Computing offsets of freeform curves using quadratic trigonometric splines

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1 Computing offsets of freeform curves using qudrtic trigonometric splines JIULONG GU, JAE-DEUK YUN, YOONG-HO JUNG*, TAE-GYEONG KIM,JEONG-WOON LEE, BONG-JUN KIM School of Mechnicl Engineering Pusn Ntionl University Geumjeong-gu, Busn, REPUBLIC OF KOREA * yhj@pusn.c.kr Abstrct: - This pper presents lgorithms for computing offsets of freeform curves. The pproch first divides the originl curve into severl segments t the inflexion points. Bsed on the obtined new control polygon nd its offsets, qudrtic trigonometric splines re constructed to pproximte the offset curves. Finlly, the shpe prmeter vlue of trigonometric spline is determined to stisfy the required tolernce. The degree of the output curve is two, independent of the originl degree. This method is ble to generte completely overestimting offset curves, which cn be uesd in NC mchining for preventing over-cut. Furthermore, it cn generte offset curves with the lowest number of control points compred with other works. Key-Words: Offsetting; NURBS; NC; Trigonometric; CAD/CAM 1 Introduction Curve offsetting is one of the most importnt geometric opertions in CAD/CAM systems due to its immedite pplictions to NC mchining [1]. In the mentime, the offsetting is one of the most difficult geometric opertions to implement. Becuse the offset curves of generic rtionl curves re non-rtionl, pproximtion techniques hve been widely used for curve offsetting. Elber [] reviewed the previous offset pproximtion methods nd clssified these pproches ccording to the following two ctegories: (1) control polygon bsed pproch, nd () interpoltion bsed pproch. Lee [3] presented n offset of the plnr curve bsed on circle pproximtion. This technique cn provide rtionl offset curves; however, the offset curve is of high degree. Li [4] presented method bsed on Legendre series pproximtion nd subdivision. Recently, Shih [5, 6] generted n one-sided offset pproximtion of freeform curves for interference-free NURBS mchining. The error mesurement only estimted the size of the convex hulls of freeform curves. The limittions of the previous works re summrized s follows: (1) The degree of the offset curve chnges with the originl degree. It is desirble to hve generic low degree offset pproximtion which is independent of the originl NURBS. () Most of the published methods re not bsed on precise error mesure, which results in ftl problem of overcut. (3) Any of previous works does not gurntee n overestimting offset curve long the whole curve, which is importnt in NC mchining for preventing over-cut. In this reserch we propose n lgorithm of curve offset using qudrtic trigonometric spline curve with shpe prmeter which hs been introduced by Hn [7]. And the offset pproximtion error control is bsed on rigorous theory. Becuse of locl control nd low degree fetures of the spline curves, this method results in the lowest number of control points compred with other works. Even more it is ble to generte high-precision offset pproximtion which completely overestimtes. This cpbility is quite useful in NC mchining for preventing over-cut. Offset pproximtion using trigono metric splines.1 Qudrtic trigonometric splines with shpe prmeters We begin by presenting the construction of qudrtic trigonometric spline functions with given points Q, Q 1, Q nd knot vector T = (,,, 1, 1, 1) introduced in Hn [7] s shown in Fig. 1. Then the prmetric formul cn be given by ISSN: ISBN:

2 T ( t) = bj ( t) Q j, t [,1], 1 λ 1 (1) j= where b j (t) re the bsis functions of T(t): b ( t) = [1 sin( t)][1 λ sin( t)], b1 ( t) = 1 [1 sin( t)][1 λ sin( t)] () [1 cos( t)][1 λ cos( t)], b ( t) = [1 cos( t)][1 λ cos( t)], nd λ is the shpe prmeter. Fig. 1 shows qudrtic trigonometric spline curves for different shpe prmeters respectively. As the increse of λ, the trigonometric spline curves pproch to the edge of the control polygon nd vice vers. Due to its locl control nd low degree properties, it cn be used to pproximte the offset curves in this pper. Q 1 λ =.3 Fig. illustrtes the process grphiclly. The curve given in Fig. () is cubic NURBS curve with six () (b) (c) progenitor curve inflexion point offset of new control polygon pproximtion curve λ =.1 Q λ = -. Fig. 1 Qudrtic trigonometric splines with different shpe prmeters Q (d) Fig. Constructions of offset pproximtion curve of NURBS curve. Construction of offset pproximtion curve In this section, we consider how to construct the offset pproximtion curve of given freeform curve of Fig. (). The min steps re s follows: 1. Divide the originl curve into severl segments t inflexion points, nd regenerte new control polygon.. Offset the new control polygon by the prescribed distnce r. 3. Construct qudrtic trigonometric splines T(t) s the offset curves with the generted offset control polygon. 4. Compute the offset error to determine the vlue of shpe prmeter of offset curve. Insert new knot into t the prmeter loction with mximl error when the shpe prmeter cnnot reduce the error ny more. This process continues until the mximum error is within llowed tolernce. te tht the degree of the output curve of this method is, independent of the originl degree. control points. The NURBS curve is divided into two segments t the inflexion point s shown in Fig. (b). Fig. (c) shows the offset of the new control polygon with the prescribed distnce r. Fig. (d) shows the constructions of qudrtic trigonometric splines T(t) s the offset curves. Most trditionl techniques subdivide t the middle of the prmetric domin when the mximum error is lrger thn the llowed tolernce ε since they do not know the loction of mximum error; however, here we insert new knots into t prmeter loctions with mximum error. This technique helps to reduce the size of curve dt nd generte it more rpidly. For exmple, Fig. 3 shows freeform curve defined with control points P,,. The offset pproximtion curve is constructed with Q, Q 1, Q. Since Ф(u) represents the exct error function (see section 3), we cn find the loction t which it mkes the mximum error. When the mximum error is bove the llowed tolernce, one time subdivision pplies to generte two sets of control points, P,,,, nd,,,1,. Then ISSN: ISBN:

