The Control Vector Scheme for Design of Planar Primitive PH curves

Transcription

1 PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND ECHNOLOGY VOLUME MAY 7 ISSN he Cotrol Vector Scheme for Desg of Plaar Prmtve PH curves Chg-Shoe Chag, Sheg-Hs sa, ad James Che Abstract he PH curve ca be costructed by gve parameters, but the shape of the curve s ot so easy to mage from the value of the parameters. O the cotract, Bézer curve ca be costructed by the cotrol polygo, ad from the cotrol polygo, we ca mage the fgure of the curve. I ths paper, we wat to use the hodograph of Bézer curve to costruct PH curve by selectg part of the cotrol vectors, ad produce other cotrol vectors, so the property of PH curve exsts. Keywords PH curve, hodograph, Bézer curve. I. INRODUCION HE curves ad surfaces computer-aded geometrc desg ca be represeted ad desged varous ways. he curves ad surfaces ca be represeted parametrc form, or mplct form. Some of the curve, such as Bézer curve, ca be represeted by ther cotrol pots. Wth smart graphc user terface, the desg of the curve ca be user fredly wth poter devce, such as mouse, so that the cotrol pots ad the curves ca be dsplay amatedly. he hodograph of a plae parametrc curve r(t)=(x(t),y(t)) s the locus descrbed by the frst dervatve r (t)=(x (t),y (t)) of the curve. A polyomal parametrc curve s sad to have a Pythagorea hodograph f there exsts a polyomal σ(t) such that x (t)+y (t)=σ (t)[]. Wth ths property, the polyomal PH curve has ratoal offset [,3,4,5,6]. he polyomal PH curve ca be represeted Berste- Bézer form, so the cotrol pots of the polyomal PH curve ca be dsplayed. However, we would le to prevew the appearace of the curve before we desg t. We ow the polyomal PH curve must be a Bézer curve, ad ot vce versa. I ths paper, we propose 3 algorthms to fd polyomal PH curves by gvg part of the cotrol vectors of the hodographs, ad aalyss the umber of the PH curves we fd each algorthms. Mauscrpt receved March, 7. hs wor was supported part by the NSC awa Grat NSC 95--E-3-3-MY. C. S. Chag s a professor the Departmet of Computer ad Iformato Scece, Soochow Uversty, awa (phoe: ext. 3; e-mal: chag@cs.scu.edu.tw). S. H. sa s a graduate studet the Departmet of Computer ad Iformato Scece, Soochow Uversty, awa. James C. Che s a Assocate Professor the Departmet of Idustral Egeerg at Chug Yua Uversty, awa. here are 5 sectos ths paper. he frst secto gves the troducto of the polyomal PH curve ad the cotets of ths paper. he secod secto defes the otato ad heorem used ths paper. he thrd secto explas the degree of freedom aalyss, gves more heorems, ad proposes 3 algorthms, the fourth secto gves expermet results, ad the fal secto cocludes the result. II. DEFINIION AND NOAION We follow the deftos ad heorems the referece [], ad lst below: Defto []: he hodograph r (t)=(x (t),y (t)) of a plaar PH curve satsfy x (t)+y (t)=σ(t), where σ (t) s a polyomal. heorem (Kubota[7,]) hree real polyomals a(t), b(t), ad c(t), where max[deg(a), deg(b)]=deg(c)>, satsfy the Pythagorea codto a (t)+b (t)=c (t) f ad oly f they ca be expressed terms of real polyomals u(t), v(t), ad w(t) the form: a(t)=w(t)[u (t)-v (t)] b(t)=w(t)u(t)v(t) c(t)=w(t) [u (t)+v (t)] heorem []: he polyomal curve correspodg to the hodograph r (t)=(x (t),y (t))=(w(t)[u (t)-v (t)],w(t)u(t)v(t)), s of degree =+λ+, where λ=deg(w) ad = max(deg(u),deg(v)). ag w(t)= ad gcd(u(t),v(t))= gves a prmtve polyomal PH curve. We are cocered prmtve polyomal PH curves ths paper ad deote the prmtve polyomal PH curve as PPH curves. Let r(t)=(x(t),y(t)) be a PPH curve whose cotrol pots are p =(x,y ), =,,, +, ad whose hodograph has cotrol vectors Δp =p + -p, =,,,. hs PPH curve s costructed by two polyomals u(t) ad v(t) gve ub Berste-Bézer form as: u(t)= vb, where B (t) = C t (-t) µ-, C = ad v(t) =!.!( )! PWASE VOLUME MAY 7 ISSN WASE.ORG

2 PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND ECHNOLOGY VOLUME MAY 7 ISSN We deote r =[u,v ], N(,)=(I,,J, )= ( u u vv, uv + uv ), ( ) C C ϕ, =, the we have the C followg heorem: heorem 3: he PPH curve costructed by two polyomals u(t) ad v(t) gve Berste-Bézer form as: u(t)= ub vb ad v(t)=, the the relatoshp betwee r ad cotrol vectors of the hodograph s: Proof: (+)Δp = ϕ (, ) (I,-,J,- ), () B m B m C C m+ C+ B m+ + From the equato =, we derve: u = u B u B = = C C = uu B+ = C+ C C = uu B = C From the defto of the hodograph of PPH curve, r (t)=(x (t),y (t))=(u (t)-v (t), u(t)v(t)), we have '( ) ( ) ( ) ( ) C C x t = u t t uu v v = B = C Wth smlar approach, we derve: '( ) ( ) ( ) ( ) C C y t = u t v t = u v u v B +. = = C Compare wth the property that r (t) has the cotrol vectors at (+)Δp, =,,,,, we prove ths heorem. Notce that φ (,)=φ (,-),N(,)=N(,),so we ca smplfy the Equato () to: ( u,u v ) Δp ( u u v, u v + u v ) Δp + Δp ( ( u u v v ) 3 ( u v ), ( u v + u v ) + 6 ( u v )) ' = 7Δ p = ( ( u u v v ) ( u u v v ), ( u v u v ) (( u v + u v ))) Δp 4 ( ( u u v ) + 3 ( u ), ( u v + u v ) + 6 ( u v )) Δp u u v, u v + u v ) Δp ( u,u v ) C (3) he cotrol vectors ca be foud drectly by u(t) ad v(t). It s ot easy to predct the geometrc varato of the PH curve through u(t) ad v(t). If we ca costruct the PH curve from the cotrol vectors or the hodograph of the curve, we ca establsh a smart graphc user terface, request for cotrol pots or cotrol vector for the PPH curve, ad fd oe or more sets of r =[u,v ],,,..,, t s helpful for us to desg a PH curve. he problems are: () How may cotrol pots (of the hodograph) we eed to costruct a PH curve? () How to fd at least oe set of r =[u,v ],,,..,, satsfed the requremet of PH curve. III. CONVERSION FROM PPH CURVE O BÉZIER CURVE We aalyze the degree of the freedom of the PPH curve frst, ad the gve a algorthm to costruct the assocated PH curve ths secto. A Aalyss of the Degree of Freedom We would le to ow how may cotrol vectors we eed to costruct the PPH curve. Cosder equato (), there are 6+4 varables the system of equatos, cludg r =[u,v ], ad Δp =(Δx, Δy ),. So the degree of freedom for the system of equatos s 6+4.here are + equatos for Δx,, ad the other + equatos for Δy,, so we have 4+ costrats for ths system of equatos. Before we wat fd all solutos, we eed to gve values to (6+4)-(4+)=+ varables. he atural selecto for these + varables s r =[u,v ],, t covert the PPH curve to Bézer curve. Or, we ca select + cotrol vectors amog Δp,=(Δx, Δy ),, ad fd other cotrol vectors wth r =[u,v ],. = (+)Δp / (, ) N(,-)+φ (,/)N(/,/) f s eve ϕ ( ) / = ϕ (, ) N(,-) f s odd. () B. heorems for the Node Value heorem 4: he solutos for u -v =a, uv=c, (a,c) (,) s: (u,v) =± (, a ) f a<, c= =± ( a, ) f a>, c= c =± ( a + c, a a a + c ) f c Proof: rval. From heorem, the relato betwee r =[u,v ] ad Δp ca be show. For example, whe = 3, we have Notce that we are worg o the real umber, so that there are two solutos for each case. hese solutos are two couples ot equal to (,) wth dfferet sg. We deote ths two solutos as (u,v) ad (u,v). Notce that (u,v) (,). Notce further that whe a=, u=v=c=. Both (u,v) ad -(u,v) always produce the same (a,c). PWASE VOLUME MAY 7 ISSN WASE.ORG

3 PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND ECHNOLOGY VOLUME MAY 7 ISSN heorem 5: Let N(,)=(I,, J, )=(a,c) (,), N(,)=(I,,J, )=(b,d) (,), the (I, -J,, I, J, )=(ab-cd, ad+bc) (,). Proof: It s easy to prove ((I, ) (J, ), I, J, )=(ab-cd, ad+bc) by carefully algebrac computato. Now, we eed to prove (ab-cd, ad+bc) (,). Assume (ab-cd, ad+bc)=(,). Let s treat b,d as varables, a,c as costats, the system of equato c has soluto (b,d)=( a a c a / c a, c a c / c a ) =(,) f a +c, whch cotradcts wth (b,d) (,). If a +c =, t cotradcts (a,c) (,). So we proved (ab-cd, ad+bc) (,). Combe heorem 4 ad 5, we have the followg corollary: Corollary : Gve N(,) ad N(,), the vectors assocated to the solutos of N(,) are two couples wth opposte sg. Proof: he proof s smlar to heorem 6. From the fact that N(,)=N(,), N(,)=N(,), more combato ca be geerated. he value for N(,) ca be geerated from () N(,), N(,) ad N(,), () N(,), N(,) ad N(,), (3) N(,), N(,) ad N(,). he graphs for the put/output odes ad umber of solutos for heorem 7 are show Fg. (d),(e),(f) respectvely. (a) corollary (b) heorem 6, < (c) corollary, < heorem 6: Let N(,) =(a,c) (,), N(,)=(e,f) (,), the N(,)= ((a,c) (e -f,ef), (-c,a) (e -f,ef)) (,). he a + c otato s the er product of two vectors. Proof: assume N(,)=(b,d), from heorem 5, we have ab-cd=e -f ad cb+ad=ef, from these two equatos wth b,d uow, we ca easly fd oe soluto for (b,d). Now we eed to prove (b,d) (,). hat s, ((a,c) (x,y), (a,-c) (y,x)) (,) where x= e -f, y=ef. Assume (b,d)=(,), we eed to solve ax+cy= ad cx+ay=. Because a +c, the system of equatos has soluto (x,y)=(e -f, ef)=(,). It mples (e,f)=(,) ad cotradcts wth (e,f) (,). So we proved N(,)=(b,d) (,). Because N(,)=N(,), the heorem ca be easly modfy as: (d) heorem 7, << (e) heorem 7, << Corollary : Let N(,) =(a,c) (,), N(,)=(e,f) (,), the N(,)= ((a,c) (e -f,ef), (-c,a) (e -f,ef)) (,). a + c Let s gve some graphs to deote the put/output ode for our heorems ad corollares. I corollary, we draw correspod graph Fg. (a), usg arrow to dcate put ode N(,) ad N(,) pots to the output ode N(,). he ode N(,) Fg. s the output ode for correspod heorems ad corollares, ad other odes whch has arrow pot out are put odes. he graphs for heorem 6 ad corollary are show Fg. (b) ad (c) respectvely. Curretly, we use two put odes to fd oe output ode. he followg heorems ad corollares use three put odes to fd oe output odes. heorem 7: Let N(,)=(e,f) (,), N(,)=(a,c) (,) ad N(,)=(b,d) (,), the (f)heorem 7, << Fg. I/O odes for heorems ad corollares Notce that whe =<, Fg. (d) degeerate to Fg. (b), whe <=, Fg. (e) degeerate to Fg. (c). heorem 7 ca be exteded by fdg oe ode from other 3 odes. For example, we fd N(,) from N(,), N(,) ad N(,), as show Fg. (d). he other 3 cases are () fdg N(,) from N(,), N(,) ad N(,), () fdg N(,) from other 3 odes, (3) fdg N(,) from other 3 odes. So there are 4 cases assocated to Fg. (d). he exteso for Fg. (e) ad Fg. (f) are smlar. Uder these extesos, there are cases for fdg oe ode from other 3 odes assumg these odes are o the left upper sde of the dagoal. hat s, these odes N(,) has the propertes. we emphass these 3 cases show Fg. (d)(e)(f) because they are the cases we used our algorthms. N(,)= e f ((e,f) (ab-cd,ad+bc), (-f,e) (ab-cd,ad+bc) (,). + PWASE VOLUME MAY 7 ISSN WASE.ORG

4 PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND ECHNOLOGY VOLUME MAY 7 ISSN Now we start the algorthms for fdg PPH curve from the cotrol vectors of ts hodographs. C. Algorthms Gve r =[u, v ], where, we wat to fd the cotrol vectors for assocated Bézer Curve. he assocated Bézer curve ca be costructed from Equato () drectly. We ca fd all (I,,J, ),, from r =[u, v ], ad fd the cotrol vectors Δp,, from Equato (). Wth the referece pot p =(x,y ), we ca draw the Bézer curve. Now we would le to costruct the PH curve the other way. It s ot so easy to prevew the curve from r,, so we would le to costruct the PPH curve by dcatg part of the cotrol vectors, ad calculate the others. As metoed secto A, we eed to ow + cotrol vectors of the hodograph order to fd r,, so that other cotrol vectors ca be foud through r ad the cotrol vectors we foud. Cosder Equato (3) as a example. Gve ay four cotrol vectors from Δp, 6, we wat to fd other cotrol vectors so that the curve ca be draw. Assume we ow Δp, 3, ad wat to fd Δp, 4 6, we ca geerate 8 degree equatos by the frst 4 row of Equato 3, ad there are 8 varables, [u, v ], 3, amog them. here are at most 8 =56 solutos the system of equato. he algorthms we proposed here try to reduce the umber of soluto wth more effcet way. We llustrate three algorthms ths secto. he frst algorthm ad secod algorthm select the frst + cotrol vectors Δp,, ad fd others. he frst algorthm fds r,, frst, ad the fds N(,)=(I,,J, )= ( u u vv, uv + uv ), the cotrol vectors Δp,, s produced by Equato () fally. he secod algorthm fds N(,) drectly, ad the derves the other cotrol vectors.he thrd algorthm selects both sde of the cotrol vectors, that s, Δp,, 3 +, ad fd others. All of these three algorthms use the Equato (). We use Equato (3) to llustrate the dea of the frst algorthm. I Equato (3), we ca fd two solutos for r =[u,v ] from heorem 4 ad the frst row of Equato (3): 7(Δx, Δx )=(u -v, u v ). We further solve r =[u,v ] by solvg system of lear equatos the secod row of Equato (3): 7(Δx, Δx )=(u u -v v, u v +u v ). Repeat the process o the thrd row ad fourth row of Equato (3), we ca fd sgle soluto for [u,v ] ad [u 3,v 3 ], assocated wth each soluto of r. Wth all of the r,,,,3, we ca fd Δp, 4 6 drectly from Equato (). We totally fd two sets of cotrol vectors ad produce at most two curves. However, these two sets of r produce the same curve. We lst the dea the followg algorthm: (Let P=,Q = ) Algorthm // Gve Δp,, fd Δp, +. Fd r =[u, v ] from the equato: (+)Δp = r Qr Pr, r =(u -v, u v ). Select ay soluto for r =[u, v ]. 3. For = to Fd r =[u, v ] from the equato: [u, v ]= + (,)( u + v ϕ ϕ,)( u + v ) [Δx ), Δy ] u v - (, ) ϕ ( u r Pr, rqr 4. For =+ to Fd Δp = ( Δx, Δy ) from the equato: (Δx, Δy )= ( ) + ϕ, Pr r, r u v Qr u he equato (4) ca be derved from Equato () wth (I,,J,- )= r Pr, r Qr. We ca smplfy ths equato to: ϕ (,) r Pr, r Qr =(+)(Δx,Δy )- ϕ (, ) r Pr, r Qr So, we ca solve [u, v ] va [u, v ], [u, v ],, [u -, v - ]. How may dfferet solutos for ths algorthm? From heorem, there are two solutos for [u, v ] the frst step of the algorthm, call them [u, v ] ad [u, v ]. he steps () gves sgle soluto o each [u, v ] ad [u, v ] for [u, v ],, ad the thrd step, we fd uque cotrol pots (Δx, Δy ), + o each [u, v ] ad [u, v ]. Observe Equato (4), we fd two sets of solutos, each [u, v ] assocated wth the other soluto [u, v ]. We have two set of solutos, they are [u, v ],, ad [u, v ],. I step 3 of Algorthm, we foud these two sets of solutos produce the same cotrol vectors Δp, +, because: ( r ) Pr, rqr =( ( r ) ) P( r ),( r ) Q( r ) So we coclude that the frst algorthm produce sgle curve. I Algorthm, we fd all r =[u, v ],, so that all of the cotrol vectors Δp, + ca be foud. I fact, we ca fd all N(,) from Δp, ad the fd Δp, + drectly from N(,) by the followg algorthm: Algorthm : // Gve Δp,, fd Δp, +. Fd N(,) from Δp. // Equato (). For = to do { Fd N(,) from Δp // Equato () For to - Fd N(,) from N(,), N(,), N(,) //hm 7 3 Fd N(,) from N(,) ad N(,) } // hm 6 3 For =+ to do Δp = ( ) + ϕ, N(,-) (4) PWASE VOLUME MAY 7 ISSN WASE.ORG

5 PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND ECHNOLOGY VOLUME MAY 7 ISSN =5 =4 =3 = = = = =9 =8 =7 =6 s ot easy to predct. So, we chage our approach to fd the cotrol vectors the mddle by gvg two sdes of the cotrol vectors, whch produce the thrd algorthm. Fg. 3(b) represets the put ad output of the cotrol polygo for the followg algorthm: (a) Example for Algorthm (b) Strategy for Algorthm Fg. Dagrams for Algorthm Let s use a example to expla Algorthm. We draw Fg. (a) to llustrate the dea of Algorthm. I Fg. (b), the arrow from upper left to the lower rght dcates the vector at the begg of the arrow s a put vectors the arrow from lower rght to the upper left dcates the vector at the ed of the arrow s a put vectors. We select =5 as the example. I Algorthm, the step fd the tal ode N(,) from Δp. I the frst terato of step (=), we fd N(,) from Δp frst (step ), ad sp the step because the for loop has tal value to the value -=, Now we ow N(,) ad N(,), we ca fd N(,) va heorem 6. I the secod terato of step (=), we fd N(,) from Equato () (step ), ad fd N(,)(step ) from N(,), N(,), N(,) va heorem 7, ad Fd N(,)(step 3) from N(,) ad N(,) va heorem 6. Now we ow all of the ode N(,), where,. Repeat step, we ca fd all N(,). Geerally, begg teratos step, we assume that the value of the ode - s ow, as mared the area () the Fg. (b). Wth the ew put value Δp, we ca fd the sgle value of ew ode N(,) from Equato (). Notce that all of the odes used ths step s ow except N(,). We further fd the sgle value of N(,), - from N(,), N(,) ad N(,) by heorem 7 step, ad fd the sgle value for N(,) from N(,) ad N(,) by heorem 6 step 3. Repeat ths process, we ca fd all value of N(,),,. After we fd all ode value,,, other cotrol vectors ca be easly foud step 3 by Equato (). Notce that we fd sgle value for all cotrol vectors. hat s, we costruct sgle curve ths algorthm. Algorthm 3: Gve the cotrol vctors Δp, 3 +, fd the cotrol vectors Δp, 3. + ad. Fd N(,),,, by Algorthm usg Δp, = as put vectors.. Fd N(,), +, +, by modfyg Algorthm usg Δp,, 3. Fd N( 4. For = + to For to +) by heorem as put vectors. Fd N(,) by heorem 7 usg N(,-), N(-,-), ad N(-,) as put. // heorem 7 5. For = + to 3, fd Δp. // Equato (). Δp N(, ) Δp 3 + N ( +, + ) N(, ) N(, ) (a) Step 3 N(, ) N ( +, + ) N(, ) N ( +, + ) N(, ) (b) Step 4, st terato Δp N(, ) (c) Step 4, after st terato Fg. 4 Dagrams for Algorthm 3 Δp N(, ) (a) algorthm (b) algorthm 3 Fg. 3 I/O cotrol vectors for algorthm ad 3 I the secod algorthm, we try to fd the last cotrol vectors (Δp,+ ) from the frst + cotrol vectors(δp, ). We use Fg. 3(a) to represet the put ad output of the cotrol vectors for ths algorthm. he dsadvatage of the algorthm s that the fal part of the curve I algorthm 3, the frst step fd N(,),,,, mared area () Fg. 4(a), usg the dea algorthm. Wth smple modfcato, we ca easly fd N(,), +, +,, mared area () Fg. 3(a). hese two odes N(, ), ad N( +, +), as mar sold crcle Fg. 4(a), are fd at the frst ad secod steps respectvely. he value of odes we dd ot fd yet s area (3). Notce that all N(,), s foud already. I ths stuato, all N(,) ca be derved by corollary. However, we fd solutos corollary for all odes area (3), PWASE VOLUME MAY 7 ISSN WASE.ORG

6 PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND ECHNOLOGY VOLUME MAY 7 ISSN whch maes the fal Δp, + 3, are hard to select, because the selecto of correct value for each ode s hard. So, we try to use corollary as few as possble. I step 3, we fd the ode N(, +) the lower rght corer of area (3) by usg corollary, as mared square Fg. 4(a), whch produce two couples wth dfferet sg. Let s select oe value for N(, +), ad cotue the algorthm, cosder the frst terato of step 4(= +), the ode N(, +),, as mared tragle Fg. 4(b), ca be foud by heorem 7, usg N(, N(, )(sold crcle), ad N(, ),, +)(square) as put. Notce that each ode produce sgle value for each value of N(, for odes area (3). +) we foud step 3. here are two sets of values Start from the secod terato (> all odes N(,), +) of step 4, assume, <, show the area (3) Fg. 4(c), produce sgle value f we select oe value for N(, +), we wat to produce the value N(, ), by heorem 7, usg N(, -), N( -, -), N( -, ) as put. Notce that N(, -) area (3) has sgle value, N( -, -) o the dagoal has sgle value, ad N( -, ) area () also has sgle value. From the observato, we ow N(,),, has sgle value. At the ed of terato, we fd sgle value for all odes, whch produce sgle cotrol vectors, whch produce sgle curve. Wth the secod value N(, +), we ca fd the secod set of values for all ode area (3) Fg. 4(a), whch produce the secod curve. So, ths algorthm produces at most curves. All 3 algorthm3 taes O() tmes, where s the maxmum degree of u(t) ad v(t)..645]. hs result produces p 5 =(4.975,5.48), p 6 =(5.855,5.8), p 7 =(6.64,4.636). he output curve s show Fg. 5(b). (a) put (b) result Fg. 5 I/O vectors ad result curves for example Example : Gve the cotrol vectors Δp =[,], Δp =[,.] wth referece pot at the org, so that the cotrol pots p =(,), p =(,), p =(,3.), the frot part of the cotrol polygo s show Fg. 6(a). Gve the cotrol vectors Δp 5 =[-.,-], Δp 6 =[-.,-.5], wth a referece pot p 5 at the org, the ed part of the cotrol polygo s show Fg. 6(b). Wth algorthm 3, we fd two solutos. he frst soluto s Δp =[-.88,.8], Δp 3 =[-.58,.4], ad Δp 4 =[-.85,-.53], ad ts assocated curve s show Fg. 6(c). he secod soluto s Δp =[.34,-.3], Δp 3 =[.58,-.4], ad Δp 4 =[-.377,-.358], ad ts assocated curve s show Fg. 6(d). IV. EXPERIMENAL RESUL We mplemet all three algorthms usg Maple. hese programs wors o PC wth Petum D.8GHz CPU ad GB RAM. hese examples 3 algorthms taes very shout tme to compute output vectors (about.5 secods for each algorthm). (a) put from left (b) put from rght Example : Gve the cotrol pots p =(,), p =(,), p =(,3.6), p 3 =(3,4.7), p 4 =(4,5.3), as show Fg. 5(a), both the algorthm ad algorthm produce the same curve, as show Fg. 5(b). I algorthm, we fd the value (u,v )=(3.365,.8),(u,v )=(.993,.478),(u,v )=(.37,.3 )ad(u 3,v 3 )=(.478,.9). Both algorthms fd the value for Δp 4 =[.975,.8], Δp 5 =[.88,-.99] ad Δp 6 =[.759,- (c) the frst result (d) the secod result Fg. 6 I/O vectors ad result curves for example PWASE VOLUME MAY 7 ISSN WASE.ORG

7 PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND ECHNOLOGY VOLUME MAY 7 ISSN V. CONCLUSION he PPH curve s a Bézer curve but ot vce versa. Whe we wat desg PPH curve from cotrol pots of Bézer curve, oly part of cotrol vectors we eed, ad produce the other parts of cotrol vectors. Varous ways we ca select the put cotrol vectors. If we select oe sde of the cotrol vectors, we ca fd sgle curve whch has PPH propertes. hs selecto maes the algorthm smple ad straght forward. However, the ed part of the curve ad the output cotrol vector s sestve ad upredctable. If we select both sde of the cotrol vectors, the assocated algorthm fd two curves whch have PPH propertes. Usg these two selectos, all of the algorthms proposed ths paper tae very short tme to costruct the PPH curves. REFERENCES [] R.. Farou,. Saals, Pythagrorea Hodographs IBM J. RES. DEVELOP. VOL 34, No. 5, September 99. [] Farou,R..,99.Pythagorea-hodograph Curves Practcal Use. I:Barhll,R.E. (Ed.), Geometry Processg For Desg ad Maufacturg. SIAM, Phladelpha, p [3] Farou,R..,Neff,99.Aalytc propertes of plae offset curves.computer Aded Geometrc Desg,7(-4): [4] Moo, H.P., Farou, R.., Costructo ad shape aalyss of PH qutc Hermte terpolats, Computer Aded Geometrc Desg,, 8, [5] W. Lü Ratoalty of the offsets to algebrac curves ad surfaces, Appled Mathematcs, 9 (Ser. B), [6] W. Lü, Offset-ratoal parametrc plae curves, Computer-Aded Geometrc Desg,, 6-66, 995. [7] K.K.Kubota, Pythagorea trples uque factorzato domas, Amerca Mathematcal Mothly, 79, 53-56, 97. PWASE VOLUME MAY 7 ISSN WASE.ORG