Chnese Journal of Aeronautcs 3(010 137-144 Chnese Journal of Aeronautcs www.elsever.com/locate/cja Constructng G Contnuous Curve on Freeform Surface wth Normal Projecton Wang Xaopng*, An Lulng, Zhou Lashu, Zhang Lyan Jangsu Key Laboratory of Precson and Macro-manufacturng Technology, Nanjng Unverstyof Aeronautcs and Astronautcs, Nanjng 10016, Chna Receved 3 July 009; accepted 19 January 010 Abstract Ths artcle presents a new method for G contnuous nterpolaton of an arbtrary sequence of ponts on an mplct or parametrc surface wth prescrbed tangent drecton and curvature vector, respectvely, at every pont. Frst, a G contnuous curve s constructed n three-dmensonal space. Then the curve s projected normally onto the gven surface. The desred nterpolaton curve s just the projecton curve, whch can be obtaned by numercally solvng the ntal-value problems for a system of frst-order ordnary dfferental equatons n the parametrc doman for parametrc case or n three-dmensonal space for mplct case. Several shape parameters are ntroduced nto the resultng curve, whch can be used n subsequent nteractve modfcaton so that the shape of the resultng curve meets our demand. The presented method s ndependent of the geometry and parameterzaton of the base surface. Numercal experments demonstrate that t s effectve and potentally useful n numercal control (NC machnng, path plannng for robotc fbre placement, patterns desgn on surface and other ndustral and research felds. Keywords: Hermte nterpolaton; normal projecton; freeform surface; G contnuty; ordnary dfferental equatons 1. Introducton 1 In numercal control (NC machnng of parametrc surface, one often needs to gude a cuttng tool to change ts tool-path from one to another wth contnuous acceleraton. In blended-wng-body (BWB desgn for both cvlan and mltary arcraft, one must frst construct two contact curves on wng surface and fuselage surface, accordng to a set of ponts along wth the necessary geometrc nformaton, such as the tangent and curvature vectors. Those ssues can be reduced to G blendng of two specfed surface curves or G nterpolaton of a sequence of ponts on surface. Actually, blendng curve [1] on surfaces s a specal case of nterpolaton curve on surfaces. As far as G nterpolaton on a gven surface s concerned, H. Pottmann, et al. [] presented a vara- *Correspondng author. Tel.: +86-5-84891678. E-mal address: levne@nuaa.edu.cn Foundaton tem: Natonal Natural Scence Foundaton of Chna (6067306, 50875130, 50805075 and 5087516 1000-9361 010 Elsever Ltd. Open access under CC BY-NC-ND lcense. do: 10.1016/S1000-9361(0960197-7 tonal approach to splne curves on surfaces. They characterzed nterpolatng and approxmatng mnmzers on surfaces of arbtrary fnte dmenson and co-dmenson. The mnmzers possess a characterzaton whch s very smlar to the famlar cubc C splnes: the fourth dervatve of a cubc vanshes and so does the tangental component of the fourth dervatve of splne segments n surface. Those authors amng at varatonal curve desgn characterzed the mnmzers of an ntrnsc geometrc counterpart to the L norm of the second dervatve, whch s ntegral of the squared covarant dervatve of the frst dervatve wth respect to arc length. M. Hofer, et al. [3] dscussed energy - mnmzng splnes n manfolds. It s good that both varatonal and energy-mnmzng approaches can, n a sense, gve an optmal soluton for curve nterpolatng data ponts on surfaces and thus surface curve desgned by these methods s overall tght. However the optmal soluton does not have any local propertes and degrees of freedom for shape control whch are mostly mportant for nteractve desgn n computer aded desgn (CAD and computer graphcs (CG. For example, to modfy a certan curve segment one must change data ponts and restart the entre process of