3 regenerte the offset pproximtion curves with Q, Q 1,, Q, nd Q,, Q 1,1, Q respectively. Q P Q 1, Q,,, Q 1 Q 1,1,1 t u i which gives the mximum error Fig. 3 Knot insertion 3 Error mesurement nd control Q Figs. 5, 6 nd Tble 1 show some experimentl results on comprisons of the two different error computtion methods. The input curve of Fig. 5 is cubic Bezier curve with 4 control points: P ( , ), ( , ), (.3, -.5), nd P 3 (.9, -.). An offset rdius 1. is used. First, we insert knot u=.5 into the input curve, divide the curve into C 1 (u) nd C (u). Then, two qudrtic trigonometric spline curves re used to pproximte segments C 1 (u) nd C (u) respectively. The shpe prmeters re.6 nd.5 for ech offset. We cn find tht the distnces mesured by Eqution (3) re lrger thn Eqution (4) for both two segments s shown in Fig. 6 nd Tble 1. The previous error computtion totlly over-estimtes the exct offset pproximtion error. P 3.1 Error function The offset pproximtion [] estimted the offset error by computing the mximum vlue of the squred distnces: ε ( u) = C( u) C ( u) r, (3) r where Cr ( u ) is n pproximtion of C r (u). However, none of these methods gurntees tht the curve Cr ( u ) hs similr speed s C r (u) in Fig. 4. Technique [8] improved the error estimtion by tight error bound only for circle pproximtion pproches. rn(u) C r (u) error C r (u) Fig. 4 Error T(t)-C r (t) In this pper, we propose different error function which is precise to be used for high precision offset pproximtion. Let be the point on the originl curve whose norml psses through point T(t) which is locted on the offset pproximtion curve s shown in Fig. 4. Point C r (u) is the exct offset of by distnce r. Thus the offset error is represented by the distnce between point T(t) nd C r (u). Then the error function is defined by Ф(u) = T(t)- -r. (4) We present n exmple of the ppliction of our error computtion method nd compre with the error function given in Eqution (3). T(t) C 1 (u) C (u) Fig. 5 Offset pproximtion fter subdivisions.5. 5 Eqution (3).5 (). C 1 (u).1.6 P 3 Eqution (4).5 1 (b). C (u) Fig. 6 Error computtion of segment C 1 (u) nd C (u) Tble 1 Error computtion results Segment λ Eqution(3) Eqution(4) C 1 (u).6 [,.44] [,.] C (u).5 [,.181] [,.161] 3. Shpe prmeter for offset curve In this section we consider how to determine the vlue of shpe prmeter of the offset pproximtion curve within desired offset errors. Fig. 7 shows the offset pproximtion error from three different cses: completely overestimtion, piecewise overestimtion nd underestimtion, nd completely underestimtion. In fct, the method of Cobb ISSN: ISBN:

