NURBS Curve Shape Modification and Fairness Evaluation for Computer Aided Aesthetic Design

Transcription

1 oceedngs of the 7th WSEAS Int. Conf. on Sgnal ocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 4-6, 7 38 NURBS Cuve Shape Modfcaton and Faness Evaluaton fo Compute Aded Aesthetc Desgn TETSUZO KURAGANO AND AKIRA YAMAGUCHI Gaduate School of Infomaton Scence, Mese Unvest -9 Nagabuch, Ome-Ct, Toko, 98-86, JAAN kuagano@e.mese-u.ac.jp Abstact: - A cuve wth a monotone dstbuton s consdeed as a fa cuve n the aea of Compute Aded Aesthetc Desgn (CAAD). But no offcal standads have been establshed. Theefoe, a cteon fo a fa cuve s poposed. A untc NURBS cuve, the fst devatve of a untc NURBS cuve, cuvatue vecto, cuvatue, and ae epessed. The concept of specfcaton to modf the shape of a NURBS cuve s llustated. The dffeence between the NURBS cuve adus of cuvatue and the specfed s mnmzed b ntoducng the least-suaes method to modf the shape of the NURBS cuve. Algebac functons such as lnea, uadatc, cubc, uatc, untc, and s degees ae appled to the dstbuton of the desgned cuve as the specfed. The dstbutons gven b these s algebac functons ae consdeed monotone, because the ndependent vaable of these algebac functons s monotone to the coespondng dependent vaable of these functons. Smlat s evaluated usng the dstbuton accodng to s algebac functons as efeences and the dstbuton of the desgned cuves as matches b usng coelaton matchng. Cuve shape smlat evaluaton s ted usng an eample. Consdeng that a cuve wth a monotone vaaton of dstbuton s fa, the smlat of the desgned cuve to a fa cuve s evaluated. Ths measued smlat epesses faness to the fa cuve. Usng ths technue, the faness of a cuve s evaluated b usng the smlat of the dstbuton. Ke-Wods: - cuve shape modfcaton, fa cuve, specfcaton, coelaton matchng, faness evaluaton Intoducton In Compute Aded Aesthetc Desgn (CAAD) [], desgnes evaluate the ualt of a desgned cuve b lookng at ts cuvatue o plots. If the ualt of a desgned cuve does not meet desgne s demands, the usuall modf the contol ponts of the cuve nteactvel. If the vaaton of the of the cuve s monotone, ths cuve s assumed to be a fa cuve []. But the defnton of a fa cuve s ambguous and no offcal standads ae gven. Theefoe, n ths pape we have ted to establsh cteon fo a fa cuve. A NURBS cuve, whch s commonl used n the feld of CAD CAM and Compute Gaphcs, s used as an epesson of a feefom cuve. A uadatc NURBS cuve s used as an epesson of a uadatc cuve usng ts weghts. In ths stud, a uadatc cuve s not used to epess the shape of a cuve. Theefoe, the weghts of a NURBS cuve ae not used. A cubc NURBS cuve s wdel used, but n ths stud, adus of cuvatue of mult segments of a NURBS cuve ae modfed based on the specfed. So, a smooth contnut s needed. Theefoe, a untc NURBS cuve s used n ths stud. ostons and gadents ae gven to the NURBS cuve euatons and fst devatve euatons of the NURBS cuve espectvel. Then, a NURBS cuve s geneated. Aftewads, f necessa, the shape of ths NURBS cuve s modfed accodng to the specfed dstbuton. Fa cuve epesson and evaluaton of faness ae descbed. As a measue of cuve faness evaluaton, dstbuton s used as an altenatve chaactestc of a cuve. Evaluaton of whethe the desgned cuve s fa o not s accomplshed b compang the desgned cuve to a cuve whose s monotone. The dstbutons of s NURBS cuves ae modfed to follow algebac functons such as lnea, uadatc, cubc, uatc, untc, and s degees to specf the dstbuton. Then, b ntoducng the coelaton matchng, the

2 oceedngs of the 7th WSEAS Int. Conf. on Sgnal ocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 4-6, 7 39 smlates of the desgned cuve to these s pedefned cuves ae eamned. Among these pedefned cuves, the hghest smlat cuve to the desgned cuve s selected as an deal fa NURBS cuve. Then, the faness of the desgned cuve to ts deal fa cuve s evaluated. Fa cuve geneaton algothms elated to cuvatue b modfng the contol ponts have been publshed. These make monotone cuvatue [3], use a clothodal cuve fo specfng the cuvatue [4], and automate a cuve fang algothm fo B-splne cuves [, 6]. Fa cuve geneaton algothms elated to eneg functon have been publshed. These ae smoothng of cubc paametc splnes b eneg functon [7], fndng the unfa poton of a cuve usng eneg functon [8], and ntoducng a low-pass flte to eneg functon [9]. Fa cuve geneaton algothms elated to cuvatue b specfng cuvatue dstbuton have also been publshed []. Thee ae man elated woks fo evaluatng smlates of polgons n two dmensonal space, especall n the aea of mage pocessng. Methods fo evaluatng smlates, whch ae based on the dstances of coespondng ponts on polgonal cuves, have been epoted [-4]. If the dstances ae close, t wll be detemned that the two polgonal cuves ae smla. Methods usng Foue descptos fo evaluatng smla polgons have been developed and mplemented [, 6]. One s to eteve the mage fles usng Foue descptos. The othe s to classf the chaactes epessed b polgonal cuves. Secton of ths pape descbes a untc NURBS cuve, the fst devatve of a untc NURBS cuve, cuvatue vecto, cuvatue, and. In secton 3, geneaton of a untc NURBS cuve whch passes though gven pont seuence and geneaton of a untc NURBS cuve usng the gven ponts and gadents ae descbed. In secton 4, NURBS cuve shape modfcaton based on the specfed s descbed. Secton descbes coelaton matchng to evaluate the smlat of the NURBS cuves. Secton 6 descbes fa cuve epesson and faness evaluaton gvng eamples. A cteon fo a fa cuve s poposed as faness. NURBS Cuve Epesson A untc NURBS cuve s used n ths stud. The objectve of feefom cuve desgn s to desgn the famewok of suface patches. Suface patches ae defned as tenso poducts, whch ae b-vaate and nomall defned b u and v. In othe wods, one knot seuence n u decton, and anothe knot seuence n v decton ae defned despte the complet of the suface patches. Theefoe, knot spacng s fed n ths stud. A untc NURBS cuve conssts of n - segments ( n ³ 6), s composed of n contol ponts such as,,, n- and n weghts such as w, w,, wn - as n E.(). R = n- å N = n- å =,6 N,6 w w whee N,6 ( =,,, n -) ae NURBS bass functons. These functons ae ecusvel defned b knot seuence t, t,, t n+ as n E.(). ì ( t t < t ) ü + N, = í î othewse ï ý t - t t+ M - t N, M = N, M - + N, M ï + - t+ M - - t t+ M - t ï + þ whee =,,, n -and M =,3,,6. The bass functons ae defned b the de Boo-Co [7] ecuson fomulas. If the knot vecto contans a suffcent numbe of epeated knot values, then a dvson of the fom N, M - /( t+ M - - t ) = / (fo some ) ma be encounteed dung the eecuton of the ecuson. Wheneve ths occus, t s assumed that / = [8]. A untc NURBS cuve wth knot vecto s epessed as n {, 4, 3,,,,,, 3, 4,, 6} E.(3). R { = ( - t) 4 3 (t t t t t 6) ( t 3t 6t 66) (t t t t t 6) t + t + t + t + t ( ) 4 + t } (3) The fst devatve of a untc NURBS cuve shown n E.(3) s epessed as n E.(4). dr( t ) dt { 4 = - ( - t) + t - t + t + t ( ) + - t + t - t 4 3 ( ) + t - t - t + t ( ) t + t + t + t ( 3 ) t } () () (4)

3 oceedngs of the 7th WSEAS Int. Conf. on Sgnal ocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 4-6, 7 4 Cuvatue vecto s epessed b E.(). ( R R ) R κ = () ( R ) 4 whee R s the fst devatve of a NURBS cuve, and R s the second devatve of a NURBS cuve. Cuvatue s the magntude of the cuvatue vecto, theefoe cuvatue s epessed as n E.(6). k = κ (6) B defnton, the cuvatue of a plane cuve s nonnegatve. Howeve, n man cases t s useful to ascbe a sgn to the cuve [9]. The choosng of the sgn s commonl connected wth the tangent otaton (n movng along the cuve n the decton of the nceasng paamete): The cuvatue of the cuve s postve when ts tangent otates counte-clockwse, the cuvatue of the cuve s negatve when ts tangent otates clockwse. Radus of cuvatue s the ecpocal numbe of cuvatue, theefoe, s epessed as n E.(7). = k 3 Geneaton of a NURBS Cuve In ths secton, a method to geneate a untc NURBS cuve whch passes though gven ponts n seuence s shown. Anothe method to geneate a untc NURBS cuve usng the gven ponts n seuence wth gadents s descbed. 3. Geneaton of a Quntc NURBS Cuve whch asses though Gven ont Seuence uttng zeo to the paamete of E.(3), E.(3) s epessed as E.(8) b defnng the geometcal knot poston coespondng to the knot of the knot vecto. R = ( ) ( =,,, 3,, m - ) Whee m s the numbe of the gven ponts, and,,, 3,, m -, m - ae the postonal vectos of the gven ponts to be assgned to R ( =,,,3,, m -) n E.(8), and,,, 3,, m+, m+ 3 ae the contol ponts of a untc NURBS cuve. When the contol ponts of a NURBS cuve ae calculated usng E.(8), the numbe of unknowns, whch ae the postons of the contol ponts, ae fou moe than the numbe of euatons whch ae epessed b E.(8). In ths case, b settng the second devatve of the NURBS cuve to zeo, and settng the (7) (8) fouth devatve of the NURBS cuve to zeo, unknown vaables become known. Theefoe, the numbe of euatons wll be eual to the numbe of unknowns. That s, a lnea sstem s detemned []. Then a NURBS cuve s geneated b solvng ths detemned sstem. In ths stud, n addton to the gven pont seuence, gadent at the gven ponts s defned. E.(9) s appled to the gadents b settng the paamete of E.(4) as zeo. dr = ( ) dt 4 ( =,,, 3,, n - ) Whee n s the numbe of gven gadents. The shown n E.(9) coesponds to the n E.(8) and s detemned stuatonall. As a magntude of the fst devatve, one thd value of the dstance of adjacent gven ponts s assgned. The defned gadents ae located at the begnnng gven pont and t s adjacent pont, and at the end gven pont and t s adjacent pont n geneal. In ths case, the ae detemned as,, n -, n - espectvel. Usng gven pont seuence and fou locaton specfed gadents, a lnea sstem becomes detemned. That s, the numbe of unknowns s eual to the numbe of euatons. The concept of a untc NURBS cuve geneaton usng the gven pont seuence and fou locaton specfed gadents ae llustated n Fg..,,, 3,, m -, m - ae gven ponts. d, d, d n-, and d n- ae the fou locaton specfed gadents. d d 3 4 d n - m -3 d n- m - Fg. Illustaton fo a untc NURBS cuve usng the gven pont seuence and fou locaton specfed gadents 3. Geneaton of a Quntc NURBS Cuve usng the Gven onts and Gadents In ths sub-secton, a NURBS cuve geneaton usng the gven ponts wth gadents s descbed. The concept of geneaton of a NURBS cuve usng the gven ponts wth gadents s llustated n Fg.. (9) m -

4 oceedngs of the 7th WSEAS Int. Conf. on Sgnal ocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 4-6, 7 4 d d d 3,,, 3, m -, m - s the gven pont seuence. d, d, d, d,, d, d ae gadents assgned to the 3 n- n- d 3 d n - 4 m -4 d d n - n - 3 m -3 m - d n - Fg. Concept of geneaton of a NURBS cuve usng the gven ponts wth gadents m - gven ponts n seuence. A NURBS cuve whch passes though the gven ponts and has the fst devatves at these gven ponts s geneated. A NURBS cuve s geneated b solvng E.(8) and E.(9) smultaneousl b makng m n E.(8) eual to n n E.(9). In ths case, the n E.(8) coesponds to the n E.(9). If the numbe of gven ponts wth gadents s 4, the numbe of NURBS cuve euatons (E.(8)) s 4 and the numbe of fst devatve euatons (E.(9)) s 4. As a lnea sstem, the total numbe of euatons s 8, wheeas the total numbe of contol ponts of a NURBS cuve s 8. Theefoe, ths lnea sstem s detemned. That s, the ank of a coeffcent mat of a lnea sstem s eual to the numbe of unknowns. The soluton to ths lnea sstem s eact. Howeve, n case the numbe of gven ponts wth gadents s 3, the numbe of euatons (E.(8)) whch pass though the gven ponts s 3, and the numbe of euatons of the fst devatve (E.(9)) s 3. In ths case, as a lnea sstem, the total numbe of euatons s 6, wheeas the numbe of contol ponts of the NURBS cuve s 7. That s, the numbe of euatons s less than the numbe of unknowns. Theefoe, ths lnea sstem s undedetemned []. Fo an undedetemned sstem, whle settng aula functon, the lnea sstem s solved unde the constant condton b selectng one soluton fom nfnte numbe of eact solutons usng Lagange's method of ndetemnate multples. In case the numbe of gven ponts wth gadents s, the numbe of euatons (E.(8)) s, and the numbe of euatons of the fst devatve (E.(9)) s. In ths case, as a lnea sstem, the total numbe of euatons s, wheeas the numbe of contol ponts of the NURBS cuve s 9. That s, the numbe of euatons eceeds the numbe of unknowns. Theefoe, ths lnea sstem s ovedetemned []. Fo an ovedetemned sstem, the dffeences between the ght and left sdes of all the euatons of the sstem ae mnmzed. The contol ponts calculated ae appomatons. Fo a sstem whee the numbe of gven ponts wth gadents s moe than, the lnea sstem s ovedetemned. Fo these sstems, n accodance wth the ncements of the dffeences between the numbe of euatons and the numbe of unknowns, the status of the appomaton wosens. 4 Cuve Shape Modfcaton based on the Specfed Radus of Cuvatue In ths secton, a method to modf a NURBS cuve shape accodng to the specfed dstbuton to ealze an aesthetcall pleasng feefom cuve s descbed. The concept of specfcaton and NURBS cuve shape modfcaton based on the specfed dstbuton s shown n Fg.3. A NURBS cuve and ts plots ae shown n Fg.3(a). A method to modf the shape of the NURBS cuve shown n Fg.3(a) to the cuve shown n Fg.3(b) s eamned. cuent NURBS cuve knot poston (a) cuent NURBS cuve and ts plots O pemete ˆ d specfed adus of cuvatue dstbuton of cuent NURBS cuve (c) dffeence between cuent and specfed shape modfed NURBS cuve (b) shape modfed NURBS cuve and ts plots knot poston O pemete specfed adus of cuvatue dstbuton (d) of shape modfed NURBS cuve and specfed (same as n (c)) Fg.3 Concept of specfcaton and NURBS cuve shape modfcaton based on the specfed dstbuton Consdeng the paamete of the NURBS cuve s dffeent fom the pemete of the cuve, the pemete of a NURBS cuve as a staght lne s set to the

5 oceedngs of the 7th WSEAS Int. Conf. on Sgnal ocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 4-6, 7 4 hozontal as, and the s set to the vetcal as as shown n Fg.3(c). Then, the adus of cuvatue dstbuton to the pemete s dawn. Afte ths, specfed s supemposed on the cuent dstbuton. Lnea, uadatc, cubc, uatc, untc, and s degee algebac functons ae appled as specfed adus of cuvatue to the cuent dstbuton to modf the shape of the NURBS cuve. To be moe n detal, coeffcents of the algebac functon ae calculated b ntoducng the least-suaes method usng the cuent adus of cuvatue dstbuton. Then, the s specfed b the detemned algebac functon. As an eample, the lnea algebac functon as a specfed specfcaton s shown n Fg.3(c). The th of dstbuton of a pemetcall epesented NURBS cuve s denoted as, the specfed at the same spot s denoted as ˆ, the dffeence d s shown b E.() and s llustated n Fg.3(c). d ( ˆ,,,,, = n- n-) - () Whee =,,,, m -, m s the numbe of specfed, and n s the numbe of NURBS cuve segments plus, whch s the degee of the cuve. S(,, n-,,, n- ) whch s the sum of the suaed dffeences fo all specfed adus of cuvatues n E.() s mnmzed b ntoducng the least-suaes method. The epesson s non-lnea. Theefoe, b Talo's theoem, E.() s lnealzed as n E.(). S(,,,,, ) n- n- m- å é ë ( ˆ,, n-,,, n-) ù û = + D n- + D n- + D n- + D n- = - S(,,,,, ) m- é = å ê (,, n-,,, n-) + D + = ë ù + D ˆ n- + D + + D n- - ú n- n- û E.() s mnmzed b euatng to zeo all the patal devatves of S( + D,, + n - Dn-,, +D, + - D wth espect to D and D ( =,,, n ) n - n - ) as n E.(3). S D S D = ( =,,, n - ) = ( =,,, n - ) () () (3) Usng these smultaneous lnea euatons, D and D ( =,,, n - ) ae calculated. Then, and ae detemned. Coelaton Matchng fo Smlat Evaluaton In ths secton, coelaton matchng fo smlat evaluaton s descbed. Two NURBS cuves ae shown n Fg.4(a) and 4(b). These cuves can hadl be dstngushed b just lookng at the gaphs. But f the plots ae dawn fo both, the dffeence between the two cuves s ecognzed mmedatel as shown n Fg.(a) and (b). Radus of cuvatue s plotted usng staght lnes dawn outwad fom and pependcula to the cuve, wth the lne length popotonal to the amount of at that spot. Cuve shape s judged b lookng at the lnes comng out fom the cuve and seeng how the lengths change along the path, not along the paamete. Theefoe, to the pemete s dawn to evaluate the smlat of the cuve shape as shown n Fg.6(a) and 6(b). (a) cuve A (b) cuve B ont maks ndcate knot poston. (a) cuve A Fg.4 Two NURBS cuves (b) cuve B ont maks ndcate knot poston. Fg. Two NURBS cuves wth plots pemete c n (a) dstbuton fo cuve A pemete (b) dstbuton fo cuve B Fg.6 Radus of cuvatue dstbuton of two NURBS cuves Radus of cuvatue dstbuton s used as an altenatve chaactestc of the shape of the cuve to evaluate the ualt of a desgned cuve. To adjust the vaous lengths of the cuve pemetes, the total length of the pemetes and ae cˆn

6 oceedngs of the 7th WSEAS Int. Conf. on Sgnal ocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 4-6, 7 43 escaled as. A pemete must be calculated accodng to the knot seuence of the knot vecto. Dscete values cn ( n =,, 3,, m) shown n Fg.6(a) whch ae to the pemete ae consdeed as the components of m dmensonal vecto fo cuve A, denotng a. In the same manne, dscete values cˆn shown n Fg.6(b) ae consdeed as the components of m dmensonal vecto fo cuve B, denotng b. Smlat between cuve A and cuve B s evaluated b E.(4). S = a b a b (4) Because cuve pemete and ae escaled as, and pemete s calculated accodng to the knot seuence of the knot vecto, the smlat s evaluated ndependent of locaton, oentaton such as otaton and eflecton, and the sze of the cuves. The evaluated smlat between cuve A and cuve B shown n Fg.6 s Fa Cuve Epesson and Evaluaton of Faness A cuve wth a monotone dstbuton s consdeed as a fa cuve n the aea of Compute Aded Aesthetc Desgn. But no offcal standads ae gven. Theefoe, cteon fo a fa cuve s poposed tentatvel. The shape of a NURBS cuve s defned b the numbe, the locaton of ts contol ponts, and the knot seuence of the knot vecto. The desgned cuve s consdeed fa f the vaaton of s monotone fo the same numbe of contol ponts and knot seuence of the knot vecto. In ths secton, fa cuve epesson and evaluaton of faness ae descbed. As a measue of cuve faness evaluaton, dstbuton s used as an altenatve chaactestc of a cuve. Fst, adus of cuvatue dstbuton of the desgned cuve s evaluated to eamne the faness of the cuve. Algebac functons such as lnea, uadatc, cubc, uatc, untc, and s degees ae appled to the dstbuton of the desgned cuve as the specfed. Then applng the cuve shape modfcaton algothm based on the specfed dstbuton, the adus of cuvatue dstbuton s modfed accodng to these s algebac functons espectvel. These dstbutons gven b these s algebac functons ae consdeed monotone, because the ndependent vaable of these algebac functons s monotone to the coespondng dependent vaable of these functons. Theefoe, cuves desgned n ths manne ae consdeed fa. These judgements ae pefomed b evaluatng the smlat technue descbed n the pevous secton. Ths smlat s evaluated b usng the dstbuton accodng to s algebac functons used as efeences and the dstbuton of the desgned cuve as a match. It s consdeed that the hghest smlat eveals that ths cuve s desgned to have ths dstbuton, and the smlat measued s consdeed as the faness of the desgned cuve. As an eample of a fa cuve geneaton and faness evaluaton, a NURBS cuve and ts dstbuton to the pemete ae shown n Fg.7. pemete NURBS cuve Fg.7 Desgned cuve and ts dstbuton pemete Fg.8 Radus of cuvatue dstbuton and s degee algebac functon to specf Algebac functons mentoned above ae appled to the dstbuton shown n Fg.7 b usng the least-suaes method. These s functons ae detemned as the specfed. These ae shown n Fg.8 togethe wth the adus of cuvatue dstbuton shown n Fg.7. Applng the cuve shape modfcaton algothm based on these s algebac functons to the desgned cuve, the shape of the cuve s modfed. Aftewads, settng these s dstbutons as efeences and the dstbuton of the desgned cuve as a match, s smlates ae evaluated. The smlates evaluated ae summazed n Table. In Table, smlat epesses the faness of the cuve. Fom Table, the desgned cuve whose adus of cuvatue s shown n Fg.7, s judged to be desgned so that the dstbuton wll be s degee. And ths measued smlat epesses the faness of the desgned cuve. s degee algebac functon

7 oceedngs of the 7th WSEAS Int. Conf. on Sgnal ocessng, Computatonal Geomet & Atfcal Vson, Athens, Geece, August 4-6, 7 44 algebac functon adus of cuvatue shown n Fg.7 Table Faness of the desgned cuve lnea uadatc cubc uatc untc s Concludng Remaks A untc NURBS cuve, the fst devatve of a untc NURBS cuve, cuvatue vecto, cuvatue, and ae epessed. The elaton of cuvatue and ae nveted. Theefoe, even f the to the pemete s lnea, the cuvatue dstbuton to the pemete s non-lnea. The s useful to undestand the ualt of the desgned cuve vsuall. The concept of specfcaton to modf the shape of a NURBS cuve s llustated. The dffeence between the NURBS cuve adus of cuvatue and the specfed s mnmzed b ntoducng the least-suaes method to modf the shape of the NURBS cuve. Algebac functons such as lnea, uadatc, cubc, uatc, untc, and s degees ae appled to the dstbuton of the desgned cuve as the specfed. The adus of cuvatue dstbutons gven b these s algebac functons ae consdeed monotone, because the ndependent vaable of these algebac functons s monotone to the coespondng dependent vaable of these functons. Smlat s evaluated usng the dstbuton accodng to s algebac functons as efeences and the dstbuton of the desgned cuves as matches b usng coelaton