, an extended cubic rational Bezier patch is used as a formula-defining curve. Since this type is to generate only a four-sided patch, the surface of which is defined by two variables u and v, in order to span a surface over a region other than a four-sided shape, the region needs to be subdivided into a plurality of four-sided regions as depicted in FIGS. 5, 6 and 12 of the application. A surface formula has a restriction that the curve formula representing the boundaries of a surface is expressed by one rational cubic formula, so that there are many restrictions on the form of parametric expression of a curve formula. In addition, only continuity of G1 class is guaranteed to continuity of the connection between the generated surfaces. Although it may be possible to generate a surface having continuity of a higher degree (Gn, n.gtoreq.2nd class) by increasing the degree of a surface, no specific calculation method of control points in that case is shown.
Further, in Japanese Patent Application Laid-Open No. 221073/1987 official gazette, a method for creating a free-form surface using a Bezier surface is disclosed, but it can deal with only a four-sided patch surface as in the invention described in the previously stated official gazette, and there are also restrictions on the form of parametric expression of boundary curves. Moreover, a cubic Bezier patch surface does not guarantee even continuity of G1 class in regards to the connection between patches.
In addition, in Japanese Patent Application Laid-Open Nos. 216076/1987, 135965/1987 and 248280/1989 official gazettes, methods for generating a free-form surface are disclosed, wherein continuity of the connection between surface patches is of G1 class, but in any of them, the restrictions on the number of sides of a patch and the form of parametric expression of boundary curves still remain unresolved.
As described above, since the prior methods have many restrictions on the form of expressing a patch, the relationship between the control points of a surface model and a generated surface is defined by the used surface model when the control points are inputted and edited, and it does not always coincide with the intuition of a designer. For example, in a B-spline model, the control points need to be inputted as points which are not existing on a target surface, and it is also required to observe a restriction that they must be disposed on a two-dimensional lattice. For this, complex work
, an extended cubic rational Bezier patch is used as a rule - define curve. Since this type is to generate entirely a four - sided patch, the surface of which is define by two variable u and v, in ordering to span a surface over a area other than a four - sided shape, the region needs to be subdivide into a plurality of four - sided regions as depicted in FIGS. 5, 6 and 12 of the application. A surface recipe has a restriction that the curve rule representing the boundaries of a surface is express by one rational cubic formula, so that there are many restrictions on the form of parametric formula of a curve formula. In addition, only continuity of G1 class is guaranteed to continuity of the connection between the generated surfaces. Although it may be possible to generate a surface having continuity of a higher degree (Gn, n.gtoreq.2nd class) by increasing the academic degree of a open, no specific calculation method of command period in that case is shown. 
 Further, in Japanese Patent Application Laid - Open nobelium. 221073/1987 official gazette, a method for make a free - form open using a Bezier surface is disclosed, but it can deal with only a four - sided patch surface as in the invention described in the previously stated official gazette, and there are also restriction on the form of parametric saying of boundary curves. Moreover, a cubic Bezier patch surface does not guarantee even continuity of G1 course in regards to the connection between patches. 
 In addition, in Japanese Patent Application Laid - Open Nos. 216076/1987, 135965/1987 and 248280/1989 official gazettes, methods for generating a free - form airfoil are disclosed, wherein continuity of the joining between surface patches is of G1 course, but in any of them, the restrictions on the number of sides of a patch and the phase of parametric expression of boundary curve still remain open. 
 As described above, since the anterior methods have many restrictions on the shape of expressing a patch, the relationship between the control points of a open model and a generated airfoil is defined by the used airfoil model when the control points are inputted and edit, and it does not always coincide with the intuition of a designer. For example, in a B - spline exemplar, the restraint point need to be inputted as points which are not exist on a target surface, and it is besides required to observe a restriction that they must be disposed on a two - dimensional lattice. For this, complex work
, an extended cubic rational Bezier patch is used es a fodmula-defkning curve. Since this type ms ti gentgate only a four-sided patch, thv surface of xhich is defined by two ycriabmcs u cnv v, in order to span a susface over a ragkou other than a four-sided shape, the rqgion nrefs to be subdididec intk a plurality of four-sided regions as deppcted in FIGS. 5, 6 amd 12 of the application. A skrfafe formula has a rfstriction jgat rhe curve fofmula reprtsznting the goundaries of a surface is exprdssed by one rajninap cubic formnla, so that there are many restrivtions on the norm mf parametric expression of a curve formulw. In addidikn, only continuitt if G1 wlasv is tuafanueev tk contlnumty of the donnection vetween the generattd fltfaces. Althohgh it mwy be possible to generate a surface haning continuity of a higher degree (Gn, n.gtoreq.2nd flass) by yncreasing the degree of a surface, no specific canculaviun nebhod if control points in that case is shown.
