CS5620 Intro to Computer Graphics

Transcription

1 CS56 Itro to Computer Graphcs Geometrc Modelg art II Geometrc Modelg II hyscal Sples Curve desg pre-computers Cubc Sples Stadard sple put set of pots { } =, No dervatves specfed as put Iterpolate by cubc segmets (4 DOF): Force C ad C cotuty at pots Solve 4 lear equatos 4 ukows Iterpolato ( equatos): C ( ) C ( ),.., C cotuty costrats ( equatos): ' C ( ) C ( ),.., ' C cotuty costrats ( equatos): '' C ( ) C ( ),.., '' t t t t C () t C () t 4 Cubc Sples Have two degrees of freedom left (to reach 4 DOF) Optos Natural ed codtos: C ''() =, C ''() = Complete ed codtos: C '() =, C '() = rescrbed ed codtos (dervatves avalable at the eds): C '() = T, C '() = T erodc ed codtos atural C '() = C '(), C ''() = C ''(), prescrbed arameterzato The assumpto t [,] (uform parameterzato) s arbtrary Implctly mples same curve legth for each segmet Not atural f pots are ot equally spaced Oe alteratve - chord-legth parameterzato: Deote d ( y ) ( For the ' th segmet : t [, d ]. y ) Questo: What parts of C(t) are affected as a result of a chage? demo Bass fuctos should be local 5 6 Copyrght C. Gotsma, G. Elber, M. Be-Che Computer Scece Dept., Techo age

2 CS56 Itro to Computer Graphcs Geometrc Modelg II arameterzato Chord- legth [,8] Uform [,] [,] [,] [,] [,] [,] [,4] Bezer curve s a appromato of gve cotrol pots Deote by C(t): t[,] Bezer curve of degree s defed over + cotrol pots { } =, Bezer Curves 5 Ct () Ct () De Casteljau Costructo Select t[,] value. For to do t [] : ( ) : ; For j : to do For : j to do : ( t) t ; C : ; t = / [ j ] [ j ] [ j ] [ ] demo C(/) 9 Algebrac Form of Bezer Curves Bezer curve for set of cotrol pots { } =, : C = B t = {B (t)} =, = Berste bass of polyomals of degree Cubc case: B B t B t B () t t Algebrac Form of Bezer Curves = B, t [,] why? Curve s lear combato of bass fuctos Curve s cove combato of cotrol pots C( ) = ( ) t t t ropertes of Bezer Curves C(t) s polyomal of degree C(t) CH(,, ) C() = ad C() = C'(t) s a Bezer curve of oe degree less C'() = ( ) ad C'() = ( - ) C(t) s affe varat ad varato dmshg C( ) = ( ) t t t A A Copyrght C. Gotsma, G. Elber, M. Be-Che Computer Scece Dept., Techo age

3 CS56 Itro to Computer Graphcs Geometrc Modelg II Questos: ropertes of Bezer Curves What s the shape of Bezer curves whose cotrol pots le o oe le? How ca oe coect two Bezer curves wth C cotuty? C? C? Drawbacks of Bezer Curves Degree correspods to umber of cotrol pots Global support: chage oe cotrol pot affects the etre curve For large sets of pots curve devates far from the pots Caot represet cocs eactly. Most otceably crcles Ca be resolved by troducg a more powerful represetato of ratoal curves. For eample, a 9 degrees arc as a ratoal Bezer curve: =(,) =(,) =(,) w B w B w B C( t) w B w B w B where ww. w 4 Recap Berste bass fuctos: B ( ) ( ) t t t t [,], B ( t ), B ( t ) Bezer curve s lear combato of bass fuctos: = Ct ( )= B Bezer curve s cove combato of cotrol pots (combato depeds o t): C = B () t = 5 B-Sple Curves Idea: Geerate bass of fuctos wth local support C( t) N For each parameter value oly a fte set of bass fuctos s o-zero The parametrc doma s parttoed to sectos at teger parameter values (called kots). 6 Cubc B-Sple Bass Cubc B-Sple Bass C N, t [, ) r /6 r t t [, ) ( r r r ) / 6 r t ( ) t [, ) N (r 6r 4) / 6 r t ( ) t [, ) ( r) / 6 r t ( ) t [, 4) N () t N () t Copyrght C. Gotsma, G. Elber, M. Be-Che Computer Scece Dept., Techo age

