Rational Ruled surfaces construction by interpolating dual unit vectors representing lines

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1 Ratonal Ruled surfaces constructon by nterolatng dual unt vectors reresentng lnes Stavros G. Paageorgou Robotcs Grou, Deartment of Mechancal and Aeronautcal Engneerng, Unversty of Patras 265 Patra, Greece Nkos A. Asragathos Robotcs Grou, Deartment of Mechancal and Aeronautcal Engneerng, Unversty of Patras 265 Patra, Greece ABSTRACT In ths aer, a new reresentatonal model s ntroduced for the ratonal famly of ruled surfaces n Comuter Grahcs. The surface arameterzaton s constructed usng the NURBS bass functons and lne geometry. The ruled surface s defned by nterolatng drectly dual unt vectors reresentng lnes, whch s a sngle arametrc surface and ts shae deends on the control lnes. All the advantages of the NURBS bass such as shae control and the local modfcaton roerty are also alcable and bequeathed to the dual NURBS ruled surface. The roblem of drawng the lnes defned by dual unt vectors s also resolved. Towards ths drecton, we roose a smle technque to calculate the surface s strcton curve n order to draw the rulngs of the surface wthn the strcton curve neghborhood. The on-screen 3D lot of the surface s realzed n a re-defned secfc regon close to the strcton curve. Wth the roosed technque a natural reresentaton of the ruled surface s derved. The shae of the surface can be ntrnscally manulated va the control lnes that ossess one more degree of freedom than the control onts. Our method can fnd alcaton not only n CAD but n the areas of NC mllng and EDM. Keywords Ruled surface, dual unt vectors, Lne Geometry, Plücker coordnates, Strcton curve. 1. INTRODUCTION A 3D surface s called ruled f through any of ts onts asses at least one lne that les entrely on that surface. The generaton of these surfaces s consdered smle snce they are created by movng a lne n 3D sace [Far97]. Alcatons of these surfaces can be found n Comuter Aded Desgn (CAD), CAGD, NC mllng, and wre Electrc Dscharge Machnng (EDM) [Yan96]. A ruled surface can be generated by lnear nterolaton between two gven bound curves or usng tensor roduct surfaces [Gra97]. Alternatvely, a ruled surface can be defned usng a lne geometry reresentaton. Ravan and Wang [Rav91] were the Permsson to make dgtal or hard coes of all or art of ths work for ersonal or classroom use s granted wthout fee rovded that coes are not made or dstrbuted for roft or commercal advantage and that coes bear ths notce and the full ctaton on the frst age. To coy otherwse, or reublsh, to ost on servers or to redstrbute to lsts, requres ror secfc ermsson and/or a fee. SHORT COMMUNICATION roceedngs ISBN WSCG 26, January 3-February 3, 26 Plzen, Czech Reublc. Coyrght UNION Agency Scence Press frst to show that ths aroach has the advantage of curve tye algorthms and they constructed ruled surfaces of degree 3m from m+1 control lnes. The dea of utlzng dual vector calculus and screw theory n CAGD s not new. Chen and Pottman [Che99], resented a method to derve an aroxmaton of ruled surface n 4-D sace by tensor roduct B-Slne reresentaton. Dng [Dn2], used the De Boor algorthm and screw theory to determne ruled surfaces n 6-D sace. Xa [Xa] resented an aroach for the roblem of moton cutter lannng for sde mllng of ruled surfaces. In ths aer, a method s ntroduced to defne a ruled surface by extendng NURBS to dual vector calculus. Most of the ublshed aroaches deal wth the defnton of a ruled surface usng aroxmaton algorthms such as the De Boor and De Casteljau. In ths aer the ruled surface s reresented by a dual NURBS structure. The reresentaton uses the same bass functon for curve defnton, where the control onts are relaced by control lnes descrbed by dual unt vectors. Snce the rulngs of the surface are reresented by dual unt vectors, the man reresentaton ssue s how to defne a secfc art of the entre surface. In ths aer we are usng the WSCG26 Short Paers Proceedngs 141 ISBN

