts, and {N j,k (t)} s the normalzed kth-order B-splne bass functons defned on the knot vector T={τ j } j=,,,n+k. The B-splne curve nterpolaton problem s formulated as follows. Gven the ponts {P } =n+, the assocatng parameter values t for each pont P can be obtaned by a parameterzaton method and the knot vector T can be computed by the average method [] as follows:

3 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts τ = = τ k τ = = τ m k m Xongbng Fang j + k + τ = t, j =, 2,, n k. k = j Then the control ponts {B j } of a B-splne curve are unknowns to be calculated to satsfy P(t )=P, =,,, n,.e. the nterpolaton curve P(t) passng through the ponts at the assocated parameter values Common parameterzaton methods There are several classcal technques to choose the parameters {t }, such as unform, chord length, centrpetal, Foley and unversal method, etc.. The unform method does not use the dstance nformaton between each par of consecutve ponts, whle the chord length and centrpetal ones utlze the nformaton. The unform, chord length and centrpetal methods can be formulated unformly as follows. Let L =P P, =,, n, D n a = L, t =, Δt =t t, then the three methods can be = rewrtten as a t = L D (2.) Equaton (2.) denotes the unform, chord length and centrpetal parameterzaton when a=, a= and a=/2, respectvely. The Foley parameterzaton modfes the centrpetal method by adoptng a Nelson metrc to defne the dstance. It s formulated as 3 ˆ θd t = t t = d + 2 ( d+ d), 3 ˆ θn-d n-2 tn = tn tn- = dn- + 2 ( dn-2+ dn-), 3ˆ θ ˆ d 3θ d + + t = t t = d ( d + d) 2 ( d+ d+ ), For more detals about the defnton of d, please refer to [6]. Lm develops a new parameterzaton approach named unversal based on the above methods[7,8].the key dea of ths method s usng the parameter t assocated wth the maxmum of the bass functon N j,k (t). Among the aforementoned methods, Foley and the unversal parameterzaton are not very popular because of ther laborous computaton. The unform method s rarely used as ts bad parameterzaton effect for the fnal nterpolaton curves. So the popular parameterzaton methods n the context of CAGD are chord length and centrpetal algorthms. PoS(CENet27) Fang s parameterzaton method Fang et al provded a refnement of the classcal centrpetal parameterzaton method by ntroducng the dscrete curvature nformaton at each pont[9]. The geometrc ntuton of the method s that the flexure of a curve at one pont wll elongate the arc length around the neghborhood of the pont. For the ponts parameterzaton, t sgnfes the parameter nterval between the pont and the next one wll be augmented. Fang s method s formulated as.5 t = L D+ e (2.2)

4 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts θl θ l Xongbng Fang = ( α + ) (2.3) 2sn( θ 2) 2sn( θ 2) e where θ and e are the exteror angle of the trangle ΔP - P P + and the parameter dsturbance quantty at pont P respectvely, the coeffcent α s determned as. by an expermental way, and l s the magntude of the shortest sde of the trangle ΔP - P P +. The characterstcs of Fang s method are summarzed as follows: () t has fully used the geometrc nformaton of dstance, angle and dscrete curvature of the gven pont set; (2) as compared wth the centrpetal and other methods, the fnal nterpolaton curve obtaned by Fang s has less devaton to the polylne constructed by the ponts and the parameters for the ponts are more optmzed. Moreover, ts computaton s also compact. 3. Curvature and torson-based parameterzaton 3.. Refnement of Fang s method [9] Fgure 2: Dscrete Curvature Crcle at Pont P A ds P In Fg. 2, Fang s method used the crcular arc P B of the crcumcrcle of the trangle ΔAP B to approxmate the arc length ncrement ds of the curve from P to B. Obvously, when use the crcumcrcle of the trangle ΔAP B to approxmate the local of the curve around P, the local of the curve s geometrcally nsde the crcle and that means the arc length of P B s bgger than ds. Based on the above observaton and the smulaton results of the method, we ve found that the dsturbance quantty e n Fang s artcle s bgger than ts optmum. Amng at the lmtaton, we gve a corrected way by usng the average of the chord length P B and the arc length P B to approxmate the ncrement ds,.e. ds = ( PB + θ r) 2. ds B PoS(CENet27)93 Suppose PB = (2.3), then one has l, the new dsturbance quantty s e e l + θ r l + θ r 2 2 = α ds nstead of e n Equaton = ( α + ) (3.) where l s the length of the shortest sde of the trangle ΔP - P P +, r =l /2snφ andφ =θ /2. Substtute them nto Equaton (3.) then t follows that: ϕ α l(+ ), = 2 snϕ ϕ ϕ = (+ ) + (+ ), = 2,, - ϕ α l (+ ), = n 2 snϕ e α l l n 2 snϕ snϕ (3.2) Thereby the parameter value for each pont P (=,, n) n the gven data set can be computed by

