Algorithm for directional projection of parametric curves onto B-spline surfaces

Transcription

1 Algorthm for drectonal projecton of parametrc curves onto B-splne surfaces Ha-Chuan Song, Jun-Ha Yong To cte ths verson: Ha-Chuan Song, Jun-Ha Yong. Algorthm for drectonal projecton of parametrc curves onto B- splne surfaces. Computer-Aded Desgn and Applcatons 3, Jun 3, Lombary, Italy. 3. <hal-9676> HAL Id: hal Submtted on 9 Dec 3 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Algorthm for drectonal projecton of parametrc curves onto B-splne surfaces Ha-Chuan Song,,3,4, Jun-Ha Yong,3,4 School of Software, Tsnghua Unversty, Bejng 84, P. R. Chna Department of Computer Scence and Technology, Tsnghua Unversty, Bejng 84, P. R. Chna 3 Key Laboratory for Informaton System Securty, Mnstry of Educaton of Chna, Bejng 84, P. R. Chna 4 Tsnghua Natonal Laboratory for Informaton Scence and Technology, Bejng 84, P. R. Chna ABSTRACT Ths paper proposes an algorthm for calculatng the drectonal projecton of parametrc curves onto B-splne surfaces. It conssts of a second order tracng method wth whch we construct a polylne to approxmate the pre-mage curve of the drectonal projecton curve n the parametrc doman of the base surface. The fnal 3D approxmate curve s obtaned by mappng the approxmate polylne onto the base surface. The Hausdorff dstance between the exact drectonal projecton curve and the approxmate curve s controlled less than the user-specfed dstance tolerance. And G T the contnuty of the approxmate curve s, where T s the user-specfed angle tolerance. Experments demonstrate that our algorthm s faster than the exstng frst order algorthm. Keywords: Drectonal projecton, B-splne, Curves on surfaces, Approxmaton INTRODUCTION Drectonal projecton of curves onto surfaces s wdely mplemented n CAD (computer aded desgn) systems [3],[]. It plays fundamental roles n many operatons, for example surface cuttng, model desgnng, etc. It can also be utlzed as a tool for constructng curves on surfaces. Gven parametrc curve Pt () and surface Suv (,) n 3D space, the drectonal projecton of Pt () onto S can be defned by the drectonal projecton of the ponts of Pt () onto S. Lettng p denote an arbtrary pont on Pt (), whch s called the test pont, the drectonal projecton of p onto S s a set of ponts q on S such that the vector ( qp ) s parallel and n the same drecton wth a gven vector Dr, whch can be expressed as follows: ( qpdr ), and( qpdr ) holds. As we move the test pont p along Pt (), the moton of q results n a set of ponts on S, and that s the drectonal projecton curve (see Fg..). Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

3 Fg. : The defnton of the drectonal projecton. Denote the drectonal projecton curve by Qt (). To get Qt (), frst order algorthm were proposed n [6]. They derved the frst order dfferental equaton systems. By solvng the system wth numercal methods, they got a sequence of ponts along Qt (). Then the approxmate projecton curve could be obtaned usng the nterpolaton method on the pont sequence. There are some drawbacks n the frst order algorthm above. Frst, the step length of the ponts they got along Qt () s not well controlled (just by a constant parametrc ncrement), whch results n the uneven dstrbuton of the ponts, and drectly mpact on the approxmaton result. Second, the approxmate curve does not le completely on S, whch s not acceptable for many CAD applcatons such as surface blendng and surface-surface ntersecton [9]. The approxmaton precson of the approxmate curve cannot be controlled. Moreover, they cannot deal wth the jumpng projecton case (ths wll be ntroduced n Subsecton 3.4). In order to overcome the drawbacks of the frst order algorthm, we provde an algorthm dealng wth the drectonal projecton of parametrc curves onto B-splne surfaces, whch approxmates the exact drectonal projecton curve Qt () wth a pecewse curve on S. And ths s smlar to our publshed method [3], n whch we deal wth the orthogonal projecton of parametrc curves onto B- splne surfaces. Accordng to the defnton of the drectonal projecton, the projecton of a curve onto a surface can be regarded as the projectons of every pont of the curve onto the surface. And ths s actually related to the ntersectons of lnes and surfaces, whch s also one of the key technques that we wll apply n our algorthm, and of whch we wll make bref survey below.. Lne Surface Intersecton The ntersecton of lne and surface s the fundamental problem n curve surface computaton, and s of great use n geometrc modelng, robotc technques, collson detectons, etc [4]. The subdvson method proposed by Whtted [],[7] was frstly utlzed to solve ths problem, whch s very easy and of hgh robustness. However, when elevatng the precson requrement, the effcency of ths method wll become low. As a transformaton of the subdvson method, Nshta et al. [9] proposed the Bézer clppng method, whch recursvely dvde the surface nto regons that may contan the soluton, and regons that do not contan the soluton. Generally speakng, the ntersecton of lne and surface can be reduced nto the solvng of a nonlnear equaton system. As a classc method, Newton-Raphson method was wdely used n calculatng the ntersecton of lne and surface [],[],[5],[8],[],[4],[5],[8]. The choce of the ntal s very mportant for Newton-Raphson method to converge relably. As a result, on one hand, some researches on whether the Newton-Raphson wll converge wth a gven ntal value were proposed. Toth [5] determned the convergng nterval of the Newton-Raphson method by usng nterval analyzng technques. Lschnsk and Gonczarowsk [4] made use of the consstency of the surface and Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

4 3 mproved Toth s method [5]. Wth the help of Kantorovch detecton, Srjuntongsr and Vavass [4] can determne whether the convergence order of Newton-Raphson method s second order or not. On the other hand, some researchers consdered how to provde a good ntal value. Generally, they subdvde the surface recursvely untl every surface patches are flat enough, and the ntersecton ponts of the lne and the boundng boxes of the surface patches are chosen as the ntal value [],[],[8],[],[8]. Wang et al. [5] combned the Bézer clppng method and Newton-Raphson method together. OUTLINE OF OUR ALGORITHM Frst, some symbols whch wll be used all through the paper wll be ntroduced. In 3D space, gven a parametrc test curve Pt () where tmn [,], a normalzed projecton drecton vector Dr and a B splne base surface Suv (,) defned by: nn u v j p q j j, Suv (,) N() unvp (), where uab [,], vcd [,], P are the control ponts, Nu () and Nv () are the pth-degree and qth-degree j, j B-splne bass functons, respectvely (see Fg..). p q Fg. : Surface of the stomach of Venus: the orgnal B-splne surface, the test curve and the projecton drecton. We denote the drectonal projecton curve of Pt () onto S by Qt (). Snce Qt () les on S, there exsts a pre-mage curve qt () ((),()) utvt T of Qt () n the parametrc doman of S and Qt () Sutvt ((),()) holds. We suppose that Qt () s at least G contnuous and qt () s n Bézer representaton. We provde an algorthm whch generates an approxmate curve of Qt () on S. Frst, we construct a polylne (a contnuous curve composed of one or more lne segments) to approxmate qt () n the parametrc doman of S, takng the Hausdorff dstance and angle tolerances nto account. Then the fnal 3D pecewse approxmate curve s obtaned by mappng the approxmate polylne onto S. The Hausdorff dstance between Qt () and the fnal approxmate curve s less than D. And the contnuty of the fnal approxmate curve s G, whch means that for any par of adjacent curves n a T Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