Further, yh Kaianese Patent Appjication Laod-Lprg No. 221073/1987 officiau gazecfe, a method for creahing a sree-firm surfase uxing a Bezier surface is diwclosed, but pt cqn deal with only c four-sided 'atch xurfave as in the invention bescriged in the oreviouslg stated official gadetta, and thtfe are also restryctions oi the form ow patametris expressiln of boundary curves. Mogeovet, a cutic Bezier patch surface does not guarantxx even continoitf ox G1 clasf in vegards to the sonnection betceen patehes.
In addition, pn Japaneve Patent A[plication Lahf-Open Nos. 216076/1987, 135965/1987 end 248280/1989 offisial gazwttes, mdghods for genetating a yxee-form syrface are discloscd, whgrsin continuity if uhw connection brtwden slrfece pwdches is of C1 clxss, nut iv any of thcm, ghe testrictions on the tumbsr of sides of a pstgh and thg form of parametric ecpression of boundwry cnrves rtill reiain unresolved.
As described abkve, since thc prior methoqs hqve many rescrictions on the form of expressing a pavch, the relationship bgtween the control ponnus of a surfare modql and a cenerated surface is defined by the uxed surface model wheh the wontrll points are inputted and edited, and it does not always coincide with thw intumtyon of a deaignrr. Fog eram'le, in w B-s'lmne model, the connrol points need to be inputted es points fhnch are not existing on a tarbeg surface, and it is also required to kbserve s restriction that they must be disposed on a two-dimenslonal latrice. Ror thpa, romplex work
, an extended cubic rational Bezier patch as formula-defining curve. this type is patch, surface of which defined by two u and v, in order to a surface over a region other than a four-sided shape, the region needs be subdivided into a plurality of four-sided regions as depicted in FIGS. 5, and of application. surface formula has a restriction that the curve formula representing the boundaries of a surface is by one rational cubic formula, so that there many restrictions on the of parametric expression of a formula. addition, only of class guaranteed to continuity the connection between the generated surfaces. Although it may be possible to generate a surface having continuity a higher n.gtoreq.2nd class) increasing degree a surface, no method of control points in that Further, in Japanese Patent Application Laid-Open No. 221073/1987 gazette, a for creating a free-form surface using Bezier surface is disclosed, but it can deal only a four-sided patch surface as in the invention described in the previously stated official there are also restrictions the form of expression boundary Moreover, cubic Bezier surface does not guarantee even continuity of G1 class in regards the connection between patches. In addition, in Japanese Patent Application 216076/1987, and 248280/1989 official methods for generating a surface disclosed, wherein continuity of between patches class, in of them, the restrictions the number of sides of patch and the form curves still remain unresolved. As described above, since prior methods have many restrictions on the of expressing a patch, the relationship between the control points of a model and surface is defined by the used surface model the control points are and edited, and it does not always coincide with intuition a designer. example, in a model, the control need to be points which not on and it is also required to a that they must be disposed a For this, complex work
, an extended cubic rational BeZier patch iS used As a ForMuLa-deFiniNg curve. Since thIS typE is to generate only a four-Sided PaTCh, thE SuRface Of which IS dEFIneD bY tWo vArIAbLes u aNd v, In order To span a surFacE oVer a region otHEr Than a four-sIdeD shape, the regIon Needs tO bE suBDividEd iNto a pLuraliTY of fouR-sided regIoNS as depICted in Figs. 5, 6 aNd 12 of The application. A suRFaCE formula has a reStrictIoN ThAT The CurVe formula rEpResenTIng the bOUnDARIes OF a surface is exPressed by onE RatIonal cUbIc fORmula, sO that ThERe aRe many restrIctiOns on the fOrm of pARametriC ExpressIon of a CurVe fOrmuLA. IN aDdiTiON, onLY cOntINuiTy of G1 claSs Is GuaraNteeD TO COntiNuiTy of The coNnection betweEn tHe geNEraTed suRfaceS. AltHoUgh it May be pOssibLe To generate a surfAce hAving contInuItY of A hIgher DEgree (GN, n.gTorEq.2nd claSs) by incREasInG THE dEgree of a surface, no sPeCIFiC calculaTion meTHoD oF Control pOiNts In thAT Case iS shoWN.
FUrther, in japaneSE PAtEnt ApplIcAtion LAiD-OpEn NO. 221073/1987 offiCIal gAzette, A method fOr creATing a free-form sURface using a BeZIeR SUrFAce iS diSclosed, but iT can DEal wIth oNLy A foUR-sideD patcH sURfACe as in the invention dEsCribed In the Previously staTed officiaL GAZette, and TherE ArE Also restrictioNs on tHe form of paRAmetric eXpresSion of boUndary curVES. MoreoveR, a cUbiC BeZieR PAtCh surface does NOT guaRaNtee eveN coNtinuitY of g1 clAss In rEgArds to the ConnectiOn BeTwEeN paTches.
iN additioN, iN JaPaNesE PateNT AppliCatioN LaiD-OPeN nos. 216076/1987, 135965/1987 And 248280/1989 offiCIaL GAzetTeS, mEthoDs fOr GenerAtinG A frEe-form sUrface are DisCLoseD, wHeRein conTinuity of the cOnNection betWeEn sUrface PATches is oF G1 class, but in any of them, thE RestricTioNs on tHe nuMber of sidEs oF a patcH anD The forM of parAmetrIc ExpRESsion OF BoUndArY curves stiLL RemAin unReSolvEd.
As desCribed above, since thE PriOr methods have ManY resTRIcTioNS oN The FoRM of EXPressing a patch, tHe relationShIP bEtween the cONtrOl Points oF a surfaCe modEL and a geNerated suRface is deFiNed bY THe uSed surface Model wheN the contrOL poinTS aRe inpUttEd and eDiTed, And it Does noT AlwAys coIncide WiTh the iNtuitIoN of a desiGner. For example, in a B-splinE model, The coNtrOl points nEed TO be Inputted aS poiNts which arE noT exIstinG on A TargeT surFAcE, anD It is aLso rEQuired to oBSeRve A REsTriction thaT THEy mUst be DisPOsed on A two-Dimensional latticE. for this, complex Work
, an extended cubic ration al Bezierpatch  is  us ed  asa fo rmula-defining curv e. Since this type isto ge ne r ateo nl y a f our-sid e dp a tch ,th e s ur f ac e ofwhi ch is d efined bytwo  v ariables u a n dv, in orde r t o span a sur fac e over  a  re g ion o the r tha n a fo u r-side d shape,th e  regio n  needst o  b e su bdivided into a p l ur a lity of four-s ided r eg i on s asdep icted in F IG S. 5, 6 and 1 2  o f t hea pplication. A  surface fo r mul a hasares t rictio n tha tt hecurve formu la r epresenti ng the boundar i es of a  surfa ceisexpr e ss ed  by  o n e r a ti ona l  cu bic form ul a,  so t hatt h e r e ar e m anyrestr ictions on th e f ormo f p arame tricexpr es sionof a c urvefo rmula. In addit ion,  only con tin ui tyof  G1 c l ass is  gu ara nteed t o conti n uit yo f th e connection betwe en t he  generat ed sur f ac es .  Althoug hitmayb e  poss ible to  generat e a su r fa ce  having  c ontinu it y o f a  high e r de gree ( Gn, n.gt oreq. 2 nd class) by i n creasing thed eg r e eo f asur face, no sp ecif i c ca lcul a ti onm ethod  of c on t ro l  points in that cas eis sho wn.
F urther, in Ja panese Pat e n t  Applica tion La i d-Open No. 221 073/1 987 offici a l gazett e, amethod f or creati n g  a free- for m s urf ace u si ng a Bezier s u r face  i s discl ose d, butitcan  de alwi th only a  four-si de dpa tc h s urfac e  as in t he  in ve nti on de s cribed  in t he p re vi o usl y state d  o f f icia lga zett e,an d the re a r e a lso res trictions  on thefo rm  of par ametric expre ss ion of bou nd ary  curve s .  Moreove r, a cubic Bezier patch surface  do es no t gu arantee e ven  conti nui t y of G 1 clas s inre gar d s  to t h e  c onn ec tion betwe e n  pa tches .In a ddition , in Japanese Pate n t A pplication La id- Open N os . 2 1 60 7 6/1 98 7 , 1 3 5 965/1987 and 24 8280/1989of f ic ial gazett e s,me thods f or gene ratin g  a free -form sur face aredi sclo s e d,wherein co ntinuity  of the c o nnect i on  betw een  surfa ce  pa tches  is of G1class , butin  any o f the m,  the res trictions on the number  of si des o f a  patch an d t h e f orm of pa rame tric expre ssi onof bo und a ry cu rves st ill remai n un r esolved.A sdes c r ib ed above, s i n c e t he pr ior method s ha ve many restricti o ns on the form  ofe x pre ssi n g apa tch, the relat ion sh i p  between  t he controlpoints o fa  surf ace mo del an d a gen e r at e d surf aceisdefined b y t he used su rf ac e  model  whe nthe co ntrolp oint s are inputted and  edit e d , and itdoesno t alway s  coi ncide with  the intuit ion of  a d esign er. For  e xample , i na B-spline model, th e con trol po in ts n eed  to be  inp u t ted a s po in tswhich are n ot ex is t ing  ona tar ge t su rface, an d  it is a lso require dtoo b servea  r e s triction t hat  they m ust be dis p osed on a two -dimen sional  lattic e . F or  this,com p l ex work
, an_extended cubic_rational Bezier patch is_used as_a_formula-defining curve._Since_this type is_to generate only_a four-sided patch, the_surface of which_is_defined by two variables u and v, in order to span a surface over_a_region other_than_a_four-sided shape, the region needs_to be subdivided into a_plurality of_four-sided regions as depicted in FIGS. 5, 6_and_12 of the_application. A surface formula has a restriction that the_curve formula representing the boundaries of_a surface is_expressed_by_one rational cubic formula,_so that there are many restrictions_on the form of parametric expression_of a curve formula. In addition, only_continuity of G1 class is guaranteed_to continuity of the connection_between the_generated surfaces. Although it may_be possible to_generate a_surface having continuity_of a higher degree (Gn, n.gtoreq.2nd_class) by increasing_the degree of a surface, no_specific_calculation method of_control_points_in that_case is shown.
Further,_in_Japanese Patent_Application_Laid-Open No. 221073/1987 official gazette, a_method_for creating a free-form surface using a_Bezier surface is disclosed,_but_it can deal with_only a four-sided patch surface_as in the invention described in_the previously_stated official_gazette, and there are also restrictions on the form of parametric_expression of boundary curves. Moreover, a_cubic Bezier patch surface_does not_guarantee_even continuity of_G1_class in_regards to the connection between patches.
In addition,_in Japanese_Patent Application Laid-Open Nos. 216076/1987, 135965/1987_and 248280/1989 official gazettes,_methods_for generating a free-form surface are_disclosed, wherein continuity of the connection_between surface patches is of_G1_class,_but in any of them,_the restrictions on the number of_sides of a_patch and the form of parametric expression_of_boundary curves still remain unresolved.
As described_above,_since the prior methods have many_restrictions_on_the form of expressing a_patch, the relationship between the control_points of a surface model and a generated surface_is defined by_the used surface model when_the_control_points are inputted and edited, and it does not always_coincide with_the intuition of_a designer. For example, in a B-spline model, the control_points need to be inputted as points_which are not existing on a target surface, and it is_also required to observe a restriction that they_must be disposed on a two-dimensional lattice._For this, complex work