5 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Xongbng Fang.5 t = t + L D+ e (3.3) where t = Analyss for the two methods From Fg. 3, we can get the followng conclusons by analyzng the formulatons of Equaton (2.3) and (3.). () If ϕ [, π 2], then [,.57] sn ϕϕ, ϕ + snϕ [,.235]. It shows that the range of 2snϕ the value of Equaton (4) s smaller than that of Fang s. (2) If ϕ ncreases, then sn ϕϕ and ϕ + snϕ also ncrease, but the change rate of the 2snϕ latter s more slowly than that of the anteror one. For a same φ, the quantty computed by Equaton (3.) s smaller than that of Equaton (2.3). Consequently, the parameter correcton computed by Equaton (3.2) s smaller than that of Fang s. Fgure 3: Comparson of the Proposed Method and Fang s Method 3.3. Dscrete curvature and torson-based method e Gven a set of 3D ponts randomly, the fttng curves usually have torson as the ponts are not coplanar. The common parameterzaton methods lke chord length and centrpetal, and Fang s method usually take the geometrc nformaton of pont poston, lengths of lne segments, angles between the lne segments and the dscrete curvature at each pont nto account rather than the hgher-order geometrc nformaton of torson. Accordng to the local dfferental geometrc characterstcs of a space curve, the shape of the curve around a pont s only determned by the curvature and torson at the pont. Based on the method n Secton 3., a dscrete curvature and torson-based parameterzaton scheme s proposed for 3D ponts. There are a lot of methods to calculate the dscrete curvature and torson[-5]. In the paper, we choose the followng approach to estmate the torson at each pont[4, 5]. Let Δ =[L, L +, L +2 ], =2,, n-2, and [U, V, W] be the scalar trple product of the vectors U, V and W, then one has Fang's method method n secton PoS(CENet27)93 ˆ τ + = 3 L + L L L L L + L + L (3.4) where M N denotes the cross product of the vectors M and N, Δ denotes the volume of the hexahedron constructed by a trple of consecutve vectors L, L +, L +2, and ˆ τ s computed