5 4 pecewse curve, the angle between the end dervatves at the mutual pont of them s less than T. Here and are the user-specfed tolerances. The man algorthm flow s descrbed as follows: D T. Compute the drectonal projecton of the startng pont of Pt () onto S wth the lne surface ntersecton method [5].. Based on the projecton pont, wth a second order tracng method, an array of ponts along qt () s derved. Set the array of ponts as the vertces, and the ntal approxmate polylne s obtaned n the parametrc doman of S. 3. Accordng to the user-specfed Hausdorff dstance tolerance D, dscrete ponts are sampled at specfed postons on qt () and added nto the approxmate polylne so as to guarantee that the Hausdorff dstance between the fnal approxmate curve and Qt () s less than D. 4. Accordng to the user-specfed angle tolerance T, splt qt () at specfed poston to make sure G T that the fnal approxmate curve s at least contnuous. 5. After the steps above, the fnal approxmate polylne n the parametrc doman of S s obtaned. Map the polylne onto S usng blossomng technques [6],[7] and the 3D approxmate curve of Qt () s obtaned. The rest of the paper s organzed as follows. Secton 3 ntroduces the second order tracng method, and constructs the ntal approxmate polylne n the parametrc doman of S. In Secton 4, the approxmaton precson and the contnuty of the fnal approxmate projecton curve are controlled, usng our publshed method [3]. Secton 5 shows the expermental results and Secton 6 concludes the paper. 3 CONSTRUCTING THE INITIAL APPROXIMATE POLYLINE OF THE DIRECTIONAL PROJECTION IN THE PARAMETRIC DOMAIN OF S Frst, the ntal approxmate polylne s to be constructed n the parametrc doman of S wth our second order tracng method, whch can be summarzed n the followng steps:. Compute the frst drectonal projecton of the ponts on Pt () onto S, and take the projecton pont as the current tracng pont.. Compute the frst and second order dervatves of qt () at the current tracng pont n the parametrc doman of S and locally approxmate qt () wth the second order Taylor polynomal. 3. Get the parametrc ncrement usng the constant arc length ncrement strategy. Then the new test pont on Pt () and the estmated poston of the next tracng pont are obtaned. 4. Refne the accuracy of the estmaton wth the Newton-Raphson method. Connect the result pont of the refnement and the current tracng pont wth a lne segment, and add t at the end of the approxmate polylne. Take ths refnement result pont as the new tracng pont. 5. Repeat steps ~ 4 untl the test pont we get reaches the end of Pt () and the ntal approxmate polylne s obtaned. 3. Preparaton for Tracng Suppose P to be the startng pont of Pt (). To start the tracng we need to compute the drectonal projecton ponts of P onto Suv (,). For these ponts les on a lne L, whch passes P and s parallel wth Dr, so we derve these ponts utlzng lne surface ntersecton method. We frst dvde the B-splne surface Suv (,) nto several Bézer surface patches (see Fg.3.). Then we can just take advantage of the dvson result to compute the drectonal projecton of P. And t can be fgured out n the followng three steps: Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

6 5. Frst we take the surface patches, whose control polygons ntersect wth L, as the canddate surface patches. Accordng to the convex hull ablty of Bézer surfaces, other surfaces can be pruned. So we can reduce the range of the computaton.. Then for every canddate surface patch, use the lne surface ntersecton method [5] to compute the ntersecton of L wth t. And each ntersecton pont s taken as one of the canddate ntersecton ponts of L wth the base surface. 3. Elmnate the ntersecton ponts, whch do not satsfy the condton ( ppdr ), from the canddate ntersecton ponts. Fg. 3: Surface of the stomach of Venus: (a) the orgnal B-splne surface; (b) the dvson result and the canddate surface patch. So we can get the ntersecton ponts of L and the base surface. Assume that one of the projecton ponts of P onto S s Q, meanwhle n the parametrc doman of S, we can get the correspondng pre-mage pont q on qt (). To control the tracng process, we preserve two ponts cur_ PR and cur_ QR to store the values of the current test pont on Pt () and the correspondng current tracng pont on qt (). cur_ P and cur_ Q have the same parameter value on Pt () and qt (), respectvely. After the drectonal projecton of P, the pont q s taken as the current tracng pont cur_ Q and P s taken as the current test pont cur_ P. To keep on tracng, we need to get the next tracng pont accordng to cur_ Q. Note that there may exst more than one ntersecton pont of L and the base surface, and for each ntersecton pont, we take t as a seed pont, and trace the correspondng projecton curve, respectvely. Takng one projecton curve for example, now we wll ntroduce how to construct the approxmate curve of the projecton curve. 3. The Frst and Second Order Dervatves of the Drectonal Projecton Curve We estmate the next tracng pont wth the second order Taylor expanson of qt () at cur_ Q, so we need to compute the frst and second order dervatves of qt () at cur_ Q. Now we wll deduce the frst and second order dfferental equaton systems satsfed by the drectonal projecton. Assume that S s at least C contnuous locally. Frst, recall the defnton of the drectonal projecton: ( qpdr ), (3.) where p s the pont on the test curve Pt (), q s the correspondng pont on the projecton curve Qt (), Dr s the projecton drecton vector, Suv (,) s the base surface, and Qt () Sutvt ((),()) holds. 3 Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

