Introduction to Computer Graphics 10. Curves and Surfaces

Transcription

1 Inroduion o Comuer Grahis. Curves and Surfaes I-Chen Lin, Assisan Professor Naional Chiao Tung Univ, Taiwan Tebook: E.Angel, Ineraive Comuer Grahis, 5 h Ed., Addison Wesley Ref: Hearn and aker, Comuer Grahis, rd Ed., Prenie Hall Prof. S.Chenney, Comuer Grahis ourse noe, Univ. Wisonsin

2 Limiaions of Polygons Inherenly an aroimaion Planar faes and silhouees. Oherwise, i needs a very large numbers of olygons. Fied resoluion No naural arameerizaion Deformaion is relaively diffiul Hard o era informaion like urvaure or o kee smoohness Figures from MIT EECS 6.87, Durand and Culer

3 Subdivision Subdividing a olygon an alleviae he roblem of olygonal mesh reresenaion. E.g. Loo s subdivision Sli a riangle ino four smaller ones. Choose loaions of new veries by weighed average of he original neighbor veries. Figure from Zorin & Shroeder SIGGRAPH 99 Course Noes

4 Wha are Parameri Curves? Define a maing from arameer sae o D oins A funion ha akes arameer values and gives bak D oins D for urves; D for surfaes. The resul is a arameri urve or surfae E.g. F:,y F, F y

5 Why Parameri Curves? Inended o rovide he generaliy of olygon meshes bu wih fewer arameers for smooh surfaes. Faser o reae a urve, and easier o edi an eising urve. Easier o animae han olygon meshes. Normal veors and eure oordinaes an be easily defined everywhere.

6 Polynomial funions as urves We an use olygonal funions o form urves. X value of ubi urves f The roblem is how o effiienly find ou he oeffiien i. You may have learned leas square urve fiing

7 Hermie Curves A Hermie urve is a urve for whih he user rovides: The endoins of he urve The arameri derivaives of he urve a he endoins angens wih lengh df/d, dfy/d, where f and fy are funions of. For, we have onsrains: The urve mus ass hrough when = The derivaive mus be when = The urve mus ass hrough when = The derivaive mus be when =

8 Hermie Curves f C i are unknown, = ~ df d The urve mus ass hrough when = f = = The derivaive mus be when = f = = The urve mus ass hrough when = f = = The derivaive mus be when = f = + + =

9 Solving for he unknowns gives: Afer rearranging, we ge Eending o D Hermie Curves z z z z y y y y z y f

10 The lending Weighs A oin on a Hermie urve is obained by weighed blending eah onrol oin and angen veor '. ' Weighs of eah omonen

11 ezier Curves Two onrol oins define endoins, and wo oins onrol he angens. loaed a / of he sar angen veor loaed a / of he end angen veor ' / ' / sloe sloe

12 The endsie ondiions are he same. f = = f = = Aroimaing derivaive ondiions f = - = f = - = +* +* Or relaing he orignal Hermie mari. s, e, s, e ezier and Hermie urves e s e s e s e s f d df 6

13 A ezier urve value beomes ezier Curves 6 T The blending weighs

14 Eamles of ezier urves

15 ernsein Polynomials The blending funions of ubi bezier urves are a seial ase of he ernsein olynomials d:degree, k:inde b kd d k d! k! d k! d k b = ~ kd k These olynomials give he blending olynomials for any degree ezier form For any degree hey all sum o They are all beween and inside, k

16 Conve Hull Proery The roeries of he ernsein olynomials ensure ha all ezier urves lie in he onve hull of heir onrol oins ezier urve onve hull

17 ezier Curve Subdivision Subdividing onrol olylines rodues wo new onrol olylines for eah half of he urve defines he same urve all onrol oins are loser o he urve M P P M M M M M P Figure from Prof. S.Chenney, Comuer Grahis oursenoe, Univ. Wisonsin P

18 de Caseljau s Algorihm You an find he oin on a ezier urve for any arameer value by subdivision If you wan =.5, insead of aking midoins ake oins.5 of he way M P P =.5 M M P P Figure from Prof. S.Chenney, Comuer Grahis oursenoe, Univ. Wisonsin

19 ezier Coninuiy We an make a long urve by onaenaing mulile shor ezier urves. P, P, P, J P, How o kee he oninuiy? P, P,

20 Coninuiy Proeries C oninuous :urve/surfae has no breaks G oninuous : angen a join has same direion C oninuous : angen a join has same direion and magniude Cn oninuous : urve/surfae hrough nh derivaive is oninuous P, P, P, J P, P, P,

21 How o reah boh C oninuiy and loal onrollabiliy? Slighly loose he endoin onsrains. -slines do no inerolae any of onrol oins. Uniform ubi -sline basis funions -slines j j j

22 -sline Mari j j j P

23 -sline urves Sar wih a sequene of onrol oins Sele four from middle of sequene i-, i-, i, i+ ezier and Hermie goes beween i- and i+ -Sline doesn inerolae ouh any of hem bu aroimaes going hrough i- and i. Figures from CG leure noe, U. Virginia

24 The lending Weighs Conve hull roery Figures from MIT EECS 6.87, Durand and Culer

25 ezier Pah ezier urves an be eended o surfaes {from o u,v}. u, v n n j k jk n j u n k v

26 ezier Pah Edge urves are ezier urves. Any urve of onsan u or v is a ezier urve Eah row of 4 onrol oins defines a ezier urve in u Evaluaing eah of hese urves a he same u rovides 4 virual onrol oins The virual onrol oins define a ezier urve in Evaluaing his urve a v gives he oin u,v v u

27 Mari Form of ezier Pah 6 6, v v v u u u v u v u T T n k n j n k n j jk V M G M U Pah lies in onve hull

28 ezier Pahes Inerolaes four orner oins Conve hull roery Wa, D Grahis, Figure 6.

29 ezier Surfaes C oninuiy requires aligning boundary urves Wa, D Grahis, Figure 6.6

30 ezier Surfaes C oninuiy requires aligning boundary urves and derivaives Wa, D Grahis, Figure 6.6

31 -Sline Surfae Pah u, v i j bi u b j v ij u T M S PM T S v defined over only /9 of region

32 Aliaions of Slines and Surfaes Modeling and ediing D objes. Smooh ahs e.g. amera views Key-frame animaion. E.