6 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Xongbng Fang by Equaton (3.4) at ponts P 2, P 3,..., P n-, and equals zero at P, P, and P n. Fnally, the dscrete torson at each pont P (=2,,n-) s estmated as τˆ ˆ - + τ, > τ = +, (3.5) When the ponts P -2, P -, P and P + are coplanar, τ =; Otherwse, τ. The sgn and absolute value of τ determne the drecton and degree of the tortlty of the fnal nterpolaton curve at P, respectvely. Hence, the parameter correcton nduced by the torson at a pont can be expressed as f =β τ, where the scale β can be set as a constant or varable. Thus the parameter t assocatng wth P can be formulated as.5 = + L + + (3.6) t t D e f where e s computed by Equaton (3.). For 2D ponts, all the dscrete torsons calculated by Equaton (3.5) are zero, and thus the Equaton (3.6) s equvalent to Equaton (3.3). Consequently, the proposed method depcted by Equaton (3.6) s approprate for both 2D and 3D cases Crteron of nterpolaton precson measurement The knot vector and nterpolaton curve can be constructed after the parameter values beng computed by Equaton (3.6). Dfferent parameter values usually result n dfferent shapes of curves and the devaton values of those curves to the polylne of the data ponts may be dfferent. There are many crtera to measure the dvergence [6-8]. In the paper, we utlzed the curvature and Euler dstance-based crteron depcted n [9] to compare dfferent methods Value range for α and β Expermental results show that the optmum value of α has relaton wth the shortest sde length of the trangles constructed by each trple of consecutve ponts,.e. the bgger the sde length s, the smaller the value of α wll be. The value of the torson augmented coeffcent β has relaton wth the ponts torsons,.e. the bgger the torsons are, the smaller the optmum value of β wll be. In ths work, we recommend usng.5 as an ntal value for β. PoS(CENet27)93 4. Numercal experments and dscusson 4.. Numercal experments We compare our approach developed n Secton 3.3 wth the chord length [], centrpetal [,4] and Fang s methods [9] by four examples. For the sake of justce, the curvature and Euler dstance-based crteron for the devaton of the fnal nterpolaton curve from the data ponts s chosen as the unfed comparson standard for the four methods. We choose the cubc B-splne curve as nterpolaton curve and the average method to calculate the knot vector T={τ j }. Once the parameters {t } s get by one of the four approaches and the knot vector T s acheved, the nterpolaton curve wll be obtaned by algorthms for calculatng a system of lnear equatons [,,9] or by a robust progressve teraton algorthm[2,2] wth a small threshold, such as.* -8. Example. The data set contans 2 planar ponts,.e. P =[;], P =[2;.2], P 2 =[3;], P 3 =[4;.3], P 4 =[6;8], P 5 =[7;.25], P 6 =[7.5;2], P 7 =[8;.2], P 8 =[8.2;-3.2], P 9 =[;.3], P =[2;4], P =[5;-2]. The polylne connectng the ponts s shown n Fg. 4(a), and the cubc B-splne

7 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Xongbng Fang curves obtaned by Fang s, the proposed, the centrpetal and the chord length methods are shown n Fg. 4(b) to Fg. 4(e) respectvely. Dstance devatons computed by the four methods wth dfferent values of α are lsted n Table and the centrpetal method and the chord length one are rrelatve wth α. The Fang s method and the proposed one get the mnmum devatons.5249 and wth α=.6 and.7, respectvely. Fg. 4(e) shows that the fnal curve obtaned by chord length method has loops and spres, whch are not acceptant n CAD and CAGD domans a. Data ponts and the polylne b. Fang s method c. Proposed method PoS(CENet27)93 d. Centrpetal method e. Chord length method Fgure 4: Data ponts and nterpolaton curves by the four methods (Example ) α Fang s Proposed Centrpetal Chord length Table : Dstance devaton of the four methods n example Example 2. The data ponts are sampled from a hypotrochod curve determned by x= ((a-b) cos(t)+h cos((a-b)/b t)), y= ((a-b) sn(t)-h sn((a-b)/b t)), where a =,

8 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Xongbng Fang b=3/4, h=9/3, t=[-3π, 3π] and the samplng step=π/2. The data set contans 3 ponts and the polylne connected them s shown n Fg. 5(a). The cubc B-splne curves obtaned by Fang s method, the proposed method, the centrpetal method and the chord length method are shown n Fg. 5(b) to Fg. 5(e) respectvely. Dstance devatons computed by the four methods wth dfferent values of α are lsted n Table 2. Fang s method and the proposed one get the mnmum devatons and wth α=.7 and.8, respectvely a. Data ponts and the polylne b. Fang s method c. Proposed method PoS(CENet27)93 d. Centrpetal method e. Chord length method Fgure 5: Data ponts and nterpolaton curves by the four methods (Example 2) α Fang s Proposed Centrpetal Chord length Table 2: Dstance Devaton of the Four Methods n Example 2