matchng. The values of to the pemete ae consdeed as the components of a mult dmensonal vecto fo the cuve. Smlat between two cuves s epessed b nomalzng the dot poduct of two vectos. Cuve shape smlat evaluaton s ted usng an eample. Consdeng that a cuve wth a monotone vaaton of dstbuton s fa, the smlat of the desgned cuve to a fa cuve s evaluated. Ths measued smlat epesses the faness of the desgned cuve. Usng ths technue, the faness of a cuve s evaluated b usng the smlat of the adus of cuvatue dstbuton. In the futue, we ae plannng to establsh a defnton of a fa cuve usng a lot of cuve data that wll be gatheed. Refeences: [] C. Wene Dankwot and Ged odehl, A New Aesthetc Desgn Wokflow-Results fom the Euopean oject FIORES, CAD Tools and Algothms fo oduct Desgn, Spnge-Velag Beln (), pp.6-3. [] H. G. Buchad, J.A. Aes, W.H. Fe, and N.S. Sapds, Appomaton wth Aesthetc Constants, Desgnng Fa Cuve and Sufaces, SIAM, (994), pp.3-8. [3] W.H. Fe, D.A. Feld, Desgnng Béze conc segments wth monotone cuvatue, Compute Aded Geometc Desgn, 7, (), pp [4] M. Kuoda, M. Hgash, T. Satoh, Y. Watanabe, T. Kuagano, Intepolatng cuve wth B-splne cuvatue functon, Mathematcal Methods fo Cuves and Sufaces Ⅱ, Vandeblt Unvest ess, Nashvlle, TN, (998), pp [] H. Guan, T. To, Modfng cuvatues at desgn ponts fo conve B-splne cuves, IEEE, (997), pp.3-9. [6] J. olakoff, Y. Wong,. Thomas, An automated cuve fang algothm fo cubc B-splne cuves, Jounal of Computatonal and Appled Mathematcs,, (999), pp [7]J.A.. Kjellande, Smoothng of cubc paametc splnes, Compute Aded Desgn,, 3, (983), pp [8] C. Zhang,. Zhang, F. Cheng, Fang splne cuve and suface b mnmzng eneg, Compute Aded Desgn, 33, (), pp [9] X. Yang, G. Wang, lana pont set fang and fttng b ac splnes, Compute Aded Desgn, 33, (), pp [] W. L, S Xu, J. Zheng, G. Zhao, Taget cuvatue dven fang algothm fo plana cubc B-splne cuves, Compute Aded Geometc Desgn,, (4), pp [] M.J. Atallah, A lnea tme algothm fo the Hausdoff dstance between conve polgons, Infomaton ocessng Lettes, vol.7, (983), pp.7-9. [] D.. Huttenloche and K. Kedem, Computng the mnmum Hausdoff dstance fo pont sets unde tanslaton, oceedngs of the sth annual smposum on computatonal geomet. ACM ess, (99), pp [3] H. Alt and M. Godau, Computng the Féchet dstance between two polgonal cuves, Intenatonal Jounal of Computatonal Geomet and Applcatons. vol., Nos. and, Wold Scentfc ublshng Compan, (99), pp.7-9. [4] H. Alt, C. Knaue and C. Wenk, Matchng polgonal cuves wth espect to the Féchet dstance, oceedngs 8th Intenatonal Smposum on Theoetcal Aspects of Compute Scence, (), pp [] D.J. Lee, S. Antan and L.R. Long, Smlat measuement usng polgon cuve epesentaton and Foue Descptos fo shape-based vetebal mage eteval, oceedng of SII Medcal Imagng: Image ocessng, San Dego, CA, SIE,vol.3, at 3, (3), pp [6] E. esoon and K.S. Fu, Shape dscmnaton usng Foue Descptos, IEEE Tansactons on Sstems, Man, and Cbenetcs, vol.smc-7, no.3, (977), pp [7] C. de Boo, On calculatng wth B-splne, Jappo. Theo, 6(), (97), pp.-6. [8] D. Mash, Appled Geomet fo Compute Gaphcs and CAD, Spnge-Velag, (), pp.88. [9] Eugene V. Shkn.: Hand Book and Atlas of CURVES. CRC ess Boca Raton, Floda, (99), pp.9. [] T. Kuagano, Y. Amtsu, A. Yamaguch, A method to geneate a fa cuve wth specfed cuvatue dstbuton fo Compute Aded Aesthetc Desgn, The th Wold Mult-Confeence on Sstemcs, Cbenetcs and Infomatcs, Vol.III, (6), pp [] W. Boehm, and H. autzsch, Geometc Concepts fo Geometc Desgn, (994), pp.6-7. [] G. Fan, and D. Hansfod, actcal Lnea Algeba A Geomet Toolbo-, (4), pp.3-4.