CONVERSIONS BETWEEN PARAMETRIC and IMPLICIT FORMS for COMPUTER GRAPHICS and VISION

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1 Proceedis of ISCIS 999, Kuşadası, Turke, pp CONVERSIONS BETWEEN PARAMETRIC ad IMPLICIT FORMS for COMPUTER GRAPHICS ad VISION * Cem ÜNSALAN ** Aül ERÇĐL *Boğaziçi Uiversi, Dep. of Elecrical & Elecroics Eieeri usala@bou.edu.r **Boğaziçi Uiversi, Dep. of Idusrial Eieeri ercil@bou.edu.r ABSTRACT: Sice parameric ad implici forms have complemear advaaes wih respec o cerai eomeric operaios, i ca be useful o cover from oe form o he oher. A ew mehod based o polar/spherical coordiae represeaios is iroduced o cover parameric form of a curve o a correspodi implici form ad vice versa. Kewords: Curves i plae, curves i space, parameric form, implici form, polar/spherical coordiaes. Iroducio The developme of Compuer Aided Graphics Desi CAGD) has see wo disic approaches for represei surfaces i D space:. Parameric mehods wih a represeaio of he form u,v),u,v), zu,v)), which maps a D domai coaii u,v) o D space.. Implici mehods ha defie a surface as a se of pois {,,z) such ha F,,z) = } The use of parameric surfaces has bee quie successful for he eeral represeaio ad desi of free-form surfaces ad remais domia i compuer raphics ad eomeric modeli. The implici approach is philosophicall more closel relaed o he coceps of Cosrucive Solid Geomer CSG) ad solid modeli ad is receivi icreased aeio. Implici surface fucios aurall describe he ierior of a objec, whereas a parameric descripio of a objec usuall cosiss of piecewise surface paches. Boh approaches have lo liss of pros ad cos []. Alhouh he parameric formulaio is useful for raci, rederi ad surface fii, ma operaios like surface iersecio desire oe of he surfaces o be represeed implicil. Moreover, he implici represeaio ca be used for esi wheher a poi lies o he boudar ad o represe a objec as a semi-alebraic se ad implici forms are fidi wider applicaios i compuer visio, mail i he area of objec recoiio [,,,5] Sice parameric ad implici forms have complemear advaaes wih respec o cerai eomeric operaios, i ca be useful o cover from oe form o he oher. Coversio from parameric o implici form is kow as impliciizaio ad ever raioal surface ad curve ca be represeed implicil as he zero se of a irreducible homoeeous polomial f,,z,w)= for surfaces, ad f,)= for D curves [6]. If a chae of variables cao reduce he deree of he polomial epressio ha i is assumed o be irreducible

2 Proceedis of ISCIS 999, Kuşadası, Turke, pp Sederber [6] applies resulas o elimiae parameers from polomials; Hoffma [7] deails he use of he Gröber bases for he same purpose; ad Hoffma [8] describes he Wu-Ri mehod. The coversio from implici o parameric form is kow as parameerizaio. Parameerizaio is o alwas possible, however; for eample, implici surfaces ha are defied b cerai polomials of fourh ad hiher deree cao be parameerized b raioal fucios [9]. Coversio is alwas possible for odeeerae quadrics ad for cubics ha have a siular poi. I his paper a ew mehod based o polar coordiae represeaios i D ad spherical coordiae represeaios i D are iroduced o cover parameric form o a correspodi implici form. Parameric form is represeed i polar or spherical coordiaes ad coversio is achieved. Also a ew mehod is iroduced o cover implici form o a correspodi parameric form. I his mehod polar ad spherical coordiae represeaios are used. The laou of he paper is as follows. I secio wo, coversio from parameric o implici form is ive for D ad D curves. I secio hree, coversio from implici o parameric form is ive for D ad D surfaces. Coclusios are ive i secio four.. Coversio from Parameric Form o Implici Form There are hree kow echiques for impliciizaio of parameric forms. All of hese echiques reduce he problem of impliciizi raioal surfaces o elimiai wo variables from hree parameric equaios. I eeral, i is believed ha echiques based o elimiaio heor ca resul i eraeous facors alo wih he implici represeaio ad heir separaio ca be a difficul ask. Furhermore, hese alorihms are o able o impliciize parameric surfaces like bicubic paches i a reasoable amou of ime ad space [9]. A ew mehod based o polar represeaio for D curves ad spherical represeaio for D surfaces is iroduced o cover a parameric form o a correspodi implici form i his secio. The echique is illusraed wih several eamples.. Coversio i D I his secio a mehod based o polar coordiae rasformaios will be used o carr ou implici o parameric coversio i D. The mehod is oriiaed from defii slope fucio of a curve i parameric form ad equai his form o is implici correspodece. Usi sadard chae of variable formulas from recaular o polar coordiae represeaios, impliciizaio is simpl achieved. The followi heorem ives mahemaical basis for his coversio mehod. Theorem : Le α be a paramerized curve i R s.. α : a, b) R wih α ) = h ), )), where a, b) is a ope ierval i R. The implici represeaio of his curve is ive as: h f )) + f )) = + ) ) where f ) is a ijecive fucio of he form f ) = h )

3 Proceedis of ISCIS 999, Kuşadası, Turke, pp Proof: See [] Eample : Le h ) = si ad ) = si cos f ) = cos = cos ) The correspodi implici form will be: + = si cos ) + si cos ) cos cos ) Simplifi his equaio will ield: + = The parameric curve for his eample is ploed i Fiure. The correspodi implici form obaied usi heorem is ploed i Fiure Fiure. Plo of he parameric Fiure. Plo of he implici curve of eample. curve of eample.. Coversio i D The mehod preseed i he previous secio is applied o curves i D i his secio. The differece is ha i D we will have wo slope fucios for a space curve. The same equaliies will be used for hese wo slope fucios. Usi sadard chae of variable formulas from caresia o spherical coordiae represeaios, impliciizaio is simpl achieved. The followi heorem ives mahemaical basis for his coversio mehod. Theorem : Le α be a paramerized space curve i R s.. α : a, b) R wih α ) = h ), ), k )), where a, b) is a ope ierval i R. The implici represeaio of his curve is ive as: z h f )) + f )) + k f )) = + + z ) + where f ) is a ijecive fucio of he form ) f ) = h ) ad f ) is a ijecive fucio of he form k ) f ) = ) + h ) Proof: See [] Eample : If we have a space curve, represei a clidrical spiral i D ive as α ) = h ), ), k )).

4 Proceedis of ISCIS 999, Kuşadası, Turke, pp h ) = a cos ) = a si k ) = si f ) = = arca ; cos + + z = a cos + si ) + which simplifies o + = a f ) = a = az +.Coversio from Implici Form o Parameric Form I his secio, a ew mehod is iroduced o cover a implici form o a correspodi parameric form. This mehod basicall depeds o rewrii implici polomial i polar form for D ad spherical form i D ad solvi he radius i erms of ales. For his reaso he proposed mehod is applicable o a deree of polomials, where roos of he polomial ca be eplicil obaied i alebraic form. I his paper, implici forms up o sum of five successive homoeeous polomials are ive.. Coversio i D I his secio, coversio from implici form o parameric form is sudied. The coversio is based o usi he polar coordiae represeaio of he implici polomial. m m) m k m) Theorem : A polomial of he form a b = ) m k + sum of wo homoeeous polomials) ca be covered o parameric form α θ ) = h θ ), θ )) as: h θ ) θ ) k m k m) bm cos θ si θ = k cosθ ) m m) a cos θ si θ k m m k m) bm cos θ si θ = k si θ 5) m m) a cos θ si θ m Parameer values o obai real valued parameric form, specifies parameer rae. Proof: See [] I he followi eamples, implici forms obaied i secio. are cosidered. To show ha covered forms correspod o he same parameric curves, suiable reparamerizaios are used for each eample. Eample : Cosider he implici curve + = Subsiui he polar o recaular coordiae coversio formulas for ad, ad solvi for r, we obai m

5 Proceedis of ISCIS 999, Kuşadası, Turke, pp r cosθ ) + r si θ ) r cosθ ) r si θ ) = cosθ si θ r = cos θ + si θ cos θ si θ h θ ) = cos θ + si θ cosθ si θ θ ) = θ [, π ] cos θ + si θ If we reparamerize his equaio usi = a We obai h ) = ) = which is he same parameric form + + ive i Eample. m l m) Theorem 5: A polomial of he form a = 6) l l = hree homoeeous polomials of cosecuive derees) ca be covered o wo parameric forms as: α ) = h ), )) 7) α ) = h ), )) 8) θ ml p p q p p q h θ ) = cosθ 9) θ ) = si θ ) s s p + p q p + p q h θ ) = cosθ ) θ ) = si θ ) s s where s = q = a m a m m m) θ θ ) p = a m cos si m m) θ θ 5) cos si m m) θ θ ) cos si Parameer values o obai real valued parameric form, specifies parameer rae. Proof: See [] Eample : + + = r rcosθ + si θ ) + = To fid real valued roos of his equaio, he followi cosrai has o be saisfied cosθ + si θ ) which is saisfied for θ, π The soluios of he above equaio are: r = cosθ + si θ si θ r = cosθ + si θ + si θ ad he correspodi wo parameric curves are obaied as: h θ ) = cosθ θ ) = si θ r r

6 Proceedis of ISCIS 999, Kuşadası, Turke, pp h θ ) = r cosθ θ ) = r si θ θ, π The implici polomial form of he curve ive i his eample is ploed i Fiure 5. Correspodi wo parameric forms obaied from heorem 5 are ploed i Fiure 6 ad Fiure Fiure 5. Plo of he implici form of curve ive i eample Fiure 6. Plo of he firs par of Fiure 7. Plo of he secod par parameric curve i eample of parameric curve i eample m l m) Theorem 6: A polomial of he form a = 6) l l = four homoeeous polomials of cosecuive derees) ca be covered o hree parameric forms as: α ) = h ), )) 7) α ) = h ), )) 8) ) = h ), ml θ ) A B) cos α )) 9) h = + θ ) A + B A B θ ) = A + B) si θ ) h θ ) = + ) cosθ ) A + B A B θ ) = + ) si θ ) A + B A B h θ ) = ) cosθ ) A + B A B θ ) = ) si θ 5) where 5

7 Proceedis of ISCIS 999, Kuşadası, Turke, pp b b a b b a = 6) B = + 7) 7 7 A + + q p p pq u a = ) 8) b = ) 9) s s 7 s s s s, p, q are defied i equaios o. u = a m m m) θ θ ) cos si Parameer values o obai real valued parameric form, specifies parameer rae. Proof: See[] m l m) Theorem 7: A polomial of he form a = ) l l = five homoeeous polomials of cosecuive derees) ca be covered o four parameric forms as: α ) = h ), )) ) α ) = h ), )) ) ) = h ), ml ) = h ), α )) ) α )) 5) p R D h θ ) = + + ) cosθ 6) θ ) si θ p R D ) = + + 7) p R D p R D h θ ) = + ) cosθ 8) θ ) = + ) si θ 9) p R E p R E h θ ) = + + ) cosθ ) θ ) = + + ) si θ ) p R E p R E h θ ) = + ) cosθ ) θ ) = + ) si θ ) Le l be a roo of he equaio q pu p v qv u l l + v) l + = ) s s s s s p q R = + l 5) s s p q pqs 8us p = R + 6) s R D p q pqs 8us p E = R 7) s R s, p, q are defied i equaios o. u is defied i equaio. v = a m m m) θ θ 8) cos si Parameer values o obai real valued parameric form, specifies parameer rae. Proof: See [] 6

8 Proceedis of ISCIS 999, Kuşadası, Turke, pp Coversio i D The mehods applied i he previous secio are also applicable i D [].. Coclusios I his paper, coversio mehods bewee parameric implici forms of curves i D ad D are cosidered. The coversio bewee hese wo forms is impora because each form has advaaes ad disadvaaes. Havi a coversio formula bewee hese wo forms allow us o use advaaes of boh forms a he same ime. A ew mehod is iroduced o cover a parameric form of a curve o he correspodi implici form. The mehod depeds o fidi slope fucio of he curve i parameric form ad implici form ad equai hem ad usi sadard chae of variable formulas from caresia o polar/spherical coordiaes. The sreh of his mehod over eisi mehods is is simplici. Coversio from implici form o a correspodi parameric form is also iroduced. This coversio mehod also depeds o polar/spherical coordiae represeaios ad fidi roos of radius fucio. The mehod is applicable o a deree of polomial ha roos of he polomial ca be obaied eplicil. Coversio formulas iroduced i his paper are applicable for broad rae of fucios i parameric form ad implici form. Oe mai advaae of hese wo coversio mehods is ha, heir ease of usae. Refereces. J. Bloomehal, ed. Iroducio o Implici Surfaces, Sa Fracisco, Mora Kaufma Publishers, Ic., H. Çivi, C. Chrisopher, A. Erçil, The classical heor of ivarias ad objec recoiio usi alebraic curves ad surfaces Techical Repor FBE-IE-6/97-7, Boğaziçi Uiversi, 997,. M. Heber, J. Poce, T. Boul, A. Gross, eds. Objec Represeaio i Compuer Visio. Sprier Lecure Noes i Compuer Sciece Series. Sprier-Verla, 995. J. Subrahmoia, D. Cooper, D. Kere, Pracical Reliable Baesia Recoiio of D ad D Objecs Usi Implici Polomials ad Alebraic Ivarias Techical Repor LEMS-7. Divisio of Eieeri, Brow Uiversi, Jue 99.W. H. Beer, CRC Sadard Mahemaical Tables, Florida, CRC Press: G. Taubi ad D. B. Cooper, Recoiio ad Posiioi of D Piecewise Alebraic Objecs Usi Euclidea Ivarias, IBM Research Divisio, T.J. Waso Research Ceer, T. W. Sederber, D. C. Aderso ad R. N. Goldma, Implici Represeaio of Parameric Curves ad Surfaces i Compuer Visio, Graphics ad Imae Processi, pp. 7-8, Academic Press, C. Hoffma, Geomeric ad Solid Modeli: A Iroducio, Sa Fracisco, Mora Kauffma, C. Hoffma, "Implici Curves ad Surfaces i Compuer-Aided Geomeric Desi", IEEE Compuer Graphics ad Applicaios, V., No., 99, pp D. Maocha ad J.F. Ca, Implici Represeaio of Raioal Parameric Surfaces, Joural of Smbolic Compuaio, Vol., pp. 85-5, 99.. W. H. Beer, CRC Sadard Mahemaical Tables, Florida, CRC Press: 979. Üsala, C. ad Erçil, A. Coversios bewee Parameric ad Implici Forms usi Polar/Spherical Coordiae Represeaios, Boğaziçi Uiversi Research Repor, FBE-IE-9/98-, 998 7