9 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Xongbng Fang Example 3. The data set contans 93D ponts,.e. P =[;-2;], P =[;;], P 2 =[;;], P 3 =[;;], P 4 =[;2;], P 5 =[;3;-3.5], P 6 =[2;3;], P 7 =[2;.5;] and P 8 =[3;.5;]. The polylne connectng the ponts s shown n Fg. 6(a), and the cubc B-splne curves obtaned by Fang s method, the proposed method, the centrpetal method and the chord length method are shown n Fg. 6(b) to Fg. 6(e), respectvely. Dstance devatons computed by the four methods wth β= and dfferent values of α are lsted n Table 3, n whch, Fang s method and the proposed one get the mnmum devaton and wth α=.3, respectvely. Dstance devatons computed by our method wth α=.3and dfferent values of β are lsted n Table 4 and our method get the mnmum value at β= a. Data ponts and the polylne b. Fang s method c. Proposed method PoS(CENet27)93 d. Centrpetal method e. Chord length method Fgure 6: Data ponts and nterpolaton curves by the four methods (Example 3) α Fang s Proposed Centrpetal Chord length 5.422

10 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Table 3: Dstance Devaton of the Four methods n Example 3 (β=) Xongbng Fang β Proposed Table 4: Dstance devaton of the proposed method n example 3 (α=.3) Example 4. The data set contans nne3d ponts,.e. P =[2;;], P =[s;s;-], P 2 =[;2;], P 3 =[-s;s;-], P 4 =[-2;;], P 5 =[-s;-s;-], P 6 =[;-2;], P 7 =[s;-s;-] and P 8 =[2;;], where s = The polylne connectng the ponts s shown n Fg. 7(a), and the cubc B-splne curves obtaned by Fang s method, the proposed method, the centrpetal method and the chord length method are shown n Fg. 7(b) to Fg. 7(e) respectvely. Dstance devatons computed by the four methods wth β= and dfferent values of α are lsted n Table 5 and also the centrpetal and chord length methods are rrelatve wth α. In Table 6, Fang s method and the proposed acheve the mnmum devaton and wth α=.2, respectvely. The dstance devatons computed by our method wth α=.2 and dfferent values of β are lsted n Table 6 and our method s the mnmum value wth β= a. Data ponts and the polylne - b. Fang s method c. Proposed method PoS(CENet27)93 d. Centrpetal method e. Chord length method Fgure 7: Data ponts and nterpolaton curves by the four methods (Example 4) α Fang s Proposed Centrpetal Chord length Table 5: Dstance Devaton of the Four Methods n Example 4 (β=) β Proposed Table 6: Dstance Devaton of the Proposed Method n Example 4 (α=.2) 4.2. Dscusson

11 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Algorthms comparson Xongbng Fang From Table to Table 3, the mnmum dstance devaton of our method s smaller than that of the Fang s, centrpetal and chord length methods when the dscrete torson nformaton at each pont s not taken nto account. From Table 5, the mnmum devaton of our algorthm wth β= s smaller than that of the Fang s about From Table 3 to Table 6, the proposed method can effcently reduce the devatons when the dscrete torsons are taken nto account. As compared wth Fang s approach, the devaton values of the proposed scheme for Examples 3 and 4 reduce about and.3876, respectvely. As compared wth the centrpetal and chord length methods, the proposed one possesses many advantages. From Table to Table 6, the proposed method can obtan a mnmum devaton value n a large range of parameter values and there s a par of values of α and β that makes the parameterzaton results of our method beng fner than the other three ways Future work There are several possbltes for the extenson of our method; we lst a few of them as follows: () The parameters of the data ponts have tght relatons wth the knot vector T and both of them have coordnatve mportance for the shape of the fnal nterpolaton curve. However, the relatons among parameter values {t }, the knot vector T and the arc length parameter of the fnal curve are not very clear and t s worthy of much more research. (2) Another extenson s to adopt dfferent α and β for each parameter nterval Δt by takng the propertes of concave and convexty of the gven ponts nto account. 5. Concluson Amng at the shortages of common parameterzaton methods and Fang s mproved centrpetal method [9], a new parameterzaton method, based on dscrete curvature and torson, s provded and llumned by the local dfferental geometrc propertes of a space curve at one pont. Expermental results ndcate that the dstance devaton between the B-splne nterpolaton curve obtaned by our method and the polylne of data ponts s smaller when compared wth the centrpetal, chord length and Fang s parameterzaton, thus the proposed method s more excellent for 2D and 3D data ponts parameterzaton. PoS(CENet27)93 References [] L.A. Pegl and W. Tller. The NURBS Book[M]. Berln: Sprnger-Verlag (997) [2] S.M. Shamsuddn, M.A. Ahmed, Y. Saman. NURBS sknnng surface for shp hull desgn based on new parameterzaton method[j]. I.J.A.M.T. 28(9): (26) [3] J.F. Huang, S.L. Wan. The parametrc method of Frendshp hull form based on desgn feature [J]. Chnese Journal of Shp Research, 7(2): 79-85(22) [4] E. Lee. Choosng nodes n parametrc curve nterpolaton[j]. CAD, 2(6): (989) [5] J. Hoschek, D. Lasser. Fundamentals of computer aded geometrc desgn[m]. London: A.K. Peters, 993. [6] T.A. Foley, G.M. Nelson. Knot selecton for parametrc splne nterpolaton[c]. Mathematcal methods n computer aded geometrc desgn, Academc Press Professonal: (989) [7] C-G. Lm. A unversal parameterzaton n B-splne curve and surface nterpolaton [J]. CAGD, 6(5): (999)

12 Dscrete Curvature and Torson-based Parameterzaton Scheme for Data Ponts Xongbng Fang [8] C-G. Lm. Unversal parameterzaton n constructng smoothly-connected B-splne surfaces[j]. CAGD, 9(6): (22) [9] J.J. Fang, C.L. Hung. An mproved parameterzaton method for B-splne curve and surface nterpolaton[j]. CAD, 45(6): 5-28(23) [] N.M. Patrkalaks, T. Maekawa. Shape Interrogaton for Computer Aded Desgn and Manufacturng[M]. Berln: Sprnger (22) [] D. Coeurjolly, S. Mguet, T. Laure. Dscrete Curvature Based on osculatng crcle estmaton[c].lecture notes n computer scence, Hedelberg: Sprnger (2) [2] G.H. Lu, Y.S. Wong, Y.F. Zhang, H.T. Loh. Adaptve farng of dgtzed pont data wth dscrete curvature[j]. CAD, 34(4), 39-32(22) [3] T. Lewner, JD Gomes Jr., H. Lopes, M. Crazer. Curvature and torson estmators based on parametrc curve fttng[j]. C&G, 29(5): (25) [4] V.P. Kong, BH Ong. Shape preservng F3 curve nterpolaton[j]. CAGD, 9(4): (22) [5] V.P. Kong, BH Ong. Shape preservng approxmaton by spatal cubc splnes[j]. CAGD, 26(26): (29) [6] WP Wang, H Pottmann, Y Lu. Fttng B-splne curves to pont clouds by curvature-based squared dstance mnmzaton[j]. ACM TOG, 25(2): (26) [7] H Park, J-H Lee. B-splne curve fttng based on adaptve curve refnement usng domnant ponts[j]. CAD, 39(6): (27) [8] CM Zhang, WP Wang, JY Wang, XM L. Local computaton of curve nterpolaton knots wth quadratc precson[j]. CAD, 45(4): (23) [9] S Gofuku, S Tamura, T Maekawa. Pont-tangent/pont-normal B-splne curve nterpolaton by geometrc algorthms[j]. CAD, 4(6): (29) [2] HW Ln, HJ Bao, GJ Wang. Totally postve bases and progressve teraton approxmaton[j]. Computers and Mathematcs wth Applcatons, 5(3-4): (25) [2] LZ Lu. Weghted progressve teraton approxmaton and convergence analyss[j]. CAGD, 27(2): 29-37(2) PoS(CENet27)93