7 6 Take the dervatve of Eqn. (3.) on both sdes wth respect to t, and t follows that: du dv dp S S Dr. u v dt dt dt Accordng to the dstrbutve law of cross product, we have: du dv dp ( S Dr) ( S Dr) Dr, u v dt dt dt multply S and S on both sdes of the equaton above, respectvely, and t follows that: u v Solve the system above, we have: du dp SSDr S Dr v u v dt dt. dv dp SSDr S Dr u v u dt dt dp S Dr v du dt dt SSDr v u. dp S Dr u dv dt dt SSDr u v Eqn. (3.) s just the frst order dfferental equaton system of the drectonal projecton, and ths was frst derved by [6]. Moreover, take the dervatve of Eqn. (3.) on both sdes wth respect to t, we have: dp du dvdp S Dr S S Dr v vu vv du dt dt dt dt dt du dv du dv S S S Dr S S v uu uv vu vvs Dr u dt dt dt dt. (3.3) dp S Dr du dvdp u dv dt S S Dr uu uv dt dt dt dt du dv du dv S S S Dr S S u vu vv uu uv S Dr v dt dt dt dt And Eqn. (3.3) s just the second order dfferental equaton system of the drecton projecton. 3.3 Searchng For the Next Tracng Pont at We estmate the poston of the next tracng pont wth the second order Taylor expanson of qt () cur_ Q, whose expresson s as follows: qt () t ( ) () () o( t) (3.) qttqt qt t, (3.4) where q( t) cur_ Q,, t s the parametrc ncrement and ot () s the second order Taylor remander, whch s a vector-valued functon ft ( ) such that remander of the expanson, the polynomal follows: t lm ft ( )/ t. By omttng the Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

8 7 qt () t, (3.5) qt ( t) qt () qt () t wth whch we approxmate qt () locally. Gven the value of t, we can get the values of qt () and qt () wth Eqn. (3.) and (3.3), respectvely. So the only unknown factor n Eqn. (3.5) s t, whch drectly mpacts on the dstrbuton of the samplng ponts and the approxmaton result. In order to obtan an evenly dstrbuted samplng durng the generaton process of the ntal approxmate polylne, we use the step length controllng method based on constant arc length ncrement used n [3] as follows. Let ds and dt denote the arc dfferental and parameter dfferental of Qt (), respectvely. It follows that: dsq() t dt. Further, accordng to the chan rule, we can get the expresson of Qt (): where the values of S u, S, du v dt, dv dt du dv Qt () S S, (3.6) u v dt dt can all be calculated wth the gven value of t, respectvely. Hence, wth the user-specfed arc length ncrement s on Qt (), the correspondng approxmate parametrc ncrement Substtute obtaned as qt ( t) (,) uv. t s determned by: s t. (3.7) Qt () t nto Eqn. (3.5), the estmated poston of the next tracng pont on qt () can be Ths step length controllng method s ncomplete n [3], because t s hard to estmate the total arc length of the projecton curve before we can generate t. And mproper arc length ncrements may leads to the wrong results. In the case of drectonal projecton, the arc length of the projecton curve Qt () can be estmated lke ths. We can frst sample on the test curve Pt (), lnk the sample ponts wth a polylne, wth whch we approxmate Pt (). Then we drectonally project ths polylne onto the control polygons, whch are only composed of planar trangles, of the Bézer surface patches we derved n Subsecton 3.. The drectonal projectons of polylne onto trangles can be computed n lnear tme and are much easer to realze, whose projecton result curve s also a polylne. Then the length of the projecton result polylne s taken as the approxmate arc length of the projecton curve Qt (), and we can choose a proper arc length ncrement accordng to the estmated total arc length. Usually, the estmated pont devates from the precse pont we desre, so an error exsts. Accordng to the defnton of the drectonal projecton n Eqn. (3.), we defne the error of the estmaton as follows: DrSuv (,) cur_ P Suv (,) cur_ P where (,) uvqt ( t) s the estmated pont generated by our tracng method n the parametrc doman, Suv (,) s the 3D mage pont of (,) uv on S, Dr s the normalzed projecton drecton. The geometrc sgnfcance of s the absolute value of the sne of the angle between Dr and ((,) Suv cur_) P. Specally, when Suv (,) concdes wth cur_ P, Eqn. (3.8) becomes nvald for the, (3.8) Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

9 8 denomnator equals to. In ths stuaton, we let the error, because cur_ P les on S now and Suv (,) s just the projecton pont of t. As a result, to get the precse pont, a refnement s needed. Take the pont Pt ( t) as the new test pont cur_ P. We ntroduce the followng Newton-Raphson method based refnement method. Multply S and S on both sde of Eqn. (3.), respectvely, and set r(( Suv (, ) p) Dr). We have u v the followng equaton system: fuv (,) (((,) Suvp) DrSuv ) (,) rsuv (,) u u, (3.9) guv (,) (((,) Suvp) DrSuv ) (,) rsuv (,) v v Solve the equaton system above wth Newton-Raphson method, where we set: uuu, vvv ff Sr SrSDrS ( ) u v uu uv u v J, ggsrsdrs ( ) Sr u v uv v u vv fuv (,) k. guv (,) Then the parameter ncrement of the teraton can be obtaned by solvng: J k. Then the convergence crtera of the teraton are as follows: ( )(,) ( )(,), u v. u usuv v vsuv Suv (,) cur_ P,. DrSu (,) v p 3., Dr Su (,) v p where (,) uv s the parameter of the th teraton, and two zero tolerances of Eucldean dstance and sne. The teraton s halted f any of the three condtons above s satsfed. In the teratons above, we set the ntal parameter value equals to (,) uv, and set pcurp _ n the above Newton- Raphson method. The convergence of the teraton depends on the errors of the estmated ntal values. As shown n the expermental results n Secton 5, the average estmaton error of our algorthm s comparable wth that of the frst order algorthms [6] mplemented wth Runge-Kutta method, whch have been wdely used. Let Connect next_ QR denote the result pont of the refnement n the parametrc doman of S. cur_ Q and next_ Q wth a lne segment L, and add L nto the approxmate polylne. Take next_ Q as the new tracng pont and we contnue seekng for the next tracng pont. The tracng wll not stop untl the test pont we obtan reaches the end of Pt (). Then we can get the ntal approxmate polylne of qt () n the parametrc doman of S. 3.4 Projecton Jumpng Checkng In the generatng process of the approxmate projecton curve, the drectonal projecton ponts of the test pont on Pt () onto S may jump, wth the ncrement of the parameter (see Fg. 4.). The Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

10 9 projecton lne L of the current tracng pont Pt () only ntersects wth surface n the back part (the pont Q n Fg. 4.). However the projecton lne L of the next tracng pont Pt ( t) ntersects both n front and back parts of the surface (the ponts Q and ' Q n Fg. 4.). And ths s called the jumpng of the ntersecton ponts. If we stll only trace the projecton curve n the back part of the surface, the should-exst projecton curve n the front part of the surface wll be mssed. We deal wth ths case as follows. Fg. 4: The jumpng of the ntersecton ponts. For each projecton curve Qt, () we record the tag Actve_ Patch representng the surface patch the cur_() Q t of Qt () les n. When we compute the next projecton pont, we construct the projecton lnes L whch start from the pont Pt ( t) and s parallel to Dr, and L whch start from the pont Pt () and s parallel to Dr. Compute the ntersecton of L and the control polygons of all surface patches. For every surface patch check whether the jumpng happens wth the followng crtera:. Inter_ Patch s not the neghbour of any of. Inter_ Patch s the neghbour of some of ( cur_ Q ) to the common edge of specfed arc length ncrement. 3. Inter_ Patch s just one of control polygon of Inter_ Patch, whose control polygon ntersects wth L, we Actve_ Patch. Actve_ Patch, but the dstance from the pont S Inter_ Patch and Actve_ Patch s bgger than s, the user Actve_ Patch, but the number of ntersecton ponts of L and the Inter_ Patch s more than that of L (In ths case, accordng to the varaton dmnshng property of the Bézer surface, the number of ntersecton ponts of L and Inter_ Patch s may be more than that of L ). 4. Inter_ Patch s just one of L and the control polygon of bgger than s. Actve_ Patch, but the dstance from any of ntersecton ponts of Inter_ Patch to the correspondng ntersecton pont of L s Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

