Two kinds of B-basis of the algebraic hyperbolic space *

Transcription

1 75 L e al. / J Zhejag Uv SCI 25 6A(7): Joual of Zhejag Uvesy SCIECE ISS h:// E-al: jzus@zju.edu.c Two ds of B-bass of he algebac hyebolc sace * LI Ya-jua ( 李亚娟 ) WAG Guo-zhao ( 汪国昭 ) (Deae of Maheacs Zhejag Uvesy Hagzhou 327 Cha) E-al: lyajua94@sohu.co Receved Oc ; evso acceed Dec Absac: I hs ae wo ew ds of B-bass fucos called algebac hyebolc (AH) Béze bass ad AH B-Sle bass ae eseed he sace Γ sa{ 3 shcosh} whch K s a abay ege lage ha o equal o 3. They shae os oal oees as hose of he Béze bass ad B-Sle bass esecvely ad ca eese exacly soe eaable cuves ad sufaces such as he hyebola caeay hyebolc sal ad he hyebolc aabolod. The geeao of eso oduc sufaces of he AH B-Sle bass have wo fos: AH B-Sle suface ad AH T-Sle suface. Key wods: Algebac hyebolc Béze bass Algebac hyebolc B-Sle bass Algebac hyebolc Béze cuve Algebac hyebolc B-Sle cuve do:.63/jzus.25.a75 Docue code: A CLC ube: TP39.72 ITRODUCTIO Béze bass ad B-Sle bass ae wo oa bases of he olyoal sace saed by { 2 } whch s a abay osve ege. Howeve as show by Maa e al. (2) hee sll exs seveal laos of Béze odel ad B-Sle odel. Che ad Wag (23) ad Wag e al.(24) cosuced C-Béze bass ad he o ufo algebac gooec (UAT) B-Sle bass of he sace saed by { 2 scos} whch ca eese exacly ascedeal cuves such as helx ad cyclod. Koch ad Lyche (99) eseed a d of exoeal sle. I hs ae we wll gve wo bases fo he algebac hyebolc sace Γ saed by { 3 shcosh} whch ca eese exacly soe eaable cuves such as he hyebola ad he caeay. ALGEBRAIC HYPERBOLIC (AH) BÉZIER BASIS * Pojecs suoed by he aoal aual Scece Foudao of Cha (o. 37) ad he aoal Basc Reseach Poga (973) of Cha (o.g22cb32) Defo of he AH Béze bass We fs ese he AH Béze bass fucos. The cosuco s ecusve sag wh he wo al fucos: sh( ) sh B() B() sh sh [ ] ( + ) () Fo > he AH Béze bass fucos {B B 2 B } of he sace Γ + saed by { sh cosh } ae defed ecusvely by: B () B ( s)ds B () ( B ( s) B ( s))ds B () B ( s)ds (2) whee B ( )d <<. The we ge he defo of he AH Béze bass: Defo {B B B } s called he AH

2 L e al. / J Zhejag Uv SCI 25 6A(7): Béze bass fo he sace Γ + saed by{ 2 sh cosh}. See Fg B 3 () B 3 () B () The lea deedece of he {B B B } wll be oved Seco 2. I acula f we use B () B () as he wo al fucos we ge he Béze bass fo he olyoal sace fo Eq.(2); f we use s o elace sh he al fucos we wll ge he C-Béze bass fo he algebac gooec sace fo Eq.(2). Poees of he AH Béze bass () Pao of uy By he defo of he AH Béze bass s easy o ove B () [] (+ )..2 (2) Syey B ()B ( ) fo []. Poof Cosde he suao whe he syey oey s obvous. Suose holds fo ; B () B ( ) We ow ove he oey by duco o : B ()d s s B ( s)ds ()d.4 B s s B ()d s s Leg we oba. Fo < < we have:.6 B ().8 B 33 () B 23 () Fg. AH Béze bass fucos of ode 2 ad 4 B ( ) B ( s)ds B ()d s s ( ( )d ) ( ( )d ) B s s B s s B () The oof fo he suao whe ad s sla. Tha s he syey oey s oved. (3) Poees of he edos A he edos he AH Béze bass has he sae oees as he Béze bass ad C-Béze bass. Tha s fo 2 (a) j j B () B ( ) (3) (b) B () B ( ) j (4) () (c) B () (5) (d) 2 2 ( ) ( ) 2 B (6) Eq.(5) ad Eq.(6) ca be oved by duco o. (4) Lea deedece Le ab () [ ]. By ag we ge fo Eq.(4) ha. Dffeeag he above foula es we deduce aga fo Eq.(4) ha fo. Theefoe B B B ae lea deede ad { B B B } s a bass of Γ +. (5) Degee elevao: Dffeeag + j j + j B () a B () es we have fo oey (3) of he AH Béze bass ha fo Thus: B () a B () + a B () (7) Usg L Hosal s Rule we have a B () a + B+ + () B () + B () a + B+ + () l + B () +

3 752 L e al. / J Zhejag Uv SCI 25 6A(7): ad a B () a B () l + B () + B () B () 2 2 () + () 2 + B ( ) ( ) 2 + ( ) B+ + ( ) whee { B ( )} s he AH Béze bass fo he sace Γ + ad s a global shae aaee. AH Béze cuve has ay oees he sae as hose of he Béze cuve: () Edos eolao ()P ()P. Fo oey () of he AH Béze bass we ow B () [ a B () + a B ()] B ( ) + By he lea deedece of he AH Béze bass we have a + a+ + I fac +. Thus we have he degee elevao foula: () ( + ) B () B () + B () B () () () () + + B + + B + () B+ + () Posvy B ()> fo () so AH Béze bass s a bledg syse. Poof Cosde a abay AH Béze basc fuco B () 2. By he Rolle s Theoe ad oey (4) of he AH Béze bass we deduce ha B () has ad oly has zeos a [] cludg he -fold zeo a ad he ( )-fold zeo a so B () s ehe osve o egave o he eval ( ). Fo oey (5) of he AH Béze bass we ge ha he AH Béze bass s osve fo (). AH Béze cuves A ode + AH Béze cuve () wh cool os s defed as: () B () [ ] ( + ) (8) (2) Covex hull oey The ee AH Béze cuve Eq.(8) us le sde s cool olygo saed by (Fg.2) (3) Devave The devave () of a AH Béze cuve Eq.(8) s clealy a ode cuve: q () B () [ ] whee q ( + ). I acula we have Fg.2 AH Béze cuve ad cool olygo ( ) ( ) B () () (4) Degee elevao By he elevao of he AH Béze bass fucos we gve he degee elevao of he AH Béze cuve easly as: + + () B () B () q (9) hee q () () B () B () q () -+ () B + () B + () q +

4 L e al. / J Zhejag Uv SCI 25 6A(7): I fac a degee elevao ocedue s a coe cug ocedue jus as hose of he Béze cuve. I ca be vefed ha he sequece of cool olygos we ge ecusvely fo Eq.(9) coveges o he AH Béze cuve. (5) Vaao dshg (V. D.) oey o lae esecs a AH Béze cuve oe ofe ha esecs he coesodg cool olygo. We wll ove Seco 3. (6) Covexy esevg oey If he cool olygo s covex he he coesodg AH Béze cuve s also covex. (7) The l of he AH Béze cuves As he l of a AH Béze cuve he sace Γ + aoaches a Béze cuve he sace saed by { 2 }. I Fg.3 Fg.3b s obaed by ag he evals ad oo he eval (a) Béze Poof Reaaezg by τ/ s easy o ove ha he esul holds fo. Suose holds fo ad by ducve hyohess: B () B ( τ ) l B ( τ ) b ( τ ) Béze (b) Fg.3 Béze cuve ad AH Béze cuve (a) fo ; (b) fo hee b (τ) τ s a Béze bass. We gve he oof by duco by : B ()d s s B ()d s s B () [ B () s B ()]d s s B ()d s s B ()d s s B B l B ( τ ) l B ( τ ) B ( τ ) ( τ )d τ ( τ )dτ b ( τ) b ( τ) b ( τ)d τ b ( τ)dτ b ( ( τ) b ( τ)) b ( τ ) By he oees (3) ad (4) of he AH Béze bass we have B ()b () heefoe: l B ( τ ) b ( τ ) Fo he defos of AH Béze cuve ad Béze cuve we ge he oey (8). (8) The AH Béze bass s B-bass By he oees () (5) ad (6) we have ha AH Béze bass s a oally osve bass. I s easy o ge f{b ()/B j () B j () } by L Hosal s Rule. Fo Pooso 3.2 (Cace ad Peña 994) AH Béze bass s B-bass so has oal shae esevg oees [Chae 4 of (Peña 999)] ad oal sably oees fo he evaluao [Chae 5 of (Peña 999)]. AH Béze suface A AH Béze suface ca be cosuced usg eso oduc: uv ( ) B ( ub ) ( v) j j j u [ ] v [ β] β ( + ) I whch B (u) B j (v) ae he AH Béze bass fucos ad j ae he cool os. I should be oed ha oe ca choose a dffee aaee β he v deco. Is oees ca be deduced by hose of he AH Béze cuve.

5 754 L e al. / J Zhejag Uv SCI 25 6A(7): ALGEBRAIC HYPERBOLIC B-SPLIE BASIS (AH B-SPLIE) Defo of he AH B-Sle bass + LeT be a gve o sequece { } wh + we fs gve a se of al fucos: sh( ) / sh( + ) < + 2( ) sh( + 2 ) / sh( ) + < + 2 ohewse () Hee we defe ha /. The he AH B-Sle bass fucos of ode he sace Γ sa { 3 shcosh} ca be defed ecusvely as: () ( ( s) ( s))ds () whee ( )d. If () ad (). We have fo Eq.() he followg: ()d s s < < + + so we ge he defo of he AH B-Sle: Defo 2 { + +2 } s called he AH B-Sle bass of he sace Γ saed by { 3 shcosh} fo [ + ]. The lea deedece of he { + +2 } wll be oved Seco 3. I acula f we elace sh by Eq.() we ge he B-Sle bass fo he olyoal sace fo Eq.(); f we elace sh by s Eq.() we wll ge he UAT B-Sle bass fo he algebac gooec sace fo Eq.(). If he o sequece s ufo we wll ge he ufo hyebolc olyoal B-Sle bass descbed (Lü e al. 22). The sequece of () has he followg oees jus he sae as hose of he B-Sle. Poees of he AH B-Sle bass () Local suo () [ + ]. (2) Pao of uy + () fo all 3 ad all. (3) Devave () () (). + + (4) Zeo fuco () f ad oly f + +. (5) Posvy ()> fo ( ). Hee <. () fo all. Ths ca be oved he sae way as ha fo AH Béze bass. (6) Dffeeal () s ( j ) e couously dffeeal a he o j wh j he ube of es j aeas + he o sequece { }. j (7) Le ax{s +s }. If 2 we he have: j j ( ) j (8) Lea deedece + () +2 () () ae lealy deede o [ + ] wh < + fo all 2. As a cosequece of hs we ge ha () ± ae lealy deede o (+ ) f ad oly f he ullcy of each o of T s less ha +. Tha s hee s o zeo fuco () ± (9) Relao wh AH Béze bass I he case + +2 < () () ae jus he AH Béze bass of ode o [ + ]. Fo he defos of he wo bases he above oey ca be oved by duco o. Iseg a ew o I s he sae as he B-Sle ad he UAT B-Sle we have he o seg heoe of he

6 L e al. / J Zhejag Uv SCI 25 6A(7): AH B-Sle: + Theoe Le T( ) be a gve o sequece wh + <+ ad le ( + T ) be a ew o sequece obaed by seg a ew o u o T wh u< + j () ad j () ae defed as Eq.() fo he o sequece T ad T esecvely. If s ae ullces of he os u T esecvely we have fo all j 2 j () j j () + β j + j + () (2) whee fo <: j j j j j < j < + j + j β j β j + j j+ < j < + j + ad fo j j + j β j j > j > + ad j sh( u ) j 2 j sh( + ) j + (3) j sh( + u) β j 2 j sh( + ) j + whe ad s we have + (Fg.4). + + ( ) Poof Le 2 we oba Eq.(3) by dec calculao. Suose he ooso holds fo : (+) Fg.4 Iseg a o a j () () + β () fo all j. j j j+ j+ ow we wll ove he case fo. By he defo we he have () () fo j ad j j+ j j () () fo j + whch les j β j+ fo j ad j β j+ fo j +. (). I s easy o ove ha j β j+ fo j ad j β j+ fo j +. (2) <. By assuo we have ( () () ) ( ) () ( () ())d j j j j+ j+ j j j β j+ j+ j+ j+ j+ j+ 2 j+ 2 () β () d A() + A2() + A3() fo + j whee j j A () j j j () j+ j+ ()d j j j () j j+ β j+ 2 A2 () j+ 2 ( j+ j+ ( ) j+ 2 j+ 2 ( ) d j+ βj+ 2 j+ () j+ 2 3 λ j + j + A () ()d wh λ soe eal ube. Le v we have j (v)a (v) A 2 (v) whle ν 3 j + j + A () v ()d fo + j. We ge ha λ so we oba β () () () j j j+ j+ 2 j j + j+ j j+ 2 fo + j.

7 756 L e al. / J Zhejag Uv SCI 25 6A(7): I eas β j j j j 2 j β + + j+ j j+ 2 fo + j. (3). Whe s he oof s sla o he case (2). If s s also sla o he case (2) fo +<j. Fo j + we have + (). We defe: ()d + + so we have + () + + ()d < ( ) ( ) ( ) d () β ( + ) d sla o he case (2) we have β () () + () Tha s β β Fo he dscusso above we ge ha he ooso also holds fo. Ths oves Theoe also gves a ehod o coue he coeffces j β j+ fo all j. Fuheoe By he oey of ao of uy ad he lealy deedece of () ± we have he foula j +β j fo all wh () fo all. The Eq.(2) ca be ewe as () () + ( + ) + () (4) ow we gve he oof of he o-egavy oey of () ± Theoe 2 () fo all. Poof By he local suo we have ha () ca be o-zeo oly o [ + ]. If + we have () fo oey (4). I he followg suose < + we se a sees of ew os o o sequece T such ha he ullcy of each o j (j +) s. We he oba a ew o sequece deoed by T. Le be he ew sles wh he ew o sequece T he () s a covex cobao of (). By oey (9) we have ha () deeed by j (j +) s acually a AH Béze bass o each eval [ j j+ ] j+ + j < j+. Thus we have () fo all. AH B-Sle cuves AH B-Sle bass () ± has ay good oees so ca be used fo geoec odellg. Because of he local suo oey we ca defe a ece of AH B-Sle cuve () wh cool os a fe eval such as [ + ] (Fg.5). 2 3 () () [ ] + (5) whee { ( )} s he AH B-Sle bass wh he o sequece T { } + fo he sace Γ. AH B-Sle cuve has ay oees he sae as hose of he B-Sle cuve: () Devave The devave () of a ode AH B-Sle cuve () s clealy a degee cuve: 5 Fg.5 A ece of AH B-Sle cuve 4

8 L e al. / J Zhejag Uv SCI 25 6A(7): () () ( ) ( ) [ ] + The + j j (8) j u ( ) ( u) (2) Local cool oey Chage of oe cool o wll ale a os seges of he ogal AH B-Sle cuve of ode. Hece local adjuse ca be ade whou dsubg he es of he cuve. (3) Geoec vaace The shae of he AH B-Sle cuve s deede of he choce of he coodae syse because () s a affe cobao of he cool os. (4) Covex hull oey The ee AH B-Sle cuve us le sde s cool olygo (Fg.5). I follows fo he oegave ad ao of uy of he AH B-Sle bass. (5) Dffeeal () s couously dffeeal a a o of ullcy. (6) Subdvso of he cuves Subsug Eq.(4) o Eq.(5) we have ( + + ) () () + ( ) () + () (6) whee ( ) + +. (7) wh ad () as defed Seco 3. Fo Eq.(7) we ge ha he ew cool os ca be obaed fo he old cool os afe subdvso. I fac he ocess of seg a ew o s acually a coe cug ocess. If we se he sae o u <u< + eavely he we wll ge a sees of ew l cool os +l fo Eq.(7). Hee l s he es of seg o. I acula whe l sce l j ( u) j j We use () o deoe a AH B-Sle cuve of ode wh Φ [ ] [ ] T T ( ) + as he cool olygo ad o sequece. Le T ( ) ++ be he ew o sequece obaed by seg a ew o o T ad Φ [ ] Φ[ Φ [ ]] [ + ] be he cool olygo whee he cool os + + ae gve by Eq.(7). Ad le ax ++. We have he followg heoe. Theoe 3 Whe as he ube of subdvsos ceases he sequece of cool olygos Φ [] coveges o he AH sle cuve (). Poof Le () be he AH B-Sle bass fucos wh he o sequece T. By he defo of Φ [] we have + + () () () [ ] Fo he oey () of he cuves we have + () () () [ ] + whee. I ca be easly see ha () s a ece of AH B-Sle cuve of ode ad he cool olygo + ca be obaed fo afe a sees of subdvsos. Hece + s bouded by he covex hull of. Howeve oe ha + ( ) ()d ()d ( + ) +

9 758 L e al. / J Zhejag Uv SCI 25 6A(7): We have l. + Theefoe l l + (9) oe ha () whe (). I hs case we have. Fo he covex hull oey fo ay u [ + ] we ow ha (u) les wh he covex hull of + + fo soe. Togehe wh Eq.(9) we coclude he heoe. Theoe 3 eas ha ecusve subdvso of cool olygo leads o s coesodg AH B-Sle cuve so s o dffcul o ove he V. D. oey ad he covexy esevg oey below: (7) V. D. oey o lae esecs a AH B-Sle cuve oe ofe ha esecs he coesodg cool olygo. Because AH Béze bass s secal AH B-Sle bass we easly ge ha AH Béze cuves have V. D. oey oo. (8) Covexy esevg oey If he cool olygo s covex he he coesodg AH B-Sle cuve s also covex. By he o seg heoe we ca easly ge he oey (9) of he AH Sle: (9) The AH B-Sle bass s B-bass: Poof Ise a o a ad l eeaedly ul he obaed ullces of o ad l ae esecvely ad he old bass ca be exessed by he ew bass. Le us deoe he bass fucos geeaed afe he h o seg o as ( + ( ) ()... ()) s hus he + + l + whee ew bass 2 has edo eolao oey. Fo [Seco 2 of (Maa ad Peña 999) ad Seco 2 of (Maa e al. 2)] we ow ha f a bass has edo eolao oey V. D. oey ao of uy ad covex hull oees s oalzed oally osve so he ew bass 2 s oalzed oally osve. Accodg o he oey of seg a ew o (Theoe ) he asfoao ax bewee + ad s oe-baded ax so he asfoao ax bewee ad 2 ca be decoosed o a oduc of bdagoal facos. Thus we deduce ha he AH B-Sle bass s oalzed oally osve. By he defo of he B-bass we wll show he followg equao holds fo evey j: () + f [ l] j + ( ) j + () (2) If <j ad ++ < j++ le ( j )/2 we have: () + f j ( + ) j + () If ++ j++ le + + he he fe u ode of + s j ode hghe ha fe u ode of j+ so l ( ) / ( ) + j ha s Eq.(2) holds. Whe >j ad > j le ( + )/ 2 we have Eq.(2) holds. Whe >j ad j le + he he fe u ode of + s j ode hghe ha fe u ode of j+ so l ( ) / ( ) ha s Eq.(2) holds. + + j + By Pooso 3.2 (Cace ad Peña 994) ad Seco 3 AH B-Sle bass s oalzed B-bass. Theefoe AH B-Sle bass has oal shae esevg oees ad oal sably oees. AH B-Sle suface Exacly as he cosuco of B-Sle eso oduc sufaces fo B-Sle cuves a AH B-Sle suface ca be cosuced usg eso + oduc. Le U { u } { } + V v j j be wo o sequeces he a AH B-Sle suface wh cool eshes j ca be defed as uv ( ) ( u ) ( v) j j j j