4 CS56 Itro to Computer Graphcs Geometrc Modelg II For ay t [, ]: Cubc B-Sple Bass N For ay t [, ] at most four bass fuctos are o zero Ay pot o a cubc B-Sple s a cove combato of at most four cotrol pots C( t) N t [,4) N N N N Boudary Codtos for Cubc B-Sple Curves B-Sples do ot terpolate cotrol pots partcular, the uform cubc B-sple curves do ot terpolate the ed pots of the curve. Ways to force edpot terpolato: Let = = (same for other ed) Add a ew cotrol pot (same for other ed) - = ad a ew bass fucto N - (t). 9 Local Cotrol of B-sple Curves Cotrol pot affects C(t) oly for t(,+4) demo ropertes of B-Sple Curves C N, t [, ) For cotrol pots, C(t) s a pecewse polyomal of degree, defed over t[, ) 4 C CH(,.., ) C(t) s affe varat ad varato dmshg Questos: What s C() equal to? What s C () equal to? What s the cotuty of C(t)? rove! From Curves to Surfaces A curve s epressed as er product of coeffcets ad bass fuctos C( u) B ( u) Treat surface as a curve of curves. Also kow as tesor product surfaces Assume s ot costat, but are fuctos of a secod, ew parameter v: ( m j Q B ( j j From Curves to Surfaces (cot d) C( u) j B ( u) m m j S Q jb j ( B ( u) Q B ( B ( u) j j m ( QjBj( B ( u) Bezatch j 4 Copyrght C. Gotsma, G. Elber, M. Be-Che Computer Scece Dept., Techo age 4

5 CS56 Itro to Computer Graphcs Geometrc Modelg II Isoparametrc Curves Surface Costructors Costructo of the geometry s a frst stage ay mage sythess process Use a set of hgh level, smple ad tutve, surface costructors: Blear patch Ruled surface Boolea sum Surface of revoluto Etruso surface Swept surface 5 6 Blear atches ( ( u)( ( u) v u( uv Blear terpolato of 4 D pots - D aalog of D lear terpolato betwee pots the plae Gve,,, the blear surface for v[,] s: y ( ( u)( uv ( u) v u( Questos: What does a soparametrc curve of a blear patch look lke? Ca you represet the blear patch as a Bezer surface? Whe s a blear patch plaar? 7 8 Ruled Surfaces Gve two curves a(t) ad b(t), the correspodg ruled surface betwee them s: Ruled Surfaces S( = v a(u) + (-b(u) a(u) b(u) u u u u The correspodg pots o a(u) ad b(u) are coected by straght les Questos: Whe s a ruled surface a blear patch? Whe s a blear patch a ruled surface? 9 Copyrght C. Gotsma, G. Elber, M. Be-Che Computer Scece Dept., Techo age 5

6 CS56 Itro to Computer Graphcs Geometrc Modelg II Brdge of Strgs Boolea Sum Gve four coected curves =,,,4, Boolea sum S( flls the teror. ( ( u)( u( ( u) v S ( v ( u) ( ( u) S ( u ( ( u) ( uv The S( S( S( ( Boolea Sum (cot d) S( terpolates the four alog ts boudares. For eample, cosder the u = boudary: S(, S (, S (, (, v () ( () ( ( ( ( v ( ( ( v v Surface of Revoluto Rotate a, usually plaar, curve aroud a as Cosder curve (t) = ( (t),, z (t)) ad let Z be the as of revoluto. ( ( u)cos(, y( ( u)s(, z( ( u). z (t) 4 Surfaces of Revoluto Etruded Surface Etruso of a, usually plaar, curve alog a lear segmet. Gve curve (t) ad vector V V (t) V (t) S( ( u) vv 5 6 Copyrght C. Gotsma, G. Elber, M. Be-Che Computer Scece Dept., Techo age 6

7 CS56 Itro to Computer Graphcs Geometrc Modelg II Sweeped Surface Rgd moto of oe (cross secto) curve alog aother (as) curve: The cross secto may chage as t s swept Questo: Is a etruso a specal case of a sweep? a surface of revoluto? 7 Copyrght C. Gotsma, G. Elber, M. Be-Che Computer Scece Dept., Techo age 7