11 The jumpng may happen f any of the four condtons above s satsfed. Under ths condton, we need to compute the ntersecton ponts of L wth the base surface Suv (,) usng lne surface ntersecton method [5] as we dd n Subsecton 3.. For each ntersecton pont derved, match t wth the estmated pont (we got n Subsecton 3.3) of current tracng ponts cur_ Q (each estmated pont can only match to one nearest ntersecton pont wthn the dstance s ). Each matched ntersecton pont s taken as the new tracng pont of the correspondng projecton curve. And each unmatched ntersecton pont s taken as a new seed pont whch s the start pont of a new segment of projecton curve. Another specal case of jumpng s the projecton curve crosses the boundary of the surface, where the number of ntersecton ponts of L and Suv (,) s less than that of L. Ths can be detected lke ths.. The estmated pont (we got n Subsecton 3.3) of current tracng ponts cur_ Q exsts the boundary of the surface.. The number of ntersecton ponts of L and the control polygons of the surface patches s less than that of L. The jumpng may happen f any of the two condtons above s satsfed. Under ths condton, we need to compute the ntersecton ponts of L wth the base surface Suv (,) usng lne surface ntersecton method [5] as we dd n Subsecton 3.. If L does not ntersect the surface at any projecton curve, the tracng of the correspondng projecton curve should be stopped. Fg. 5. shows our processng result of ths jumpng case. Fg. 5: The processng result of jumpng. Recall that the frst order algorthm [6] only solve the frst dfferental equaton systems wth numercal method, so t cannot deal wth ths jumpng case, whch frequently appears n practcal. 4 CONTROLLING THE APPROXIMATION PRECISION AND THE CONTINUITY OF THE FINAL APPROXIMATE PROJECTION CURVE After the constructon of the ntal approxmate polylne n the parametrc doman of Suv (,), we control the approxmaton precson and the contnuty of the fnal approxmate curve wth the same method we used n [3], whch adds more samplng ponts of Qt () nto the approxmate polylne and Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

12 guarantees that the Hausdorff dstance between the fnal 3D approxmate curve and the exact projecton curve Qt () s less than the user-specfed Hausdorff dstance tolerance D, and the contnuty of the fnal 3D approxmate curve s More detals can be found n [3]. G, where T T s the user-specfed angle tolerance. Wth the fnal approxmate polylne n the parametrc doman of S, we map the polylne to the Bézer surfaces we get n Subsecton 3. usng blossomng technques [6],[7], and the correspondng 3D approxmate curve of the exact drectonal projecton curve Qt () on S s obtaned, whch s n Bézer form and les completely on S. The degree of the 3D approxmate curve s p q, where p and q are the u-drecton degree and v-drecton degree of the B-splne surface S, respectvely. 5 EXPERIMENTAL RESULTS AND COMPARISONS We present several examples for the drectonal projecton. All the experments are mplemented wth Mcrosoft Vsual Studo 5 wth wndows XP on the same PC wth Intel Core Duo CPU.53 GHz, GB Memory. In all of our experments below, where, are two convergence tolerances for Newton-Raphson method ntroduced n Subsecton 3.3. Fg. 6: Example, Drectonal projecton of a cubc B-splne curve onto the bcubc B-splne free form surface where the projecton drecton s opposte to z axs. We mplement the frst order algorthms n [6] wth fourth order Runge-Kutta method (R-K4), whch s also the recommended method n [6]. Comparsons between the frst order algorthms [6] and our second order tracng method on effcency and accuracy are performed durng the process of generatng the ntal approxmate polylne n Example, where a cubc B-splne curve Pt () s drectonally projected onto a 5 order Bézer surface, as llustrated n Fg. 6.. Specfcally, the 6 control ponts of Pt () are: T P (., 43.75,3.86) T P (., 4.3, 3.86) T P (., 3.8, 3.58) T. P (.,.87,33.86) 3 T P (.,.6,6.4) 4 T P (.,6.7,.7) 5 Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

13 And the knot vector s: (:; :; :; :; :6; :68; :; :; :; :): The 6 control ponts of the surface are: T T P ( 98.75,3.5, 4) P ( 98.75,3.5,4) T P ( 9.77,.86, 4) T P ( 9.77,.86, 4) T P ( 8.48,.3, 4) T, P ( 8.48,.3,4) T. P ( 86.57, 3.46, 4) T 3 P ( 86.57, 3.46,4) 3 T P ( 93.98, 5.36, 4) T P ( 93.98, 5.36,4) 4 4 T T P ( 97.84, 57.7, 4) P ( 97.84, 57.7,4) 5 5 For the purpose of equty, constant parametrc ncrement strategy, as Runge-Kutta method does, are utlzed n both of the two knds of algorthms. Further, accordng to ths paper, both of the two knds of algorthms generate the ntal approxmate polylne by estmatng the poston of the tracng pont. In order to measure the accuracy of the two knds of algorthms, we record the estmaton errors, defned n Eqn. (3.8), for each estmated pont of the two knds of algorthms, and compute the average estmaton errors for each parametrc ncrement of the two knds of algorthms. And the refnement based on the Newton-Raphson method ntroduced n Subsecton 3.3 s performed n both of the two knds of algorthms. The comparson results of the two knds of algorthms on Example are recorded n Tab... Increment Tme (ms) Errors Number of dscrete ponts t R- K4 [6] ours R- K4 [6] ours e-.97e e-.765e e e e-4.67e e-4.444e-4 Tab. : Tme cost and error comparson for Example. Fg. 7: Effcency comparson between our second order tracng method and the frst order algorthms for Example. As shown n Tab.., as we decrease the parametrc ncrement, the average errors decrease, the tme costs and the numbers of dscrete ponts ncrease. We frst consder the errors. For each parametrc ncrement, the average errors of our algorthm are comparable wth those of the frst order Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

14 3 algorthm wth R-K4, and are even smaller than the frst order algorthm when the parametrc ncrement decreases to 5. As for the tme costs, the dfference of the two knds of algorthms can be clearly llustrated n Fg. 7. Our algorthm s about tmes faster than the frst order algorthms mplemented n R-K4 method. And the gaps are gettng larger as the parametrc ncrement decreases. Fg. 8: Drectonal projecton of a cubc B-splne curve onto the bcubc B-splne free form surface where the projecton drecton s opposte to z axs, the stomach of Venus; (b) Example 3, surface of woman face. D 3 T, : (a) Example, surface of Besdes the comparsons wth the frst order algorthm durng the process of generatng the ntal approxmate polylne, we also test the performance of our algorthm for the whole projecton process, whch uses the constant arc length ncrement strategy, performs jumpng checkng and controls the approxmaton precson and the contnuty of the fnal approxmate projecton curve. Fg. 8. shows the drectonal projecton of cubc B-splne curves onto the bcubc B-splne free form surfaces. The results of our algorthm on Example and Example 3 are shown n Tab... Example Example 3 Arc length ncrement s Tolerance ( / ) D T / / / / Degree Number of segments Number of control ponts Contnuty G G G G Tme (ms) Tab. : Results of our algorthm for the Examples. As we can see from the results, the effcency of the algorthm meets the real-tme requrement. Moreover, the approxmaton precson and the contnuty of the fnal approxmate projecton curve can be controlled. Then we present two examples for jumpng case projecton, whch we ntroduced n Subsecton 3.4. Fg. 9(a). and 9(b). plot the jumpng case projectons, where two cubc B-splne curves are drectonally projected onto a tour (transformed nto NURBS form) and a free form B-splne surface, respectvely. The sold blue curves are the test curve, whle the curve wth green break ponts s the 3D approxmate projecton curve. As we can see from the results, the jumpng s correctly detected, and new segment of projecton curves are added nto the fnal projecton result, and the projecton curve stops at the boundary of the surface. Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

15 4 Fg. 9: jumpng case projecton. Then, more examples of our algorthm are shown n Fg.. and Fg... Fg. : The desgn of the front cover of a car. Fg. : Constructon of curves on surface. 6 CONCLUSION Ths paper presents an approxmaton algorthm for drectonal projecton of parametrc curves onto B-splne surfaces. The second order tracng method s appled to construct the ntal approxmate polylne n the parametrc doman of the surface. Wth the ntal approxmate polylne, further Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,