10 L e al. / J Zhejag Uv SCI 25 6A(7): u [ u u ] v [ v v ] l. + l + I whch (u) j (v) ae he AH B-Sle bass fucos wh o sequece U V esecvely. Is oees ca be deduced fo he oees of he AH B-Sle such as he covex hull oey ad he covexy esevg oey. The subdvso foulae ca be used boh decos ec. Refeeces Cace J.M. Peña J.M 994. Toally osve fo shae esevg cuve desg ad oaly of B-Sles. Coue Aded Geoec Desg : Che Q.Y. Wag G.Z. 23. A class of Béze-le cuves. Coue Aded Geoec Desg 2: Koch P.E. Lyche T. 99. Cosuco of Exoeal e- so B-Sles of Abay Ode. I: Laue P.J. Le Méhaué A. Schuae L.L.(Eds.) Cuves ad Sufaces. Acadec Pess ew Yo Lü Y.G. Wag G.Z. Yag X Ufo hyebolc olyoal B-Sle cuves. Coue Aded Geoec Desg 9: Maa E. Peña J.M Coe cug algohs assocaed wh oal shae esevg eeseaos. Coue Aded Geoec Desg 6: Maa E. Peña J.M. Sáchez-Reyes J. 2. Shae esevg aleaves o he aoal Béze odel. Coue Aded Geoec Desg 8:37-6. Peña. J.M Shae Pesevg Reeseaos Coue Aded Geoec Desg. ova Scece Publshes Coac (ew Yo). Wag G.Z. Che Q.Y. Zhou M.H. 24. UAT B-B-Sle cuves. Coue Aded Geoec Desg 2: Welcoe cobuos fo all ove he wold h:// The Joual as o ese he laes develoe ad achevee scefc eseach Cha ad oveseas o he wold's scefc couy; JZUS s eded by a eaoal boad of dsgushed foeg ad Chese scess. Ad a eaoalzed sadad ee evew syse s a esseal ool fo hs Joual's develoe; JZUS has bee acceed by CA E Coedex SA AJ ZM CABI BIOSIS (ZR) IM/MEDLIE CSA (ASF/CE/CIS/Co/EC/EM/ESPM/MD/MTE/O/SSS*/WR) fo absacg ad dexg esecvely sce saed 2; JZUS wll feaue Scece & Egeeg subjecs Vol. A 2 ssues/yea ad Lfe Scece & Boechology subjecs Vol. B 2 ssues/yea; JZUS has lauched hs ew colu Scece Lees ad waly welcoe scess all ove he wold o ublsh he laes eseach oes less ha 3 4 ages. Ad assue he hese Lees o be ublshed abou 3 days; JZUS has led s webse (h:// o CossRef: h:// (do:.63/jzus.25.xxxx); MEDLIE: h:// Hgh- We: h://hghwe.safod.edu/o/jouals.dl; Pceo Uvesy Lbay: h://lbweb5.ceo.edu/ejouals/.