Design Quadratic Patch and Cubic Patch of the Surface

Transcription

1 IOSR Journl o Mthemtics IOSR-JM e-issn: - Volume, Issue Jn - Feb, PP - Design Qudrtic Ptch nd Cubic Ptch o the Surce R B M Amer, M A Abd El- Mgeed, Deprtment o Mthemtics nd Ph Engineering, Fcult o Engineering/ Zgig Universit, Egpt Abstrct: This pper describes new method or designing qudric ptch nd cubic ptch o the surces nd show how to use Mthemtic progrms or computtions the equtions o the ptch o the surces The surce with qudrtic ptches is deined through given set o si points nd the surce with cubic ptches is deined through given set o ten points Kewords: Qudric surces, Cubic surces, Coons ptches, Beier surce Ptches, Mthemtic progrm I Introduction Deinition I: The -dimensoinl nlog o curve is surce nd the two dimensionl generlition o n curve in is ptch or locl surce Deinition I : A ptch or locl surce is dierentible mpping Where u is n open subset o : A n More generll, i A is n subset o n : U,, we s tht mp n is ptch provided tht cn be etended dierentil mpping rom u into, where U is n open set contining nd we cll U or A the trce o For emple the ptch sphere o rdius is deined on the closed rectngulr [, π ], but the dierentil etension to n open set contining [, π ], A ptch cn be written s n n-tuple o unctions u, v u,v,, nu,v, The prtil derivtives u o with respect to u: n u u,v u,v,, u,v u u The other prtil derivtives o re,,, see [] v uu v vv II Qudrtic Surces The qudrtic surces [] re the second order lgebric equtions nd it is nmed surces =, which re denoted b the qudrtic equtions which hs the orm,, b c g h p q r d Where, b, c, d, e,, g, h, p, q nd r re rel constnts nd,, re vribles nd we cn deine ptch b u, v u,v, u, v, where u nd v rnge over the domin o Emples o qudrtic surces see Fig include the cone, clinder, ellipsoid, elliptic cone, elliptic clinder, elliptic hperboloid, elliptic prboloid, hperbolic clinder, hperbolic prboloid, prboloid, sphere, nd spheroid nd these surces re sid to be qudrtic becuse ll possible products o two o the vribles,, pper in qudrtic eqution Pge

2 Design Qudrtic Ptch nd Cubic Ptch o the Surce Fig Some qudrtic surces in R Since [] qudrtic surce crries two milies o stright lines clled genertors The tpes o surces determined rom the both milies o stright lines re: - The qudrtic surces re doubl ruled or nnulr like hperbolic prboloid nd hperboloid o one sheet when the both milies re rel nd dierent - The qudrtic surces re non ruled or ovl like ellipsoid hperboloid o two sheets nd elliptic prboloid when the both milies re non rel nd dierent - The qudrtic surces re singl ruled nd degenerte like clindr nd cone when the both milies re coincide to one mil An plne section o qudrtic is conic sections III Cubic Surces The Cubic surce re the third order lgebric equtions in our homogenous vribles,,, w = ie, it is the vnishing set o homogenous polnomil o degree in this men it consist o ll : : : w in with,,, w w w w w w w w Emples o cubic surces see Fig include Clebsech digonl surce which is designing b ten points nd lines nd ever three lines meet in point [] nd lso Cle cubic surce which designing b our double points such tht ech one corresponds to set o three points on line in the plne Clebsech digonl surce Cle cubic surce Fig Pge

3 Design Qudrtic Ptch nd Cubic Ptch o the Surce IV The Methods To Design Surce Ptch There re mn methods to design surces: - Ferguson [] used prmetric equtions or the deinitions o curves nd surces in ircrt designed nd Ferguson surce ptch deined b this mtri, v r r u v u u u v v - Beier hs recombined the terms o Ferguson surce ptch in w tht mke phsicl mening o vector coeicients more pprent see [], [], [] nd Beier surce ptch deined b r r u, v T v u u u MBM, Where r r r r r r r r M, B r r r r r r r r Beier ptch see Fig is designed in terms o chrcteristic polhedron which is speciied in terms o position vectors r ij o its vertices v v Fig Beier surce ptch - Coons [] ormed coons ptchessee Fig which constructed rom inormtion given on its boundr nd certin uilir sclr unctions o u nd v b dividing the surce into n ssembl o topologicl rectngulr ptches ech o which hs its boundries two u-curves nd two v-curves Fig Coons surce ptch Pge

4 Design Qudrtic Ptch nd Cubic Ptch o the Surce The Coons ptch is deined b hs our originl curves s its boundries nd deined b this mtri r, v v r r u, v u u,,, r u r u r v v r, r, v u u,, r r v V The New Method To Design Surce Ptch Let the grph o rel vlued unction o two vribles =, is surce in see Fig Assuming surce ptch is given s the grph o unction Fig For qudrtic ptch o the surce []: The generl method is: VI Qudrtic Ptch From this eqution, there re ive unknowns,,,, with ive equtions Form set o ive equtions s ollows: To solve these unknowns there eist si points s ollows:,,,,,,,,,,,,,,,,,,,, Tke point s n initil point,, We cn be written the equtions o unknowns b mtri orm s ollows: Pge

5 Design Qudrtic Ptch nd Cubic Ptch o the Surce Pge So we cn solve this sstem to ind the vlues o,,,, nd then substitute in bove eqution to ind the eqution o the ptch o the surce B using Mthemtic progrm to input bove mtri Let it is required to design qudrtic ptch o surce through the points:,,,,,, -,,,,,,,,,,, Using Mthemtic progrm to inter these points: There re two methods to solve the liner sstem mv = s b using Mthemtic progrm

6 Design Qudrtic Ptch nd Cubic Ptch o the Surce The irst method: Since we interred the si points so rom mthemtic progrm we cn get without n clcultions nd lso mtri v s ollows: And then using liner solve commnd to get the unknowns Then =, =, =, =, = From mthemtic progrm substitute these vlues in the eqution o qudrtic surce ptch we get: Then the eqution o the qudrtic ptch o the surce rom mthemtic progrm is: = + ` ` + ` - ` ` Pge

7 Design Qudrtic Ptch nd Cubic Ptch o the Surce Or numericll: = + + The Second method: We use RowReduce commnd to get the ive unknowns We get the sme results = + + The progrm listed bove help us to compute qudrtic ptch o the surce or n si points VII Cubic Ptch For cubic ptch o the surce: The generl method: From this eqution, there re nine unknowns,,,,,,,, with nine equtions To solve these unknowns there eist ten points s ollows: Pge

8 Design Qudrtic Ptch nd Cubic Ptch o the Surce Pge,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Tke point s n initil point,, We cn be written the equtions o unknowns b mtri orm s ollows: So we cn solve this sstem to ind the vlues o nine unknowns,,,,,,,, nd then substitute in bove eqution to ind the eqution o the ptch o the surce B using Mthemtic progrm to input bove mtri

9 Design Qudrtic Ptch nd Cubic Ptch o the Surce Let it is required to design cubic ptch o surce through the ten points:,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Using mthemtic progrm to inter ten points s ollows Pge

10 Design Qudrtic Ptch nd Cubic Ptch o the Surce Then solve the liner sstem m s = v b using Mthemtic progrm to get the nine unknowns,,,,,,,, nd then substitute the results in the eqution o cubic ptch o the surce Then Pge

11 Design Qudrtic Ptch nd Cubic Ptch o the Surce The eqution o cubic ptch o the surce is: = + ` ` + ` ` ` + ` + ` ` + ` Or numericll: = The progrm listed bove help us to compute cubic ptch o the surce or n ten points VIII Conclusions From this pper: - To show how to use Mthemtic progrms or computtions the equtions o the ptch o the surces - New method or designing qudrtic ptch o the surces rom n si points - New method or designing cubic ptch o the surces rom n ten points Reerences [] D Hilbert & S Cohn-Vossen, The Second-Order Surces, in Geometr nd the Imgintion New York: Chelse, pp - [] B Hunt, The lines on cubic surce nd Cubic surces, Ch nd ppendi B in the geometr o some specil rithmetic quotients New York; Springer verlg, nd [] G Alred, Modern Dierentil Geometr o Curves nd Surces CRC Press, [] L Suresh nd W Joe, Béier representtion or qudric surce ptches, Computer-Aided Design, Vol,, - [] W Boehm nd D Hnsord, Prmetric representtion o qudrtic surces, Mthemticl Modeling nd Numericl Anlsis, Vol,, [] W Boehm nd D Hnsord, in G Frin editor, Beier ptches on qudrtics, NURBS or Curve nd Surce Design SIAM, Phildelphi, [] I D Fu nd M J Prtt, Computtionl geometr or design nd mnucture, Mthemtics nd its Applictions Ellis Horwood limited, New York-Chichester, [] R B M Amer nd M A Abd El- Mgeed, Using Stirling s Interpoltion to Find Guss nd Men Curvture or the Surce, IOSR Journl o Mthemtics,,, - Pge