16 5 samplng on the exact drectonal projecton curve s performed to control the approxmaton precson and the contnuty of the fnal 3D approxmate curve. The man contrbuton of ths paper s the dervaton of the second order dfferental equaton system of the drectonal projecton, wth whch we can compute the farthest ponts and the Hausdorff dstances between the curve segments and ther correspondng lne segments n the parametrc doman usng teraton methods, whch enables us to control the approxmaton precson and the contnuty of the approxmate curve [3]. Expermental results ndcate that the accuracy of our algorthm s comparable wth that of the frst order algorthm [6], and at the same tme our algorthm s faster than t. Moreover, our algorthm can deal wth the jumpng projecton case, and control the approxmaton precson and the contnuty of the approxmate curve, whle the frst order algorthm [6] cannot do ths. ACKNOWLEDGEMENTS The research was supported by Chnese 973 Program (CB38) and Chnese 863 Program (AA49). The frst author was supported by the NSFC (635, 6735). The second author was supported by the NSFC (6639, 67377). REFERENCES [] Alan, F.; John, B.: Chebyshev polynomals for boxng and ntersectons of parametrc curves and surfaces, Computer Graphcs Forum 3 (3), 994, 7 4. [] Barth, W.; W. Sturzlnger,: Effcent ray tracng for Bézer and B-splne surfaces, Computer and Graphcs 7 (4), 993, [3] CATIA, Dassault Systems. [4] Danel, L.; Jakob, G.: Improved technques for ray tracng parametrc surfaces, The Vsual Computer 6 (3), 99, [5] Danel, T.: On Ray Tracng Parametrc Surfaces, Computer Graphcs (SIGGRAPH 85 Proceedngs) 8 (4), 985, [6] DeRose, T. D.: Composng Bézer smplexes, ACM Transactons on Graphcs 7 (3), 988, 98. [7] DeRose, T. D.; Goldman, R. N.; Hagen, H.; Mann, S.: Functonal composton algorthm va blossomng, ACM Transactons on Graphcs (), 993, [8] Mchael, S.; Rchard, B.: Ray Tracng Free-Form B-splne Surfaces, IEEE Computer Graphcs and Applcaton 6 (), 986, [9] Nshta, T.; Sederberg, T. W.; Kakmoto, M.: Ray tracng trmmed ratonal surface patches, Computer Graphcs (SIGGRAPH 9 Proceedngs) 4 (4), 99, [] Qn, K.-H.; Gong, M.-L.; Guan, Y.-J.; Wang, W.-P.: New method for speedng up ray tracng NURBS surfaces, Computer Graphcs (5), 997, [] Rubn, S. M.; Whtted, T.: 3-Dmensonal Representaton For Fast Renderng Of Complex Scenes, Computer Graphcs (SIGGRAPH 8 Proceedngs) 4 (3), 98, 6. [] SoldWorks, Dassault Systems. [3] Song, H.-C.; Yong, J.-H.; Yang, Y.-J.; Lu, X.-M.: Algorthm for orthogonal projecton of parametrc curves onto B-splne surfaces, Computer Aded Desgn 43 (4),, [4] Srjuntongsr, G.; Vavass, S. A.: A condton number analyss of a lne-surface ntersecton algorthm, SIAM Journal on Scentfc Computng 3 (), 7, [5] Wang, S.-W.; Shh, Z.-C.; Chang, R.-C.: An effcent and stable ray tracng algorthm for parametrc surfaces, Journal of Informaton Scence and Engneerng 8 (4),, [6] Wang, X.-P.; We, W.; Zhang, W.-Z.: Projectng curves onto free-form surfaces, Internatonal Journal of Computer Applcatons n Technology, 37 (),, [7] Whtted, T.: Improved Illumnaton Model For Shaded Dsplay, Communcatons of the ACM 3 (6), 98, [8] Yang, C.-G.: On Speedng Up Ray Tracng Of B-splne Surfaces, Computer Aded Desgn 9 (3), 987, 3. [9] Yang, Y.-J.; Cao, S.; Yong, J.-H.; Zhang, H.; Paul, J.-C.; Sun, J.-G.; Gu, H.-j.: Approxmate computaton of curves on B-splne surfaces, Computer Aded Desgn 4 (), 8, Computer-Aded Desgn & Applcatons, (a), 3, bbb-ccc 3 CAD Solutons, LLC,