138 Wang Xaopng et al. / Chnese Journal of Aeronautcs 3(010 137-144 No.1 curve desgn. Another method beng able to use n nteractve desgn was gven by E. Hartmann [1]. However n practcal applcaton, we found that, n addton to the complex mplementaton process (frstly construct a blendng surface and then conduct surface-to-surface ntersecton or trace an mplct plane curve, the other drawback of the method s that the resultng curve sometmes unlkely preserves G 1 contnuty at nner ponts between nterpolaton ponts, snce the new blended surface does not always ntersect the base surface transversally even f the base surface s specal surface such as sphere or other quadrcs. An mprovement was made n Ref.[4], where surface curve desgn problem s reduced to that of fndng the zero set of a bvarate polynomal of relatvely low degree n the parameter space. However, mplementaton process s stll complcated for t nvolves tracng an mplct curve n plane. Moreover, J. Pegna, et al. [5] once developed a method of orthogonally projectng a space curve onto the surface, whch can be comprehensvely used n surface trmmng, blendng and patchng. However the method cannot be used for truly desgnng a curve on a surface, for example, fttng a sequence of data ponts on surface to create a surface curve. We once mproved ths orthogonal projecton method [6]. Other related theoretc researches can be found n Refs.[7]-[11]. In ths artcle, we develop a new method to overcome the drawbacks n the exstng methods [1, 4]. As t s well known, a curve on a surface s usually expressed n the form of dfferental equatons. Here, makng use of tradtonal dfferental geometrc method, we wll gve a new soluton for desgnng G contnuous curve on a surface. We focus our nvestgaton on the problem of G contnuously nterpolatng a range of data ponts on a surface wth prescrbed tangent drecton at every pont.. Mathematcal Prelmnares Let us begn wth a dfferentable parametrc surface that s descrbed by a vector-valued functon of two varables as follows: S( uv, xuv (, yuv (, zuv (, ( uv, D R (1 where x(u, v, y(u, v, z(u, v are dfferentable bvarate functons of u and v, whch are called the surface parameters, and D denotes the surface doman. The partal dervatves of the vector valued functon S wth respect to u and v are S u (u, v, S v (u, v, (u, v D. The vector N = S u S v / S u S v s called the unt normal vector (or just normal vector for short of the surface S at the correspondng pont. We assume the surface S s regular,.e., S u S v 0 for any (u, v D. A curve on the surface can be descrbed by parametrc equatons u= u (t, v= v (t, t [ a, b ]n the surface doman. Its equaton n R 3 can be wrtten as the vector-valued functon P( t S ( u(, t v( t t[ a, b] ( Takng the dervatve of P (t wth respect to tto lnearze Eq. (, we get the followng correspondng equaton n the tangent space of the surface: P( t Sud u/dts vd v/d t t [ a, b] (3 Let S be a C regular parametrc surface. It follows from classcal dfferental geometry [1] that (d u,d v (d P E(d u Fd udvg(d v (d u,d v N d P e(d u fdud vg(d v where Iand IIare known as the frst and the second fundamental form respectvely, whle ndcates the scalar product (same as below. In addton, along the curve P (t, the normal vector N s N( t N ( u(, t v( t Then we have the followng equatons: Nu a11su a1sv (4 Nv a1su asv where a11 ( ff eg/( EGF a1 ( gf fg/( EGF a1 ( ef fe/( EGF a ( ff ge/( EGF Further there s du N'( t ( a11su a1sv d t dv ( a1su as v (5 d t Now let us consder mplct surfaces f(x, y, z=0, where the frst partal dervatves f f / x, f f / y, f f / z are contnuous and not all y z zero, everywhere on the surface,. e., the surface s regular. The vector f = [ f x f y f z ] s called the gradent of the mplct surface at the pont (x, y, z. The vector N f / f s the unt normal vector of the mplct surface at the pont ( x, y, z. The Jacoban of the gradent f s called the Hessan of f. Wrte t as H (f =J ( f. A curve on the mplct surface s descrbed by parametrc form P ( t x( t y( t z( t. It s characterzed by the followng equaton f(x(t, y(t, z(t=0. Lnearzng t, we get x
No.1 Wang Xaopng et al. / Chnese Journal of Aeronautcs 3(010 137-144 139 f d x/dt f d y/dt f d z/dt 0 (6 x y z In addton, smlarly to Eq. (5 we have T N '( t P'( t H( f[ I N ( t N( t] / f (7 Fnally one vector dentty should be mentoned here for applcatons n the followng dscusson. For any three vectors a, b, c t follows that ( ab c ( ac b( bc a (8 3. Computng Normal Projecton Curve Consderng the complexty of the base surface, the projecton curve may be comprsed of several dsconnected segments. Moreover, for a space curve, the normal projecton may result n a set of dfferent projecton curves on the surface. We manly descrbe how to trace one such connected curve. In addton, f not stated otherwse, we focus our dscusson on a knd of base surface wthout boundary. S. The projecton curve P(t thus nherts the parameter of the space curve C(t and s contnuous. J. Pegna [5] once dscussed the computaton of frst ordered projecton curve. Here we deduce the equaton of the projecton curve n another way. Assume that the space curve C (t s contnuous and dfferentable. The normal projecton s characterzed by [ C( t P( t] N ( t 0 (9 Takng the dervatve of Eq.(9 wth respect to t and usng Eq. (3 and Eq. (5, we obtan d u a11( PC NSu a1( PC Sv dt d v a1 ( PC Su a ( PC NSv C' N dt (10 Calculatng the cross product of the coeffcent vectors of du/dvand dv/du n Eq. (10 wth both sdes of Eq. (10 respectvely, usng Eq. (8 and then takng the dot product of N and both sdes of the resultng equaton, we fnally have Let C(t be the parametrc representaton of a space curve that we want to project normally onto a surface d u a1 ( C P N( C' N Su a ( C P N 1 ( C' N Sv d t ( a11a a1a1 ( C P N ( a11 a ( C P N1 Su Sv d v a1 ( C P N( C' N Sv a11( C P N 1 ( C' N Su d t ( a11a a1a1 ( C P N ( a11 a ( C P N1 Su Sv f the space curve C(t does not pass through the centers of prncpal curvature of the surface at the correspondng pont P(t (. e., the denomnator n Eq. (11 s non-zero. For system Eq. (11 to be completely determned, the followng ntal-valued condtons on the surface doman Dmust be added. u(0 u0 (1 v(0 v0 Once the system of Eqs. (11-(1 s solved by numercally ntegratng uand v, the space parametrc equaton of the projecton curve can be got by substtutng uand vnto Eq.(. In the mplct case, the normal projecton s also characterzed by Eq.(9. Takng the dervatve of Eq.(9 wth respect to t, takng the cross product of N and both sdes of the resultng equaton and usng Eqs.(6-(8, we obtan P'( t [ C'( t ( C'( t N N] I ( C( t P( t N H(f T -1 ( 1 N N/ f (13 f the space curve C (tdoes not pass through the centers of prncpal curvature of the surface at the correspondng pont P(t (. e., the exstence of the (11 nverse matrx n Eq.(13 s guaranteed. Smlarly to the parametrc case, addng an ntal value condtons to Eq. (13 as follows: x(0 x0 y(0 y0 (14 z(0 z 0 we get an ntal value problem for a frst-order system of ordnary dfferental equatons. Numercally ntegratng t, the projecton curve can be obtaned. 4. Surface Curve Desgn 4. 1. G contnuous curves Specfy a sequence of the ponts P ( =1,,, n on a C r (r3 regular surface S. Assume that ts correspondng parameterzatons are descrbed by a sequence of real numbers t ( =0, 1,, n. Also specfy a sequence of tangent drectons T ( =0, 1,, n on the surface S and gve a sequence of vectors k. Construct a curve lyng on the surface S and passng the ponts P ( =1,,, n on condton that the curve s tangent drectons and curvature vectors at the ponts P are T and k respectvely. We only consder one
140 Wang Xaopng et al. / Chnese Journal of Aeronautcs 3(010 137-144 No.1 par of trplets, such as(p, T, k and(p +1, T +1, k +1. Our tactc s to construct a space curve segment C (t wth the correspondng trplets(p, T, C (0and(P +1,T +1,C +1 (1 and then project t onto the surface S to get the curve segment P (t. Frst we would lke to gve the followng proposton: Lemma 1 For a space C curve r(t, ks curvature vector f and only f the followng holds r( t r(/ t r( t r( t k (15 where s an arbtrary constant. Let ( uv, xuv (, yuv (, zuv (, S be a C regular parametrc surface. We have the followng conclusons. Lemma Let T correspond to du/dv and N be a unt tangent vector and a unt normal vector of S at the pont P respectvely. Then k (Tk can be the curvature vector of a curve lyng on S, passng P and havng T as ts tangent vector at P f and only f the followng holds II(d u,d v kn I(d u,d v P From Lemma we know that k descrbed n the frst paragraph n Subsecton 4.1 cannot be arbtrary vectors and must satsfy II (d u,d v k N I(d u,d v Lemma 3 Let T and N be a unt tangent vector and a unt normal vector of S at the pont P respectvely. k s the curvature vector of some or other curve lyng on S, passng P and havng T as ts tangent vector at P. r(ts an arbtrary curve lyng on S, passng P and havng T as ts tangent vector at P. Then t follows at P that r( t kn r( t N Theorem Let T and N be a unt tangent vector and a unt normal vector of S at the pont P respectvely. k s the curvature vector of some or other curve lyng on S, passng P and havng T as ts tangent vector at P.c(ts a C space curve passng P and havng T as ts tangent vector at P. r(ts the normal projecton curve of c(tonto S. Then at P the followng equaltes hold: 1 c( t r ( t c( t r( t ( c( t N r( t kn N Proof The projecton s characterzed by c( t r( t ( t N ( t (16 By takng the dervatve of Eq. (16 wth respect to t, the relatonshp between the correspondng tangent vectors of space curve and ts projecton curve on the surface S can be deduced as follows: P c( t r( t ( t N( t ( t N ( t (17 From Eq.(16, we have ( t 0at P. Thus from Eq.(17, at P t follows that c( t r( t ( t mn Therefore ( t 0and at P there s c( t r ( t (18 In addton, from Eq.(17, the relatonshp between the correspondng second dervatve vectors of space curve and ts projecton curve on the surface S can be descrbed as follows: c( t r( t ( t N( t ( t N( t ( t N( t Further, at P t follows that c( t r( t ( t N (19 From Eqs.(18-(19 and Lemma 3, the equalty can be easly proved. Let the normal vectors of the surface S at P, P 1 be N, N 1 respectvely. We would rather assume that the P, P 1 are neghborng n the followng sense [1].The absolute value of the angles between the normal vectors N, N 1 should be less than 180. For a fxed, take C ( t P, C ( t 1 P 1, C ( t 1T and C ( t1 T 1, where s a postve constant. From Theorem, we further take C( t 1T 1k 1k NN C ( t1 T1 k1 k 1 N1N1 Then takng F 0 (, s F 1 (, s G 0 (, s G 1 (, s H 0 (, s H 1 ( s as one group of fve degree Hermte blendng functons, we desgn a space quntc Hermte curve C ( t as follows: C ( s F ( s P F ( s P G ( s T 0 1 1 0 1 G1( s T1 H0( s 1T 1k 1k N N H1( s T1 k 1[ 1 1] k N N1 where s ( tt/( t1 t and t [ t, t 1]. Wth the method descrbed n Subsecton 3.1, we then project C ( t normally onto surface S,.e., we can obtan the frst-order ordnary dfferental equatons of the projecton curve P ( t wth ntal condtons for both parametrc and mplct representatons of the base surface S. For a fxed, the projecton curve P ( t s just the desred curve that nterpolates the data ponts P, P 1 wth the prescrbed tangent drectons T, T 1 and curvature vectors
No.1 Wang Xaopng et al. / Chnese Journal of Aeronautcs 3(010 137-144 141 k, k 1 at correspondng ponts. As a matter of fact, from Theorem we have P( t C ( t 1 T and P"( t 1 T 1 k. Then from Lemma 1 the curve P ( t has the curvature vector k at the pont P. Smlarly the conclusons hold at the pont P 1. Lettng 1,,, n1, we get a sequence of space curve segments C (.Then projectng every t curve segment C ( t normally onto the surface S, we fnally obtan a sequence of projecton curve segments P ( t, 1,,, n1. The composte curve n1 0 s (0 0 P( t P ( t t [ t, t ] s just what we want. 4.. G blendng curves The method for nterpolatng a sequence of ponts on surface can be used drectly n constructng a G blendng curve (transton curve between two gven curves parametrcally or mplctly on a parametrc surface. The G blendng curve s completely determned by the tangents and curvatures at the two ends of the transton curve segment and has nothng to do wth the global geometry and representaton of two gven curves. In contrast to general nterpolaton problem, we must frst specfy two ponts on two curves, compute the tangent drectons or curvatures of two surface curves at the two ponts respectvely, and use them as nterpolaton condtons. Then the remander for us to do s smlar to dealng wth the nterpolaton ssue. As for curvature computaton of surface curves wth all knds of expresson form, one can use the formula gven out n Ref. [1]. 4.3. Degenerate cases How do we do when one of curve segments C ( t happens to pass one center of prncple curvature of the surface at a pont of ts orthogonal projectve curve P ( t, whch means the correspondng system of equatons collapses? We gve two ways as answers to the queston above. One way s to nsert addtonal nterpolaton data nto the sequence of ponts P, 0,, s. For example, f C ( t0 happens to be one center of prncpal curvature of the surface S at the pont P ( t0, then nsert P ( t0 between the ponts P and P 1. Let C( t P( t ( t N ( t. If one of the followng conclusons holds: ( C( t0 N( t0 Su ( t0 ( C( t0 N( t0 Nu ( C( t0 N( t0 Sv ( t0 ( C( t0 N( t0 Nv then take the unt projectve vector T 0 of C ( t0 onto the tangent plane of surface at P ( t0 as tangent drecton. Of course, here we cannot use Eq. (11 or Eq.(17 to compute T 0. Obvously the drecton of the orthogonal projectve vector of C ( t onto the tangent plane of surface at the correspondng pont s determned by du/dv or dv/du. For example, from Eq.(3 we have du P (// t Su S v (1 dv Moreover, take du du 1 ( ( ( 1 ( lm dt a C P N CN Su a C P N CN S d v C P N CN S C P N CN S dt d v tt0 a 0 1 ( ( a11( 1 ( t t v u tt v 0 ( Then T 0 can be obtaned wth Eqs.(1-(. Otherwse we estmate a tangent vector at P ( t0 wth those at P and P 1 through the method presented n Ref.[1].We must addtonally estmate the curvature vector at P ( t0 wth the method [1]. Fnally, restart the above-mentoned nterpolaton process between data ponts P and P ( 0, and P ( 0 and P 1. t t However, another way s to make the space curve segment far enough from the surface S, whch has two non-zero prncpal curvatures everywhere. Take (0 and (1 as postve or negatve constants, whose absolute values are large enough, and let C (0 and C (1 be perpendcular to N and N1 respectvely. Set
14 Wang Xaopng et al. / Chnese Journal of Aeronautcs 3(010 137-144 No.1 C(0 (0 N P C(1 (1 N 1P 1 C(0 (0 N(0 1T C(1 (1 N1(1 T 1 C(0 1T 1k [ (0 N N1 ( k N] N (0 N(0 C(1 T1k1[ (1 N 1N 1 ( k 1 N 1] N 1(1 N 1(1 to construct Hermte curve C ( t as above and then project t onto S, where N (0 and N 1 (1 can be calculated by 1 Su a11 a1 0 Su N( t P( t a1 a 0 Sv Sv S 0 0 1 u Sv Su Sv for the parametrc case or by Eq.(7for the mplct case. N (0 and N 1 (1 can be calculated by takng the second dervatve of Eq.(5 and Eq.(9 wth respect to t,and usng Eq.(5 for the parametrc case, or by takng the second dervatve of Eq.(7 and Eq.(9 wth respect to t, and usng Eq.(7 for the mplct case. Remark 1 The constants, provde some freedoms n desgnng curve. Consequently, they can be used as control parameters to modfy the desred curve so that ts shape meets our demand better. However one must pay partcular attenton to the fact that the overlarge value of these parameters mght cause an undesrable cups or loop on the curve. Remark If the base surface s a sphere, then the orthogonal projecton curve of a space curve (not passng the sphercal center onto the sphere can be descrbed by an explct equaton nstead of by a procedural curve [13]. 5. Numercal Integraton The presented method makes use of numercal ntegraton technques feasble for frst-order explct ordnary dfferental equatons (ODEs systems assocated wth ntal value problems. The desred surface curves for all cases can be obtaned by solvng ther correspondng ntal value problems. Nevertheless, there are no analytcal solutons for the system of frst order ODEs presented n the artcle except some specal surfaces such as sphere [13], etc. In general, to deal wth the most typcal surface n CAD we can solve the systems very effcently by usng standard numercal technques. For example, the ODEs solver of MATLAB [14] or Numercal Recpes [15], based on Runge-Kutta, Adams-Bashforth and other numercal methods can be used to solve these systems. In addton, these solvers provde user wth good controls of tolerance [16]. Generally speakng, the presented method works well for any parametrc or mplct surface. It should be emphaszed, however, that some specal cases need a careful analyss. For nstance, f the surface s composed of several pecewse contnuous surface patches, some knd of contnuty condtons must be consdered to ensure that the dfferental model s stll vald n the neghborhood of each patch boundary. In practcal applcaton, we may demand that the resultng curve should be descrbed n the standard form such as B-splne or non-unform ratonal B-splnes (NURBS. However, the precedng numercal ntegratng yelds an array of ponts n parametrc doman and hence n the surface. Fortunately, usng, for example, a cubc B-splne to nterpolate those ponts on the surface we can create a closed form B-splne approxmaton of projecton curve. Here the partcular approxmaton method [17-18] developed by F.E. Wolter and J. Qu can be used to obtan good accuracy. Moreover, based on the method [19] we can also get the B-splne approxmaton of the resultng curve wth the array of ponts n the parametrc doman. 6. Implementaton Examples Actually the presented method can be appled to any mplct or parametrc surface, ncludng Bezer, B-splne and NURBS surface that are popular n CAD and CG. However, for the sake of smplcty, we only take a Bézer surface as the base surface n the parametrc and mplct surface cases respectvely to demonstrate ther effectveness. The control ponts of Bzer surface are( 4, 4, 1, ( 4, 0, 3, ( 4, 4, 1, (0, 4, 3, (0, 0, 6, (0, 4, 3, (4, 4, 1, (4, 0, 3, (4, 4, 1. Fg.1 shows how to desgn a G contnuous curve on Bzer surface S, whch nterpolates only a par of ponts, such as P 0, P 1, wth the prescrbed unt vectors T 0, T 1 and curvature vectors k 1,k at correspondng ponts on Bzer surface S, where the curve P(ts the desred curve. We frst desgn a space curve C(twth Hermte nterpolaton method and then project orthogonally the curve C(tonto Bzer surface to create the curve P(t.Fg. shows a G contnuous curve on Bzer surface S. The curve s defned by three ponts P 0, P 1, P wth the prescrbed unt vectors T 0, T 1, T and curvature vectors k 0, k 1, k at correspondng ponts. Fg.3 llustrates the G contnuous
No.1 Wang Xaopng et al. / Chnese Journal of Aeronautcs 3(010 137-144 143 blendng curve of two curves on surface S,.e., the blendng curve segment 01 and curves 0 have common tangent drecton and common curvature vectors at P 0, 1, and the same s true for 01 and curves 1 at P 1. Fg.4 ndcates a G contnuous ntal path desgned by the presented method for robotc fbre placement n producng fbre-renforced composte components. Fg.1 Desgn procedure for G nterpolaton curve on Bézer surface. 7. Conclusons Ths artcle develops a method for constructng a G contnuous curve lyng on a surface. (1 The method has good local propertes and reasonably good flexblty n shape control of the resultng curves, n contrast to the most related methods for curves desgn on surfaces. Compared wth the exstng method for G nterpolaton and blendng on surfaces, the presented one gves drectly the frst-order ordnary dfferental equatons of the desred nterpolaton curve, thus avodng usng any surface/surface ntersecton algorthms(whch s usually a troublesome process, and can guarantee the exstence of the nterpolaton curve. ( The method s also applcable to the feature desgn or pattern desgn on surfaces n some related ndustral felds.other potental applcatons mght appear n path plannng for robotc fbre placement toward producng fbre-renforced composte components. References Fg. G contnuous curve on Bézer surface. Fg.3 G blendng curve of two curves on Bézer surface. Fg.4 An ntal path P 0 (tdesgned by our method over a mould surface. [1] Hartmann E. G nterpolaton and blendng on surfaces. The Vsual Computer 1996; 1(4: 181-19. [] Pottmann H, Hofer M.A varatonal approach to splne curves on surfaces. Computer Aded Geometrc Desgn 005; (7: 693-709. [3] Hofer M, Pottmann H. Energy-mnmzng splnes n manfolds. ACM Transactons on Graphcs 004; 3(3: 84-93. [4] Wang X P, Zhou R R, Yu Z Y, et al. Interpolaton and blendng on parametrc surface. Journal of Software 004; 15(3: 451-460. [5] Pegna J, Wolter F E. Surface curve desgn by orthogonal projecton of space curves onto free-form surfaces. Journal of Mechancal Desgn 1996; 118(1: 45-5. [6] [Wang X P, Zhang W Z, Zhou L S, et al.constructng G 1 contnuous curve on a free-form surface wth normal projecton. Internatonal Journal of Computer Mathematcs (n press. [7] Flöry S, Hofer M. Constraned curve fttng on manfolds. Computer-Aded Desgn 008; 40(1: 5-34. [8] Gu X F, He Y, Qn H. Manfold splnes. Graphcal Models 006; 68(3: 37-54. [9] Lamn A, Mraou H, Sbbh D. Recursve computaton of Hermte sphercal splne nterpolants. Journal of Computatonal and Appled Mathematcs 008; 13(: 439-453. [10] Popel T, Noakes L. Bzer curves and C nterpolaton n Remannan manfolds.journal of Approxmaton Theory 007; 148(: 111-17. [11] Popel T, Noakes L.C sphercal Bzer splnes. Computer Aded Geometrc Desgn 006; 3(3: 61-75. [1] Do Carmo M P. Dfferental geometry of curves and
144 Wang Xaopng et al. / Chnese Journal of Aeronautcs 3(010 137-144 No.1 surfaces. Englewood Clffs, NJ: Prentce-Hall, 1976. [13] Wang X P, Zhou R R, Ye Z L, et al. Constructng sphercal curves by nterpolaton. Advances n Engneerng Software 007; 38(3: 150-157. [14] The Math Works, Inc.Usng MATLAB. Verson 6.Natck,MA: The Math Works, Inc., 000. [15] Press W H, Teukolsky S A, Vetterlng W T, et al. Numercal recpes.nd ed. Cambrdge:Cambrdge Unversty Press, 199. [16] Pug-Pey J, Glvez A, Iglesas A. Helcal curves on surface for computer aded geometrc desgn and manufacturng. Lecture Notes n Computer Scence 004; 3044: 771-784. [17] Wolter F E, Tushy S T. Approxmaton of hgh-degree and procedural curves. Engneerng wth Computers 199; 8(: 61-80. [18] Qu J, Sarma R. The contnuous non-lnear approxmaton of procedurally defned curves usng ntegral B-splnes. Engneerng wth Computers 004; 0(1: -30. [19] Renner G, We V. Exact and approxmate computaton of B-splne curves on surfaces. Computer-Aded Desgn 004; 36(4: 351-36. Bography: Wang Xaopng Born n 1964, he receved Ph.D. degree n Aerospace Manufacture Engneerng from Northwestern Polytechncal Unversty. Afterwards, n 00, he fnshed hs postdoctoral research work n Research Center of CAD/CAM Engneerng, Nanjng Unversty of Aeronautcs and Astronautcs and became an assocate professor there. Currently hs man research nterests nclude geometrc modelng, CAD/CAM, robotc fbre placement and so on. He has publshed a lot of scentfc artcles on both domestc and foregn perodcals. E-mal: levne@nuaa.edu.cn