Tom Duff Computer Division Lucasfilm Ltd. Technical Memo No Dec 1983

Transcription

1 Famles of Local Matrx Splnes Tom Duff Computer Dvson Lucasflm Ltd Techncal Memo No 04 Dec 983 Presented as tutoral notes (by Alvy Ray Smth) at the 983 SIGGRAPH, July 983 Ths document was reentered n Mcrosoft Word on Dec 999 Spellng and punctuaton are generally preserved, but trvally mnor spellng errors are corrected Otherwse addtons or changes made to the orgnal are noted nsde square brackets or n footnotes Abstract Jm Clark's Cardnal splnes are a famly of local nterpolatng splnes wth an adjustable tenson parameter The famly may be descrbed by a matrx whch yelds splne coeffcents as lnear functons of knot values We characterze the Cardnal splne matrx n a way whch suggests a method of addng tenson to B- splnes, and show that ths tenson corresponds to the β parameter of Bran Barsky's β -splnes The smlarty between the tenson parameters of these two splnes suggests lookng for an nterpolatng splne famly whch ncorporates a bas parameter analogous to Barsky's β We demonstrate two such famles wth dfferent contnuty propertes (one s G, the other C ) Fnally, we develop a 0 fve-parameter characterzaton of all C, G translaton nvarant cubc matrx splnes and ndcate that all the famles we have developed are sub-famles of t CR Categores and Subject Descrptors: G [Numercal Analyss] Splne and pecewse polynomal nterpolaton, G [Numercal Analyss] Splne and pecewse polynomal approxmaton, I35 [Computer Graphcs] Curve, surface, sold and object representatons General Terms: Algorthms Addtonal Keywords and Phrases: β -splne, Cardnal splne, matrx splne, local splne, computer aded geometrc desgn Introducton A local splne s one for whch changng the value of a sngle knot affects only a bounded number of splne segments n the knot's vcnty Ths s a partcularly useful property for geometrc desgn systems and computer anmaton systems, snce t means that a desgner can adjust the appearance of a partcular part of hs desgn wthout fear of global complcatons Many popular local splnes may be characterzed by a system of lnear equatons relatng the splne's knots to ts polynomal coeffcents These lnear equatons may be wrtten as a rectangular matrx

2 Dec 83 Matrx Splnes The Matrx Splne Notaton Gven a sequence of control ponts or knots, K 0 < m and an n + by s matrx M we can defne the matrx splne S M of degree n and support s by n s n j M jk + k j= 0 k= 0 S ( t + ) = t M K 0 t <,0 < m s+ We wll usually omt the subscrpt M when the matrx s clear from context, and we wll use the notaton S () t = S ( + t) In computer graphcs we are partcularly concerned wth the case n = 3, s = 4, whch makes M one of the 4 by 4 matrces so famlar to computer graphcs hardware and software In ths case the above equaton may be rewrtten as K 3 K + SM ( t + ) = t t t M 0 t <,0 < m 3 K + K + 3 Most (but not all, see for example [Knuth]) of the well-known local splnes may be expressed n ths form For example, the unform cubc B-splne [Resenfeld] s represented by M Bsplne = Wthout loss of confuson [sc], we wll dentfy matrces wth ther splnes Thus, when we speak of the dervatve of M, we wll mean the dervatve of the splne whose matrx s M The frst use of the matrx splne notaton of whch I am aware s [Catmull] Famles of Splnes Bran Barsky ntroduced a two-parameter famly of matrx splnes called the β -splnes (see [Barsky]) The β -splnes are contnuous and have contnuous unt tangent and curvature vectors In partcular, the β -splnes satsfy S() = S+ (0) β S () = S (0) + βs () + βs () = S + (0) Barsky calls the last two of these condtons G and G They are geometrc generalzatons of the parametrc contnuty condtons C and C The β -splne ma- trx s Here () the rght S s the dervatve from the left and (0) S + s the dervatve at the same pont from Tech Memo 04 Pxar

3 Dec 83 Matrx Splnes β ( β+ β + β + β) ( β+ β+ β+ ) 3 3 6β 3( β + β + β) 3β + 6β 0 M Beta = β + β+ 4β+ 4 β+ 6β 6( β β) 6β 0 3 β β+ 4β + 4β 0 β bases or slews the curve to the left or rght (parametrcally) of the unbased B- splne As β ncreases, the curve becomes more tense and more closely approxmates ts knots Another useful famly of splnes s the Cardnal splnes [Clark] The Cardnal splnes are nterpolatng splnes wth frst dervatve contnuty They are defned by the constrants S (0) = K + S() = K+ S (0) = ck ( K ) + S () = ck ( + 3 K+ ) Solvng these equatons for the splne coeffcents yelds the matrx c c c c c c 3 3 c c c 0 c Usng the notaton lerp( M, N, α) = ( α) M + αn (lerp s an abbrevaton for lnear nterpolant), we can rewrte ths as lerp( MEase, Mc=, c) where M Ease =, M c = = () Thus, each of the Cardnal splnes s a weghted average of the c = splne, whch has S (0) = K+ K and S () = K+ 3 K+, and the Ease splne, whch has S (0) = S () = 0 3 The Cardnal splnes may be thought of as varable-tenson local nterpolatng splnes whch take ther corners more sharply as c 0 An nterestng member of the Cardnal splne famly has c = 5 Ths s the Catmull-Rom cubc splne, descrbed n [Catmull-Rom] and later n [Brewer-Anderson] Equaton () mmedately suggests a method of adjustng the tenson of B- splnes usng Note that lerpng two splne matrces s equvalent to lerpng the coeffcents of the polynomals they generate, whch n turn s equvalent to lerpng the ponts on the splne curves and ther dervatves 3 The Ease splne curve s the polygon that nterpolates ts knots Its velocty decelerates to zero as t passes through each knot Thus, ts moton as a functon of t `eases' n and out of each knot The term ease s taken from the lexcon of conventonal (not computer) anmators Tech Memo 04 Pxar

4 Dec 83 Matrx Splnes 4 M M M () TenseBsplne = lerp( Ease, Bsplne, ) = My exctement at dscoverng ths famly of splnes was tempered by Alvy Ray Smth's revelaton (about an hour later) [Smth] that t s the subfamly of the β - splnes wth β = and β = ( )/ The man advantage to my formulaton s that vares over a ncer range than β In partcular 0 β < corresponds to > 0 The case = 0, whch yelds the Ease splne (the only nterpolatng splne of the famly) s unattanable n the β -splnes The substtuton of for β can be generalzed to other values of β If we 3 substtute β = ( β + β + β+ )( )/ and β = /( + ) nto the β - splne matrx, we get: ( + ) ( + ) ( ) lerp( M, Ease, 3 ) 6 3 3( + + ) 3 ( + ) ) ( + ) 0 Thus, we see that for any fxed β (equvalently ), β (eqv ) has the effect of pckng a matrx (and therefore a curve) that s some weghted average of the Ease splne and the β = 0 (eqv = ) splne, n the same manner that the Cardnal splnes are derved from the c = splne Interpolatng Splnes wth Tenson Control The analogy between the Cardnal splnes and the β parameter of the β - splnes suggests lookng for a generalzaton of the Cardnal splnes that ncorporates a parameter analogous to β There at least two ways of dong ths β expresses the rato of the lengths of a β -splne's dervatve vectors as we approach a knot from ether sde Therefore let us consder the two nterpolatng splnes whose frst dervatves match the c = splne's at one end, and are zero at the other end These have the matrces M SkewLeft = and M SkewRght = M SkewLeft has zero dervatve at the left end, whle M SkewRghtt has zero dervatve at the rght end Thus, Tech Memo 04 Pxar

5 Dec 83 Matrx Splnes 5 lerp( M, M, τ ) (3) SkewLeft SkewRght s a G famly of nterpolatng splnes When τ = 5, formula (3) yelds M CatmullRom Lerpng ths formula wth the Ease matrx we get ττ ( τ ) τ + ττ ( τ) τ ττ ( τ) τ 3 3 ττ ( τ ) τ lerp( M Ease,lerp( M SkewLeft, M SkewRght, τ), τ) = (4) ττ 0 ττ Ths s a two parameter famly of G nterpolatng splnes wth varable tenson and bas, whch we wll call the τ -splnes Instead of applyng the β -splne bas concept drectly to the Cardnal splnes, we could note that the Catmull-Rom cubc splne s defned by S (0) = K + S() = K+ S (0) = lerp( K K, K K, δ ) S () = lerp( K+ K+, K+ 3 K+, δ) where δ = 5 When δ = 0 and δ = the matrces satsfyng (5) are M Left = and 0 0 M Rght = respectvely Varyng δ from 0 to causes the splne's dervatve at K to vary from K + K to K+ K+, causng the curve to slew to the left or the rght The bas that δ causes s C rather than G Lerpng the δ famly of splnes wth the Ease splne produces a two parameter famly of C nterpolatng splnes wth varable tenson and bas The matrx for ths famly, whch we wll call the δ - splnes, s ( δ ) δ δδ ( δ) δ δδ ( δ) δ (3δ ) δ 3 3 δ δδ lerp( M Ease,lerp( M Left, M Rght, δ), δ) = (6) ( δ ) δ ( δ) δ δδ A Fve-Parameter Famly of G Local Splnes For a gven cubc polynomal, S() t s unquely specfed f we know S (0), S (), S (0), and S () Each of these s a lnear combnaton of the coeffcents of (5) Tech Memo 04 Pxar

6 Dec 83 Matrx Splnes 6 S () t, and therefore s a lnear combnaton of the knots If we restrct our attenton to those matrx splnes whch are C, G and translaton nvarant, how are 0 our choces for S (0), S (), S (0), and S () restrcted? By translaton nvarant, we mean that addng some constant D to each of K has no effect on S() t other than to add D to t everywhere 4 That s K + D 3 K+ + D t t t M = S() t + D K+ + D K D 0 The C condton means that S+ (0) = S() Ths mples that S () cannot depend on K, snce K s not one of the knots of S () + t Therefore, S () and S (0) + may be expressed as ak+ + bk+ + ck + 3 Ths mples that S(0) = ak + bk+ + ck+ Translaton ndependence mples that ak ( + D) + bk ( + + D) + ck ( + + D) = ak + bk+ + ck+ + D Snce ths must be true ndependent of K, K +, and K +, t s true when K = K+ = K+ = 0, and therefore a+ b+ c= Smlarly, G mples that S (0) cannot depend on K + 3 and S () cannot depend on K, and therefore, for some d, e, f, and S (0) = ( )( dk + ek+ + fk+ ) S () = ( dk+ + ek+ + fk + 3) Translaton ndependence requres that S () not change when D s added to the knots Therefore ( dk ( + D) + ek ( + D) + f( K + D)) = ( dk + ek + fk ) and thus d + e+ f = 0 Therefore, we can substtute a = ι( σ) b = ισ to yeld c= a b= ι d = τδ ( ) + + e = τ( δ) f = d e= τδ S (0) = lerp( K,lerp( K, K, σ), ι) S() = lerp( K+,lerp( K+, K+ 3, σ), ι) (7) S (0) = ( )lerp(0,lerp( K K, K K, δ), τ) S () = lerp(0,lerp( K+ K+, K+ 3 K+, δ), τ) 0 It should be clear from the dervaton that ths famly ncludes all the C, G, translaton nvarant cubc matrx splnes The ntutve functons of the fve parameters are as follows: 4 Note that all matrx splnes are nvarant under scales and rotatons about the orgn Tech Memo 04 Pxar

7 Dec 83 Matrx Splnes 7 ι controls how close S() t comes to nterpolatng K + and K + When ι = 0, S(0) = K +, and S() = K + When ι =, S (0) and S () are ndependent of K + and K +, respectvely σ controls how much S (0) and S () slew parallel to the lnes ( K, K + ) and ( K+, K+ 3) When σ = 5, S (0) wll le symmetrcally between K and K +, as wll S () between K + and K + 3 controls the geometrcty of the splne When = 5, the splne wll be C For other values of, the splne wll be gven a kck n the drecton of ts tangent as t passes through S (0) τ controls the tenson on the splne When τ = 0, S () t goes to zero as the curve passes each knot As τ get larger, the curve gets less tense, passng ts knots at greater speed δ controls the drecton the curve heads as t passes each knot When δ = 0, S (0) heads left, parallel to the lne ( K, K + ) When δ =, S (0) heads rght, parallel to ( K+, K+ ) The matrx for ths famly of splnes, whch we wll call the M-splnes 5, s δτ+ δτ+ τ ισ + ι τ 3δτ δτ τ + ισ 4ι+ τ + 3δτ + δτ + τ + ισ + ι δτ ισ δτ δτ τ+ 3ισ 3ι+ τ 5δτ+ 4 δτ+ 3 τ 3ισ+ 6ι τ 3 4δτ δτ τ 3ισ 3ι+ 3 3ισ δτ τ( δ+ δ+ ) τ(δ δ + ) δτ( ) 0 ι( σ) ι ισ 0 All the splne famles we have dscussed above are subfamles of the M-splnes Our tense B-splne famly has σ = 5, ι= /3, = 5, δ = 5, and τ = The Cardnal splnes are the subfamly wth ι = 0, = 5, δ = 5, and τ = 4c (The value of σ s rrelevant when ι = 0 ) The τ -splnes are the class wth ι = 0, = τ, = 5, and τ = τ The δ -splnes have ι = 0, = 5, δ = δ, and τ = δ Examples Fgure shows several members of each of the splne famles developed n ths paper These pctures were drawn by a '50s Formca boomerang desgn system wrtten n Ideal [VanWyk] and Emacs Each of the llustratons uses the same set of sx knots and vares one of the parameters of the famly from 0 to n steps of /4 Curves drawn wth longer dashed lnes correspond to larger parameter values, wth the sold curve showng the parameter set to The upper left llustraton shows tensed B-splnes wth = 0(5) The upper rght shows Cardnal splnes wth c = 0(5) The lower left shows τ -splnes wth τ = 0(5) and τ = 5 The lower rght shows δ -splnes wth δ = 0(5) and δ = 5 The llustratons clearly show the effects of the tenson and bas parameters As or c approaches zero, the curves more closely approach the polygon connectng the knots As τ or vares from zero to one, the curves slew from left to rght Conclusons The orgnal observaton on whch ths work s based s that the Cardnal splne tenson parameter s equvalent to lerpng the c = and Ease splnes Ths 5 M s for Matrx, snce ths class ncludes most of the useful cubc matrx splnes Tech Memo 04 Pxar

8 Dec 83 Matrx Splnes 8 suggested an analogous method of applyng tenson to B-splnes, whch Alvy Ray Smth demonstrated was equvalent to the β parameter of Barsky's β - splnes Ths n turn suggested lookng for an extenson of the Cardnal splnes that have a bas parameter analogous to β There are two famles of nterpolatng splnes wth bas and tenson controls, one of them G (the τ -splnes) and the other C (the δ -splnes) We have constructed a fve-parameter G famly whch subsumes the β -, c-, τ -, and δ - splnes and whch exhausts the translaton nvarant G cubc matrx splnes All of ths work was made easy by the matrx splne notaton Lnear condtons on polynomals and ther dervatves are equvalent to correspondng condtons on ther coeffcents (because the dervatve s a lnear operator) Snce the coeffcents of a matrx splne are lnear combnatons of ts knots, these condtons are therefore equvalent to correspondng condtons on the matrx Ths equvalence makes the proofs of contnuty crtera trval (to the pont where I haven't bothered ncludng any n ths paper), and frees the magnaton n ts search for splnes wth nterestng and useful propertes Acknowledgements Ths work owes much to conversatons wth Alvy Ray Smth durng the sprng of 983 He dscovered the mappng β = ( )/ whch shows that my tense B-splnes are just a sub-class of the β -splnes He also encouraged the research by ncludng parts of t n hs notes for a tutoral on splnes gven at Sggraph '83 [Smth] Tom Porter's crtcal readng of multple drafts of ths paper aded the presentaton greatly Bblography [Barsky] Bran A Barsky, The Beta-Splne: A Local Representaton Based on Shape Parameters and Fundamental Geometrc Measures, PhD dssertaton, Department of Computer Scence, Unversty of Utah, Salt Lake Cty, Dec 98 [Brewer-Anderson] J A Brewer and D C Anderson, Vsual Interacton wth Overhauser Curves and Surfaces, Computer Graphcs, Vol, No (Sggraph '77 Proceedngs), Summer 977, 3-37 [Catmull] Edwn Catmull, A Subdvson Algorthm for Computer Dsplay of Curved Surfaces, PhD dssertaton, Department of Computer Scence, Unversty of Utah, Salt Lake Cty, Dec 974 [Catmull-Rom] Edwn Catmull and Raphael Rom, A Class of Local Interpolatng Splnes, Computer Aded Geometrc Desgn, edted by Robert E Barnhll and Rchard F Resenfeld, Academc Press, San Francsco, 974, [Clark] James H Clark, Parametrc Curves, Surfaces and Volumes n Computer Graphcs and Computer-Aded Geometrc Desgn, Techncal Report, Computer Systems Laboratory, Stanford Unversty, Palo Alto, Calforna, Nov 98 [Knuth] Donald E Knuth, Mathematcal Typography, Bulletn (New Seres) of the Amercan Mathematcal Socety, Mar 979, [Resenfeld] Rchard Resenfeld, Applcatons of B-splne Approxmaton to Geometrc Problems of Computer-Aded Desgn, Techncal Report UTEC-Csc-73-6, Department of Computer Scence, Unversty of Utah, Salt Lake Cty, Mar 973 Tech Memo 04 Pxar

9 Dec 83 Matrx Splnes 9 [Smth] Alvy Ray Smth, personal communcaton, Apr 983 [Smth] Alvy Ray Smth, Splne Tutoral Notes, Techncal Memo #77, Computer Graphcs Project, Lucasflm Ltd, Introducton to Computer Anmaton Tutoral Notes, Sggraph '83, Jul 983, [VanWyk] Chrstopher J Van Wyk, Ideal User's Manual, Computer Scence Techncal Report #03, Bell Laboratores, Murray Hll, NJ, Dec 7, 98 Fg Examples of Splne Famles Tech Memo 04 Pxar