4 [9] completely underestimtes the exct offset. Interpoltory scheme [1] is in generl piecewise overestimting nd underestimting the exct offset. It is sometimes desirble to hve n offset pproximtion which completely overestimtes. When the centre of the bll-end of the tool follows the offset pproximtion, completely overestimting offset will prevent over-cut. As shown in Fig. 8, in this pper, the offset method is ble to do this. The computing procedure of shpe prmeter λ is shown in Fig. 9. It strts from λ i = -1. If there exist negtive error checked by error function Ф(u) on the prmetric domin, jump to λ i+1. Record the shpe prmeter vlue until find ll the errors on the prmetric domin re greter thn or equl to zero. If more deciml digits re needed for the shpe prmeter λ (which mens more precise), redefine the initil vlues of rry λ[i]. Insert new knot into t the prmeter loction with mximum error if it is greter thn the defined tolernce ε. Ф(u) Overestimtion Piecewise Underestimtion i++ Insert new knot into t u i Strt Given curve, tolernce ε, offset curve T(t) Define =-1, b=1, n=, j=1 λ[i]= +i*(b-)/n; i n λ=λ[i] whether ll u i [,1], Ф[u i ]>=? j<deciml digits of ε? Define =λ[i-1], b=λ[i], n=1, j++, i n, λ[i]= +i*(b-)/n Record λ, mximum vlue of Ф[u i ], nd the corresponding u i mximum vlue of error[u i ]<= ε? End Fig. 9 Determintion of shpe prmeter with overestimted error scheme u P Fig. 7 Under nd over estimtions knot insertion P 3 λ i -overestimtion exct offset Fig. 8 Shpe prmeter with overestimted offset 4 Implementtion nd exmples The method described erlier hs been implemented to offset vriety of freeform curves. We continue to complete the Bezier curve exmple which is presented in section 3. (This exmple cn be found in Les [1].) The result for ε=1 - is shown in Fig. 1, using qudrtic spline curves with two times of knot insertion. Les [1] published comprison of vrious offset pproximtions. We dd our column to their tbles. Tble shows the number of control points compred with other methods. The input curve of Fig. 11 is uniform cubic B- spline curve with 7 control points: P ( ,.34143), ( ,.84), (-1.745,.787), Fig. 1 Bezier curve exmple: ε=1 - Tble Number of control points ε Cobb Til Lst Elb M- P&T Gu [9] [11] [1] [1] [3] [1] P 3 ( , -.775), P 4 ( ,.99), P 5 (.9416, ) nd P 6 (.87, 3.775). An offset rdius.5 is used for the exmple. We first remove the intersecting loop s shown in Fig. 11. The curve is subdivided into six segments t the inflexion points. And then the qudrtic trigonometric spline curves re constructed with the offset control polygon to pproximte the offset curve. The result for ε=1-1 is shown in Fig. 1. Tble 3 shows the number of control ISSN: ISBN:

5 points of the proposed method compred with other methods. Ntionl Core Reserch Center Progrm from MEST/KOSEF (.R ). Fig. 11 Inflexion points insertion of Cubic B-spline Fig. 1 Construction of offset curve with 1-1 Tble Number of control points ε Cobb Til Lst Elb M- P&T Gu [9] [11] [1] [1] [3] [1] As the results clerly indicte, the method compres very well with existing techniques. 5 Conclusion In this pper we presented n lgorithm for pproximting offsets of NURBS curves using trigonometric splines with shpe prmeters. The proposed method is cpble of obtining n overestimting offset curve long the whole curve, which results no over-cut in NC mchining. The degree of the output curve is two, independent of the originl degree, nd the error control is bsed on rigorous theory, which insures the stbility of this method. Additionlly, comprisons with other methods show tht this method produces the fewest control points. References: [1] Elber, G nd Cohen, E. Error bounded vrible distn ce offset opertor for free curves nd surfces. Comp ut. Geom. Applict. 1991;1(1): [] Elber G, Lee IK, Kim MS. Compring offset curve pproximtion method. Computer Grphics nd Applic tion 1997;17(3):6-71. [3] Lee IK, Kim MS, Elber G. Plnr curve offset bsed on circle pproximtion. Computer Aided Design 199 6;8(8): [4] Li YM, Hsu V. Curve offsetting bsed on Legendre s eries. Computer Aided Geometric Design 1998;15(7): [5] Shih JL, Chung SH. One-sided offset pproximtion of freeform curves for interference-free NURBS mchi ning. Computer Aided Design 8;4(): [6] Shih JL, Chung S-H. NURBS output bsed tool pt h genertion for freeform pockets. Int J Adv Mnuf T ech 6;9(7-8): [7] Hn X. Qudrtic trigonometric polynomil curves wi th shpe prmeter. Computer Aided Geometric Des ign ;19:53-1. [8] Zho HY, Wng GJ. Error nlysis of reprmetrizti on bsed pproches for curve offsetting. Computer A ided Design 7;39: [9] Cobb, B. Design of sculptured surfces using the B-s pline representtion. PhD Thesis University of Uth, Computer Science Deprtment, [1] Piegl L, Tiller W. Computing offsets of NURNS cu rves nd surfces. Computer Aided Design 1999;31: [11] Tiller, W nd Hnson, E. Offsets of two dimension l profiles. Comput. Grph. & Applict. 1984;4: [1] Hoschek, J nd Wissel, N. Optiml pproximte con version of spline curves nd spline pproximtion of offset curves. Comput. Aided Des. 1988;(8): Acknowledgment This reserch ws finncilly supported by the Ministry of Eduction, Science Technology (MEST) nd Kore Institute for Advncement of Technology (KIAT) through the Humn Resource Trining Project for Regionl Innovtion, nd by grnts-in-id for the ISSN: ISBN: