A Pecewse Ratonal Quntc Hermte Interpolant for Use n CAGD Gulo Cascola and Luca Roman Abstract. In ths paper we descrbe and analyze a new class of C pecewse ratonal quntc Hermte nterpolants for use n CAGD whch are capable of eactly representng any conc arc of arbtrary length by usng only one segment. They can also provde a varety of local/global shape parameters for ntutvely sculptng free form curves wthout affectng the C contnuty nherent n the orgnal constructon.. Introducton Durng the last two decades there has been consderable nterest n developng a C soluton to the nterpolaton problem of zeroth, frst and second order dervatves at a gven selecton of ponts. Ths s due to the fact that such geometrc propertes turn out to be of prmary concern n geometrc modellng or computer aded desgn applcatons, e.g., n the smoothng of curves. However, the work done over the years has resulted n many C nterpolaton methods [, 3,,, 8, 9, ] that cannot satsfy at the same tme all the propertes we have found vtal or smply desrable to nclude n a user-orented model, desgned to be ntegrated n a conventonal NURBS-based CAD system. For ths reason, we are gong to propose a new class of C pecewse ratonal quntc Hermte nterpolants that possesses an eplct constructon,.e., that does not nvolve the soluton of any equatons; provdes local and ntutve sculptng parameters for fast nteractve manpulaton of the shape of a curve; s capable of representng both smooth shapes and sharp shapes, and more precsely of mng smooth zones and sharp ones n the same curve, so that the transton between them always preserves the orgnal C contnuty. Mathematcal Methods for Curves and Surfaces: Tromsø M. Dæhlen, K. Mørken, and L. L. Schumaker (eds.), pp.. Copyrght Oc by Nashboro Press, Brentwood, TN. ISBN --9788--X All rghts of reproducton n any form reserved.
G. Cascola and L. Roman The paper s structured as follows. In Secton we develop and construct ths new class; n Secton 3 we show that our ratonal quntc Hermte nterpolants are capable to eactly represent any conc arc of arbtrary length by usng only one segment wth postve weghts and fnte control ponts. Fnally, n Secton we eplot our model for ntutvely sculptng C -contnuous free form curves by local deformatons whch do not compromse the orgnal C contnuty. Although our proposal s not lmted to parametrc sets of ponts, but works effcently both n the vectoral and n the scalar cases, we confne our attenton here to the parametrc formulaton only.. The Class of Pecewse Ratonal Quntc Hermte Interpolants Our objectve s to construct over an arbtrary non-trval nterval [t, t + ] R, a ratonal curve segment c (t) : [t, t + ] R ν, ν >, that satsfes the followng nterpolaton condtons at the endponts: c (t ) = F, c (t ) = D (), c (t ) = D (), c (t +) = D () +, c (t +) = D () +, c (t + ) = F +. () To fulfll our am, we wrte c (t) as the ratonal quntc Bézer curve R j,(t) = c (t) = P jrj,(t), () j= µ j B j, (t) ( k= µ k B k, (t) wth Bj,(t) (t+ t) = j) j (t t ) j (t + t ), and the control ponts P j Rν, j =,..., have to be determned. Two of them turn out to be quckly defned: P = F and P = F +. For the remanng four, we recall the frst and second endpont dervatve formulae for ratonal quntc Bézer curves: c (t ) = µ (P P ) h µ, c (t ) = µ (P P ) h µ ( ) µ µ µ (P P ) h µ, (3) ( c (t +) = µ 3 (P 3 P ) µ h µ µ ) µ (P P ) h µ, c (t +) = µ (P P ) h µ h = t + t and µ j, j =,..., are the postve weghts of the ratonal representaton (). Thus, by solvng for P, P, P 3, P, we get
A Pecewse Ratonal Quntc Hermte Interpolant 3 P = F + h µ D () µ, P = F + (µ µ )h µ P 3 = F + (µ µ )h D () µ 3 + + h µ D () µ 3 D () + h µ µ D (), +, P = F + h µ µ D () +. If a sequence of nterpolatng data F, D (), D (), =,..., N s gven, such a constructon allows to solve the nterpolaton problem by a pecewse ratonal quntc made of peces c (t) that jon together wth the so-called C ratonal contnuty (see [6]). In order to guarantee that adjacent curve segments jon eactly parametrcally C (not just ratonally C ), we set µ = µ and, wthout loss of generalty, we assume them to be. Thus, Defnton. Gven the nterpolatng ponts F R ν, =,..., N and the frst and second order dervatves D (), D () R ν, =,..., N defned at the knots t, =,..., N (wth t < t <... < t N ), a pecewse ratonal quntc Hermte nterpolant c(t) C[t,t N ] s defned for t [t, t + ], =,..., N by the epresson c (t) = P jrj,(t), () j= P = F, P = F + hd() µ, P = F + (µ )hd() µ + h D(), () µ P 3 = F + (µ )hd() + µ 3 + h D() + µ 3, P = F + hd() +, P = F +, and {R j, (t)} j=,..., are the ratonal quntc Bézer polynomals n (3) defned by the postve weghts µ j, j =,..., wth µ µ =. µ 3. Eact Representaton of Conc Arcs of Arbtrary Length Conc arcs play a fundamental role n CAD/CAM applcatons. In ths secton we wll show that a ratonal quntc Hermte segment can be used for precsely representng any conc arc of arbtrary length. Thus, the pecewse nterpolatory model presented n Secton allows us to ncorporate the class of the so-called conc secton subsplnes (see [7], page 79), whch contans all those smooth curves made of peces of conc sectons that pass through gven ponts and assume prescrbed dervatves. Such a model s of great use n engneerng applcatons, e.g. n mllng processes and n the constructon of dsk cams.
G. Cascola and L. Roman F = F = " f f y f y Parabolc Arc y = a a δ ], + [ Hyperbolc Arc y = a b b a < left branch > rght branch δ ], + [ c = cosh(δ) s = snh(δ) #» δ» ac aδ bs f y f y Ellptc Arc + y = a b a, b δ ], π] ρ = ( + tg δ ) ω = ρ 8ρ 3 + 8ρ ρ + a(ρ 8ρ 3 +ρ ) ω bρ( ρ +3ρ ) ω " # " # " # " f f f f # " # f»» D () δ as = f y aδ bsc " # " f D () = f y f y 3 D () = f y 3 D () = f y f y 3 6aρ( ρ +ρ 3 ρ +ρ ) ω b( 8ρ 6 +ρ ρ +ρ 6ρ+) ω # " f # " # f y y» " # as (c ) 8aδ bs( + c c ) 3 f y 3 f y 6a( ρ +ρ 3 ρ ρ+) ω 8b( 8ρ +ρ +8ρ 3 3ρ +ρ ) ω 3 f y 3 3 µ = µ µ = µ 3 3+c +3c ρ +ρ 3 6ρ +ρ+ ω ρ 8ρ 3 +ρ + ω Tab.. Data for eact representaton of arbtrary conc arcs va the ratonal quntc Hermte nterpolatory model. Theorem. The ratonal quntc Hermte segment c (t), t [, ] nterpolatng the data n Tab. wth prescrbed postve weghts {µ j } j=,...,, allows us to eactly represent any conc arc of arbtrary length. Proof: Wrtng the ratonal quntc Hermte nterpolatory model () by usng the data gven n Tab., we obtan an epresson of c (t) whose components (t) and y(t) turn out to satsfy the followng canoncal equatons
3 3 8 6 6 3 3 A Pecewse Ratonal Quntc Hermte Interpolant 8 6 6 8.8...6.8...6 8 6 6 8 3 3 6 Fg.. Eamples of arbtrary conc arcs eactly represented va a ratonal quntc Hermte segment: parabolc arc, hyperbolc arc, ellptc arc/full ellpse, crcular arc/full crcle. n case of parabolc, hyperbolc, or ellptc arcs, respectvely: y(t) = a (t) (t), a y (t) (t) b =, a + y (t) b =. Remark. The postve free parameter δ allows us to eactly represent conc arcs of arbtrary length. Notce that n the case of ellptc arcs of ampltude δ, by choosng the half-angle δ equal to π, we can precsely reproduce full ellpses of rad a, b R that pass through the ponts F F = ( a, ) and assume the frst and second order dervatves D () D () = (, b), D () = (6a, 8b), D () = (6a, 8b) (see Fg. ). As a specal case we can thus represent full crcles and crcular arcs by specfyng the radus r a = b and the half-angle δ ], π] (see Fg. ). We remnd the reader that, as was proven n [], the ratonal quntc Bézer form s the mnmum degree representaton that allows us to obtan a full crcle/ellpse usng only postve weghts and fnte control ponts.. Sculptng of Free Form Curves by Local/global Deformatons We now rearrange equaton () n the equvalent cardnal form and c (t) = I jφ j,(t), j= I = F, I = D (), I = D (), I 3 = D () +, I = D () +, I = F + φ,(t) = R [,(t) + R,(t) + R,(t) ] φ,(t) = h R,(t) + µ R,(t) φ,(t) = φ 3,(t) = µ [ µ h µ µ R,(t) h µ 3 R 3,(t) φ,(t) = h R3,(t) + R µ,(t) φ,(t) = R3,(t) + R,(t) + R,(t) µ 3 ]
6 G. Cascola and L. Roman s the ratonal formulaton of the well-known quntc Hermte polynomals. Snce φ,(t)+φ,(t), t follows that whenever φ j,, j =,..., vansh, the curve segment c (t) concdes wth the lne through F, F +. As h s never zero, ths s easly verfed whenever µ, µ approach nfnty and µ, µ 3 approach nfnty faster than µ, µ respectvely. Ths last condton s trvally satsfed by settng µ = α(µ ) β and µ 3 = α(µ ) β, wth α, β >. Here we choose µ = (µ ), µ 3 = (µ ) and reformulate Defnton n the followng way. Defnton. Gven the nterpolatng ponts F R ν, =,..., N and the frst and second order dervatves D (), D () R ν, =,..., N assgned at the knots t, =,..., N (wth t < t <... < t N ), we defne over the nterval [t, t + ] the pecewse ratonal quntc Hermte nterpolant c(t) C[t,t N ] by the epresson c (t) c (t, v, w ) = C = F, C = F + hd() v C 3 = F + (w )hd() + w C jrj,(t) (6) j=, C = F + (v )hd() v + h D() + w + h D(), (7) v, C = F + hd() + w, C = F + and {R j, (t)} j=,..., are the ratonal quntc Bézer polynomals n (3) defned by the postve weghts µ = µ :=, µ := v, µ := v, µ 3 := w, µ := w. Remark. In ths way, by choosng w = v =,..., N, the pecewse ratonal quntc Hermte nterpolant c(t), represented as NURBS on a sngle knot-partton wth nternal -fold knots, becomes parametrcally C -contnuous at t = t and thus can be represented on a new knot-partton wth only nternal 3-fold knots. Although the shape of a NURBS curve can be modfed by the manpulaton of ts weghts, the possblty of controllng the shape through the classcal change of the weght vector may sometmes be confusng, as the modfcatons of two adjacent weghts are mutually cancelled, and not very effectve, snce the curve s forced to stay n the conve hull of ts control ponts. For ths reason we have proposed an nterpolatory ratonal quntc splne nvolvng two sculptng parameters per nterval that, although correspondng to the weghts of the ratonal representaton, nfluence the control ponts defnton and thus have large scale effects on the shape of
A Pecewse Ratonal Quntc Hermte Interpolant 7 the curve. In partcular, such free parameters provde a varety of local and global shape controls, lke pont and nterval tenson effects. In order to analyze such effects on the shape of the curve, we consder here the lmtng behavor of the ratonal pecewse nterpolant whenever each shape parameter approaches nfnty, we assume that the other parameters are held constant wth respect to each lmt process (note that ths s possble only because every weght modfcaton do not nfluence ts neghborng weghts). Thus the followng shape deformatons mmedately follow by nspecton of equaton (6). Theorem. [Pont Tenson] Let v and w approach nfnty. Snce lm c (t) = F = lm c (t), (8) v w we have a pont tenson parameter controllng the curve tenson from both rght and left of the pont F, the pecewse ratonal nterpolant c(t) wll appear to have a corner. Proof: To prove the frst (second) equalty n (8) we nsert the equaton of c (t) (c (t)), dvde both ts numerator and denomnator by v (w ) and compute lm v(w ). Whle two shape parameters per nterval are necessary for provdng pont tenson effects, we are gong to show now that only one shape parameter per nterval s enough when nterval tenson s requred. To ths end we assume the parameters v, w satsfy the relaton v = λ w, wth λ ], + [, and we show that the parameter w plays the role of nterval tenson parameter for the sngle pece c (t, λ, w ). Lemma. Let v = λ w wth λ ], + [. If w approaches nfnty, then the ratonal quntc Hermte nterpolant c (t, λ, w ) converges unformly to the ratonal lnear nterpolant of F, F + wth weghts λ, : lm c (t, λ, w ) l (t, λ ) =, w wth l (t, λ ) = λ ( θ )F + θ F + λ ( θ ) + θ, t [t, t + ], θ = t t h [, ]. Proof: By smple computatons on c (t, λ, w ) t follows that the pecewse ratonal quntc Hermte nterpolant c (t, λ, w ), t [t, t + ] defned by (6) can be decomposed n the followng way: c (t, λ, w ) = l (t, λ ) + e (t, λ, w ),
8 G. Cascola and L. Roman h w (λ w )h D () w f(t) g(t) h(t) (F F + ) w D () +h D() (w )h D () + +h D() + w w (F + F ) h w w λ w (F F + ) + h w (λ w )h D () w D () +h D() (w )h D () + +h D() + w D () + h w D () + w λ w λ w (F + F ) w Tab.. Bézer coeffcents of the quntc polynomals f(t), g(t), h(t). and l (t, λ ) = λ ( θ )F + θ F + λ ( θ ) + θ, e (t, λ, w ) = f(t)λ B,(θ ) + g(t)b 3,(θ ) h(t)(λ B, (θ ) + B 3, (θ )), wth f(t), g(t), h(t) beng quntc Bézer polynomals defned by the coeffcents n Tab.. Thus, snce lm w e (t, λ, w ) =, t trvally follows that lm w c (t, λ, w ) l (t, λ ) =. Theorem 3. [Interval Tenson] Let v = λ w, wth λ ], + [. If w approaches nfnty, then the ratonal quntc Hermte nterpolant c (t, λ, w ) s pulled towards the lne segment through F and F +. Proof: By standard NURBS theory, t follows that when ν >, whatever we choose λ ], + [, the ratonal lnear nterpolant l (t, λ ) concdes wth the lne segment through F and F +. As a consequence, when all the nterval tenson parameters approach nfnty, the pecewse ratonal lnear nterpolant c(t) s pulled towards the polylne through the ponts F, but practcally s never a pecewse lnear nterpolant because the parameterzaton s C here, as t s only C for lnear nterpolants. Corollary. [Global Tenson] Let l(t), t [t, t N ] denote the pecewse ratonal lnear nterpolant defned t [t, t + ] by l (t, λ ). Suppose w w and v = λ w, wth λ ], + [, =,, N. Then the pecewse ratonal quntc Hermte nterpolant c(t) C [t, t N ] converges unformly to l(t) as w approaches nfnty,.e. lm w c(t) l(t) =, t [t, t N ] and thus c(t) s pulled towards the polylne through the ponts F.
A Pecewse Ratonal Quntc Hermte Interpolant 9 Proof: The result follows by applyng Theorem 3 over each nterval [t, t + ]. Whle we have just shown that for suffcently bg values of the nterval tenson parameters we are able to ntroduce straght lne segments nto a gven pecewse curve, we are gong to show now that for suffcently small values of the parameters we can pull out a bump or push n an ndentaton n a curve segment (see Fgs., 3). Theorem. [Interval Warpng] Progressvely decreasng the shape parameters v and w towards zero, the ratonal quntc Hermte nterpolant c (t) produces a looser and looser curve segment. Proof: If we look at the behavor of the control ponts C j, j =,, 3,, and hence of the Bernsten-Bézer conve hull, when the shape parameters v and w progressvely decrease towards zero, t s a smple matter to see that we wll get a looser and looser curve. Corollary. [Global Warpng] Let v and w progressvely decrease towards zero =,, N. Then the pecewse ratonal quntc Hermte nterpolant c(t) C [t, t N ] progressvely becomes looser and looser over each nterval [t, t + ]. Proof: We apply Theorem over each nterval [t, t + ].. Conclusons and Future Work In ths paper we have presented a new class of pecewse ratonal quntc Hermte nterpolants whch provde a varety of local and global shape parameters for ntutvely sculptng free-form curves, wthout affectng the C contnuty nherent n the orgnal constructon. In addton, the proposed model s capable of producng ether a sharp C -nterpolaton or a smooth C -nterpolaton, that s, although t always produces C -nterpolants, t enables the creaton of a varety of shape effects lke angular ponts, sharp edges, bumps and ndentatons (see Fgs., 3). The ablty to ncorporate eact conc arcs of arbtrary length and to m smooth curve segments, sharp corners and flat peces n an unrestrcted way, makes the pecewse ratonal quntc Hermte nterpolant model a canddate of choce for many applcatons. Our net step wll be to show that usng a non-lnear optmzaton procedure for determnng the sculptng parameters, we can use the proposed class also for appromatng any trgonometrc curve wth the desred order of precson. Acknowledgments. Ths work has been supported by FIRB.
G. Cascola and L. Roman....8.8.8.6.6.6.........6.8..........6.8..........6.8........8.8.8.6.6.6..............6.8........6.8........6.8... Fg.. Eamples of local/global deformatons on an open 3D curve: no tenson (v = w =, =,..., ), nterval tenson (v = w = ), nterval warpng (v 3 = w 3 =.), pont tenson (v 3 = w = ), global tenson (v = w =, =,..., ), global warpng (v = w =., =,..., ).....8.8.8.6.6.6..................6.8.......6.8.......6.8.....8.8.8.6.6.6..................6.8.......6.8.......6.8. Fg. 3. Eamples of local/global deformatons on a closed 3D curve: no tenson (v = w =, =,..., 8), nterval tenson (v 3 = w 3 = ), nterval warpng (v 3 = w 3 =.), pont tenson (v 7 = w 6 = 6), global tenson (v = w =, =,..., 8), global warpng (v = w =., =,..., 8). References. Cascola G., Roman L., Ratonal Interpolants wth Tenson Parameters, n: Lyche T., Mazure M.-L. and Schumaker L.L. (Eds.), Curve and Surface Desgn: Sant-Malo, Nashboro Press (3), -. Chou J.J., Hgher order Bézer crcles, Computer Aded Desgn, 7() (99), 33-39
A Pecewse Ratonal Quntc Hermte Interpolant 3. Costantn P., Cravero I., Mann C., Constraned Interpolaton by Frenet Frame Contnuous Quntcs, n: Lyche T., Mazure M.-L. and Schumaker L.L. (Eds.), Curve and Surface Desgn: Sant-Malo, Nashboro Press (3), 7-8. Gregory J.A., Sarfraz M., A ratonal Cubc Splne wth Tenson, Computer Aded Geometrc Desgn 7 (99), -3. Gregory J.A., Sarfraz M., Yuen P.K., Interactve Curve Desgn usng C Ratonal Splnes, Computers & Graphcs 8()(99), 3-9 6. Hohmeyer M.E., Barsky B.A., Ratonal Contnuty: Parametrc, Geometrc, and Frenet Frame Contnuty of Ratonal Curves, ACM Transactons on Graphcs, 8() (989), 33-39 7. Hoschek J., Lasser D., Fundamentals of Computer Aded Geometrc Desgn, A K Peters (993) 8. Peters J., Local generalzed Hermte nterpolaton by quartc C space curves, ACM Transactons on Graphcs 8(3) (989), 3-9. Sarfraz M., Interpolatory Ratonal Cubc Splne wth Based, Pont and Interval Tenson, Computers & Graphcs 6() (99), 7-3. Sarfraz M., Balah M., A Curve Desgn Method wth Shape Control, Lecture notes n Computer Scence, Sprnger 669/3, 67-679 Gulo Cascola and Luca Roman Dept. of Mathematcs - Unversty of Bologna P.zza Porta San Donato, 7 Bologna, Italy cascola@dm.unbo.t roman@dm.unbo.t