2 locus of the surface s strcton onts n order to reresent the rulngs wthn the neghborhood of ths locus. The roosed calculaton of the strcton curve s focused on the roertes of dual lne reresentaton and s qute smle. The rulngs are rojected on the screen ultmately by keeng them under a secfc xel range from the strcton curve. In ths way we avod the extra vectors needed for the comuter grahcs reresentaton of the dually defned rulngs. 2. BACKGROUND Dual unt vector reresentaton of a lne n sace. A lne n R 3 can be reresented by a dual unt vector, whle the elements of ths vector are known as Plücker coordnates. Dual vectors are based on the theory of dual numbers nvented by Clfford [Cl73]. A dual unt vector ˆL s defned as: ˆL = L+ ε L where L s the rncal vector, L the rncal moment and ε the dual unt. The reader who s unfamlar wth dual numbers and vectors s referred to the Aendx. The vector L = ( L, M, N) defnes the drecton of the gven lne and the vector L = ( L, M, N ) s the moment of the lne about the orgn. The dual numbers L+ ε L, M + ε M, N + ε N are the dual drecton cosnes of the lne whch are the comonents of the dual unt vector ˆL. The defnton of an arbtrary lne n sace s shown n Fg. 1 where r s the oston vector of a ont on the lne. Snce L s the moment of the lne around the orgn we have L = r L and obvously L L = L L = ( r L) L =. The symbol declares the dot roduct and the symbol the cross roduct. The vectors L and L are always orthogonal. Addtonally, L, M, N are chosen such that L L = L + M + N = 1. Snce L L = and L L = 1 these equatons are mosed to the sx Plücker coordnates resultng to only four ndeendent varables of degrees of freedom for the defnton of a lne n 3D sace. Fgure 1: The Plücker coordnates of a lne n the sace. Thus, the three dmensonal sace can be 4 consdered to be comosed of lnes nstead of 3 onts. A lne has four degrees of freedom whle a ont has only three but n some cases t s convenent to treat a 3D sace as a set of lnes and to reresent ts roertes n terms of these lnes [Roo78]. 3. NURBS NURBS curves In ths art of the aer we brefly outlne the roertes of NURBS. NURBS curves are ratonal and resent some nce roertes such as the ablty to reresent conc sectons and free form surfaces on a rch geometrc doman. NURBS also resent the roertes of the B-Slnes, such as the strong convex hull and the local modfcaton roerty. In fact, a NURBS curve s transformed to a B-Slne curve f all ts weghts are set equal to 1. The weght whch s the addtonal arameter for the defnton of a NURBS curve s qute advantageous. Ths extenson rovdes one more degree of freedom for shae desgn and therefore makes NURBS curves more owerful than B-Slnes. A NURBS curve s defned by the equaton: n 1 Cu ( ) = ( ) n N uwp=, = N ( u) w, (1) = n = = R, ( u) P where N ( u ) s the -th bass functon of degree, defned by: 1 f u [ u, u ) + 1 N ( u) =, otherwse WSCG26 Short Paers Proceedngs 142 ISBN

u u N ( u) = N ( u) +,, 1 u u + u u u + + 1 N + 1, 1 u u + + 1 + + ( ) where N ( u) w R u = n, ( ),, n j = N j, ( u) w j (2) are the NURBS bass functons and w the weght to control ont P.

3 u u N ( u) = N ( u) +,, 1 u u + u u u N + 1, 1 u u ( ) where N ( u) w R u = n, ( ),, n j = N j, ( u) w j (2) are the NURBS bass functons and w the weght to control ont P. For the NURBS curve we have: a) n+1 control onts P ( n), b) a knot vector U that holds m + 1 knots and u u u... u u 1 = =, c) a degree 1 2 m 1 satsfyng m = n NURBS curves are ratonal snce R ( u) are ratonal functons., m control lne, mantanng local control at the same tme. For reasons of smlcty, n the rest of ths aer, all examles are constructed utlzng weghts equal to one. Moreover, for llustraton uroses, throughout ths aer the control lnes are colored red and the strcton curve n bold blue. In Fg. 3 and 4 two dually defned NURBS ruled surfaces wth the control hull and ther strcton curve are shown defned by 5 and 4 control lnes resectvely. In Fg. 3 and 4, t can be notced that the frst and last control lnes, concde wth the rulngs of the surface. In both these examles, the boundares of the control hull are marked wth a dashed red lne. From these llustratons t s evdent that the dually defned ruled surface nherts two of the most mortant roertes that NURBS curves ossess: The local effect of weghts to the shae of the surface and the restrcton of the surface by the hull that the control lnes form. Dual defnton of NURBS ruled surface The dual defnton of the NURBS ruled surface s straght-forward. The control onts are relaced by dually defned lnes usng the corresondng dual unt vector L ˆ = L+ ε L. The equaton: n Mˆ ( u) = ( ) ˆ R u L (3), = defnes a NURBS ruled surface where L ˆ are the dual unt vectors reresentng the control lnes and R ( u ) are the NURBS bass functons gven by, Eq (2). For the dual defnton of the NURBS ruled surface we have the followng: a) n+1 control lnes reresented by the corresondng dual unt vector L ˆ,( n ), b) a knot vector U that holds m + 1 knots and = u u u... u u = 1, 1 2 m 1 m c) a degree satsfyng m = n In order to fully understand the meanng of the weght, let us consder a control lne L ˆ and a weght w. The multlcaton of the control lne by the weght w gves: w Lˆ = w L + ε ( w L ) whch s the same lne as ths oeraton has no affect to ts oston and to ts drecton due to the use of unt vectors. However, t affects the shae of the ruled surface snce we can assgn a secfc weght on one or more control lnes. In Fg. 2, the same ruled surface s rojected but wth the use of weghts w = 1.1 and w = 1.4 assgned to the control lne 2 2 reresented by ˆL 2. The result s aarent; the ruled surface has been ulled from the corresondng Fgure 2 Ruled Surface wth alternatve weghts ( w = 1.1 n the second case and w = 1.4 n the 2 2 thrd) WSCG26 Short Paers Proceedngs 143 ISBN

(3) returns a dual lne vector: ˆ M ( u) = M( u) + ε M ( u) We normalze the M ( u) vector to obtan the unt vector: M ( u) Su ( ) = M ( u) whch s the drecton of the lne.

4 Fgure 3 Control hull of 5 dually defned control lnes and strcton curve. Fgure 4 Control hull of 4 dually defned control lnes and strcton curve. Snce NURBS curves can reresent conc sectons and a crcle, t s clear that usng Eq.(3) reresentatons of concal surfaces, generalzed cylnders etc can be derved easly. For a secfc value of u, Eq.(3) returns a dual lne vector: ˆ M ( u) = M( u) + ε M ( u) We normalze the M ( u) vector to obtan the unt vector: M ( u) Su ( ) = M ( u) whch s the drecton of the lne. Then we subtract the roduct of the tch ( u) and M ( u) from M ( u) : S ( u) = M ( u) ( u) M( u) where ( u) s gven by: M ( u) M ( u) ( u) =. M ( u) M( u) The derved dual unt vector: S ˆ ( u ) = S ( u ) + ε S ( u ) satsfes the Plücker condtons and reresents a rulng of the ruled surface for a gven value of u. The man roblem wth ths knd of reresentaton s that lne segments cannot be easly reresented and therefore lne to ont transformatons are needed [Rav91]. In ths aer we ntroduce another aroach for the reresentaton of the rulngs by utlzng the strcton curve of the surface. The strcton curve lays an mortant role n the desgn and n the dfferental geometry of ruled surfaces snce t ossesses some nterestng roertes. The maxmum Gaussan curvature of the generators s located on each strcton ont of the surface. The ont on the rulng whch s closest to the successve rulng s called the strcton ont and s reference system ndeendent. The locus of the strcton onts, known as strcton curve, resents nterestng roertes and s useful for the evaluaton of ruled surfaces n general. A noncylndrcal ruled surface can be re-arameterzed usng a strcton-drector curve reresentaton takng nto account that the strcton curve does not deend on the choce of the base curve. Furthermore, the strcton curve s always n contact wth all the rulngs of the surface and all the sngulartes of the surface are located on ths curve. 4. DETERMINATION OF THE STRICTION CURVE The locus of the strcton onts s called strcton curve and has attracted the nterest of researchers due to the role that lays n dfferental geometry of ruled surfaces. Pottman [Pot96] calculated the strcton curve va the De Casteljau algorthm and lne geometry. The authors of [Sch98], resented a technque for the calculaton of the strcton curve of a ruled surface utlzng ts classcal defnton and dual numbers. The defnton was derved utlzng the arc length of the ndcatrx curve as the arameter and was deended on an arbtrary drectrx of the ruled surface. In ths aer, the strcton curve s calculated usng elements that are ncluded n the roosed lne reresentaton of the ruled surface.the strcton onts are determned as follows: Let Nu ˆ (, u) be the common erendcular between two successve rulngs Su ˆ( ) and Su ˆ( + u), u R,where u Rs very small as t s shown n Fg. 5. WSCG26 Short Paers Proceedngs 144 ISBN

5 Let and ˆ( ) ( ) Su = Su + ε S ( u ) ˆ( ) ( ) Su+ u = Su+ u + ε S ( u+ u ). Then the common erendcular s gven by: Nu ˆ (, u ) = Su ( ) Su ( + u ) + (4) ε ( Su ( ) S( u+ u) + S( u) Su ( + u)) n dual vector form. The reader who s not famlar wth the dual number oeratons s referred to the aendx. Utlzng the straght forward defnton of the dual ont [Aza1] by two dual unt vectors we get for the strcton ont S ( u ): S ( u) = S( u) S( u + u) + (5) ( Nu (, u) N( u, u) Su ( )) Su ( ) where Nu ˆ (, u) = Su ( ) Su ( + u) and N ( u, u) = ( S( u) S ( u+ u) + S ( u) S( u+ u)) Two dual unt vectors ntersect when the dual comonent of ther dual scalar roduct vanshes. Ths roerty s used for the above defnton of the strcton ont. The resultng constructon has enough freedom to hold three ont vectors smultaneously. Thereby, extra nformaton can be stored nto the exresson of the strcton ont such as the tangent vector, the curvature vector etc. In Fg. 5 the strcton ont S ( u ) s llustrated for two successve rulngs on a ruled surface along wth the common erendcular Nu ˆ (, u). of drawng an nfnte lne reresented by a dual vector has been addressed va several ways. Ravan and Wang [Rav91] ntroduced a method that utlzed the centre ont of a lne to defne a lne segment. They used two addtonal arameters on the lne coordnates to do so. However, the locaton of the centre ont cannot be controlled durng the nterolaton. Dng [Dn2] used the dual De Boor algorthm for the on screen reresentaton of screws. Srott and Ravan [Sr97] roosed a method that used a ont on the frst control lne as a reference n order to defne a lne through that ont. Then they derve a surface constructon nterolatng the control screws wth the reference lne. These methods manly utlze a lne-segment scheme or aroxmaton algorthms n order to ass from the dual number world to the screen. The technque roosed n ths aer, s not based on an aroxmaton algorthm such as De Boor or De Casteljau but on the exact calculaton of the strcton onts. The strcton onts always le on the common erendcular among successve rulngs and ther locus forms a curve of mnmal length that meets all the rulngs. Usng ths roerty, we draw each one of the rulngs of the surface by lottng the straght lne segments n the neghborhood of the strcton curve. More recsely, we roject onto the comuter screen a secfc number of xels from the strcton curve along a rulng. Thus, the need for the two ont defnton of a lne segment s elmnated snce the only necessary arameter s the dstance from the strcton curve. In the followng llustrated examles, we demonstrate the dfference between two rojectons of the same ruled surface and of alternatve strcton curve neghbourhoods. In Fg. 6 and 7 a dual NURBS ruled surface s shown along wth the strcton curve. It s obvous that n Fg. 7 only the xels n one sde of the strcton curve are rojected. Fnally, two comlete shaded models of two ruled surfaces va the dual NURBS defnton are reresented n Fg. 8 and 9. Fgure 5 The Strcton Pont S ( u ). 5. RULED SURFACE DRAWING Our method s based on the calculaton of the strcton curve for the renderng of dual NURBS ruled surfaces. In the current lterature, the roblem Fgure 6 Dual NURBS ruled surface (5 xels for the frst rulng) and strcton curve S ( u ). WSCG26 Short Paers Proceedngs 145 ISBN

The calculaton s based uon the dual defnton of the surface and s erformed utlzng the rchness of the dual vector reresentaton.

6 Fgure 7 Dual NURBS ruled surface (25 xels for the frst rulng) and strcton curve S ( u ). unt vectors, the NURBS bass functons and the strcton curve. The roblem of rojectng straght lnes reresented by dual unt vectors s resolved by the utlzaton of the strcton curve. We rovde a way for the calculaton of the surface s strcton onts. The calculaton s based uon the dual defnton of the surface and s erformed utlzng the rchness of the dual vector reresentaton. Moreover, the defnton of the strcton onts s used to address another roblem of the dual world : the on screen rojecton of dual unt vectors reresentng lnes. Our reresentaton, can fnd alcaton n the areas of CAD and CAGD. In the last decade, lne geometry and dual vector calculus has attracted the nterest of many researchers and has been studed from the desgn ont of vew. However, ont-based reresentatons are stll domnatng CAD and CAGD. Ths s due to the fact that modern comuter grahcs systems can only roduce 3D to 2D rojectons ont wse. Future research should nclude the utlzaton of lne geometry by modern APIs. Moreover, the study of the transformatons of these surfaces resents great nterest. Dual quaternons and dual matrces that have been roved to be smooth transformaton oerators of dual vectors, are two natural canddates for future nvestgaton. 7. REFERENCES Fgure 8 Dual NURBS ruled surface and strcton curve S ( u ). Fgure 9 Dual NURBS ruled surface and control lnes 6. CONCLUSIONS In ths aer, a new method for the reresentaton of ruled surfaces usng lne geometry has been roosed. We have develoed an alternatve way for the defnton of ruled surfaces utlzng dual [Aza1] Azarads P, Asragathos A.N. Comuter grahcs reresentaton and transformaton of geometrc enttes usng dual unt vectors and lne transformatons, Comuters & Grahcs 25(2): , 21. [Che99] Chen, H. Y., and H. Pottmann Aroxmaton by ruled surfaces, J. Comut. Al. Math. 12, , [Cl73] Clford WK. Prelmnary sketch of bquaternons, Proceedngs of the London Mathematcal Socety, 1s.,4(64/65):381-95, [Dn2] Dng, R. Drawng Ruled Surfaces Usng The Dual De Boor Algorthm. In Proceedngs Of Comutng: The Australasan Theory Symosum Cats 22. Jan. Melbourne, Australa, Elsever Scence B.V, 22. [Far97] Farn G Curves and surfaces for comuter aded geometrc desgn (4rth ed.): a ractcal gude, Academc Press Professonal, Inc., San Dego, CA, 1997 [Gra97] Gray A. Modern Dfferental Geometry of Curves and Surfaces (2/E), CRC Press, [Pot95] Pottmann H and Farn G. Develoable ratonal Bezer and B-slne surfaces. Comut. Aded Geom. Desgn, 12: , WSCG26 Short Paers Proceedngs 146 ISBN

7 [Pot96] Pottmann H, Lü W and Ravan B. Ratonal Ruled Surfaces and Ther Offsets, Grahcal Models and Image Processng, Vol. 58, No. 6, Nov., , [Rav91] Ravan B. and Wang J. Comuter aded geometrc desgn of lne constructs, ASME J. Mech. Desgn 113, , [Roo78] Rooney J. A comarson of reresentatons of general satal screw dslacement, Envronment and Plannng, B,5, 45-88, [Sch98] Schaaf, J. A. and Ravan, B.Geometrc contnuty of ruled surfaces. Comut. Aded Geom. Des. 15, 3, , [Sr97] Srott, K. and Ravan, B. (1997) Ruled surfaces, Le grous, and mesh generaton ASME Engneerng Techcal Conferences, Se , Sacramento, Calforna,USA, [Xa] Xa, J., and Ge, Q. J. Knematc Aroxmaton of Ruled Surfaces Usng NURBS motons of a Cylndrcal Cutter. Proc. 2 ASME Desgn Automaton Conference, Baltmore, MD, Paer No. DETC2/DAC- 1428, 2. [Yan96] Yang M, Lee E. NC verfcaton for wre- EDM usng an R-ma. Comuter Aded Desgn;28:733 74, APPENDIX Dual numbers can be defned as: aˆ = a+ ε a, a, a R where ε s the dual unt wth the roertes of: 2 ε, ε =, 1 ε = ε 1 = ε, ε = The multlcaton and addton oeratons of two dual numbers aˆ = a + ε a, aˆ = a + ε a are defned as: Addton: aˆ + aˆ = ( a + ε a ) + ( a + ε a ) = ( a + a ) + ε ( a + a ) Multlcaton: aˆ aˆ = ( a + ε a ) ( a + ε a ) = a a + ε ( a a + a a ) External roduct: aˆ aˆ = a a + ε ( a a + a a ) WSCG26 Short Paers Proceedngs 147 ISBN

8 WSCG26 Short Paers Proceedngs 148 ISBN

TIME-EFFICIENT NURBS CURVE EVALUATION ALGORITHMS

TIME-EFFICIENT NURBS CURVE EVALUATION ALGORITHMS Kestuts Jankauskas Kaunas Unversty of Technology, Deartment of Multmeda Engneerng, Studentu st. 5, LT-5368 Kaunas, Lthuana, kestuts.jankauskas@ktu.lt Abstract: