Geometric Theory, Algorithms, and Techniques

Transcription

1 Geometrc Theory Algorthms and Technqes Hong Qn Deartment of Comter Scence State Unversty of New York at Stony Brook Stony Brook New York Tel: ; Fax: htt:// Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

2 Introdcton Geometrc modelng and vsal comtng Comter grahcs Vsalzaton anmaton vrtal realty CAD/CAM Engneerng manfactrng Comter vson Physcal smlaton Natral henomena Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

3 D Shae Reresentaton Ponts vertces a set of onts Lnes olylnes crve Trangles olygons Tranglar meshes olygonal meshes Analytc commonly-sed shae Qadrc srfaces shere ellsod tors Serqadrc srfaces serellse serellsod Blobby models Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

4 Basc Shaes ont lne lane crve srface trangle Crved olygon sold Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

5 Fndamental Shaes Vertex vertces Lne segments Trangle tranglar meshes Qadrlateral Polygon Crved object Tetrahedron yramd hexahedron Many more Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

6 Polygonal Meshes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

7 Shaded Model Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

8 Mechancal Part Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

9 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5 STzNY BRzzK

10 Sbdvson model NURBS model Imlct model PDE models Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

11 Bldng Strctre Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

12 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

13 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

14 Mathematcal Tools Parametrc crves and srfaces Slne-based objects ecewse olynomals Exlct mlct and arametrc reresentatons The ntegrated way to look at the shae: Object can be consdered as a set of faces each face can be frther decomosed nto a set of edges each edge can be decomosed nto vertces Sbdvson models Other rocedre-based models Sweeng Srfaces of revolton Volmetrc models Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

15 Lne Eqaton Parametrc reresentaton l - [] Parametrc reresentaton s not nqe l.5 In general [ a b ] v [ ] Re-arameterzaton varable transformaton v a b a v a q v b a v a v [ ] / b a.5 v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

16 Basc Concets Lnear nterolaton: v v t v t Local coordnates: Rearameterzaton: Affne transformaton: Polynomals Contnty v [v v] t [] f g v f g v h v f ax a b by af x bf y Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

17 Lnear and Blnear Interolaton b a b a d c e a c e f b d f v e vf a b Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

18 Fndamental Featres Geometry Poston drecton length area normal tangent etc. Interacton Sze contnty collson ntersecton Toology Dfferental roertes Crvatre arc-length Physcal attrbtes Comter reresentaton & data strctre Others! Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

19 Mathematcal Formlatons Pont: a a a Lne: l a a a T Qadratc crve: Parametrc doman and rearameterzaton: x y z [ ] [ b b b] T [ ] T [ ] T [ ] a a a b b b c c c T q x y z [ s e ]; v []; v s / e s x y z x y z Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

20 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Parametrc Polynomals Hgh-order olynomals No nttve nsght for the crved shae Dffclt for ecewse smooth crves Hgh-order olynomals No nttve nsght for the crved shae Dffclt for ecewse smooth crves n z n y n x n z y x z y x a a a a a a a a a c

21 Parametrc Polynomals Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

22 How to Defne a Crve? Secfy a set of onts for nterolaton and/or aroxmaton wth fxed or nfxed arameterzaton x y z x' y' z' Secfy the dervatves at some locatons What s the geometrc meanng to secfy dervatves? A set of constrants Solve constrant eqatons Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

23 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

24 One Examle Two end-vertces: c and c One md-ont: c.5 Tangent at the md-ont: c.5 Assmng D crve Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

25 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Cbc Polynomals Parametrc reresentaton s n [] Each comonents are treated ndeendently Hgh-dmenson crves can be easly defned Alternatvely Parametrc reresentaton s n [] Each comonents are treated ndeendently Hgh-dmenson crves can be easly defned Alternatvely c b a c b a c b a c b a z y x [ ][ ] UC z UB y UA a a a a x T

26 Cbc Polynomal Examle Constrants: two end-onts one md-ont and tangent at the md-ont In matrx form x x.5 [ ] [ ] A [.5.5 ] x'.5 x [ ]A x x.5 x'.5 x.5.75 A.5.5 A A Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

27 Solve ths Lnear Eqaton Invert the matrx A x 4 x.5 x'.5 x Rewrte the crve exresson x UM x y z UM UM z [ x.5 x'.5 x ] [ y y.5 y'.5 y ] [ z.5 z'.5 z ] T T T Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

28 Bass Fnctons Secal olynomals What s the mage of these bass fnctons? Polynomal crve can be defned by Observatons More nttve easy to control olynomals f f f f c c f c.5 f c'.5 f c f4 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

29 Lagrange Crve Pont nterolaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

30 Lagrange Crves Crve a a n n c L... a a L n a a Lagrange olynomals of degree n: L n Knot seqence:... n Kronecker delta: n L j δ j The crve nterolate all the data ont bt nwanted oscllaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

31 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Lagrange Bass Fnctons n n n n n n n n n n n n n n n j n L L L Otherwse n j j L

32 Cbc Hermte Slnes C C C C Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

33 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Cbc Hermte Crve Hermte crve Two end-onts and two tangents at end-onts Matrx nverson Hermte crve Two end-onts and two tangents at end-onts Matrx nverson z y x c A x x x x ' ' [ ] [ ] T T z z z z UM z y y y y UM y x x x x U x ' ' ' ' ' '

34 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Hermte Crve Bass fnctons Dslay the mage of these bass fnctons and the Hermte crve tself Bass fnctons Dslay the mage of these bass fnctons and the Hermte crve tself 4 f f f f ' ' 4 f f f f c c c c c

35 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Cbc Hermte Slnes Two vertces and two tangent vectors: Hermte crve Two vertces and two tangent vectors: Hermte crve ; ; d c d c v c v c ; 4 f H f H f H f H H H H H d d v v c

36 Hermte Slnes Hgher-order olynomals c v v n / c v H H n n n / v n n... n Note that n s odd! Geometrc ntton Hgher-order dervatves are reqred c v H n... v H... v n / / v ; H n n / H n n ; Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

37 Why Cbc Polynomals Lowest degree for secfyng crve n sace Lowest degree for secfyng onts to nterolate and tangents to nterolate Commonly sed n comter grahcs Lower degree has too lttle flexblty Hgher degree s nnecessarly comlex exhbt ndesred wggles Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

38 Varatons of Hermte Crve Varatons of Hermte crves c c c' c' c' / c' / In matrx form x-comonent only c c c' c' x x x x x x x x Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

39 Cbc Bezer Crves For control onts Crve geometry Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

40 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Crve Mathematcs Cbc Bezer crve Control onts and bass fnctons Image and roertes of bass fnctons Bezer crve Control onts and bass fnctons Image and roertes of bass fnctons B c B B B B

41 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Recrsve Evalaton Recrsve lnear nterolaton Recrsve lnear nterolaton c

42 Recrsve Sbdvson Algorthm Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

43 Basc Proertes Cbc The crve asses throgh the frst and the last onts end-ont nterolaton Lnear combnaton of control onts and bass fnctons Bass fnctons are all olynomals Bass fnctons sm to one artton of nty All bass fnctons are non-negatve Convex hll both necessary and sffcent Predctablty Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

44 Dervatves Tangent vectors can easly be evalated at the end-onts c' ;c' Second dervatves at end-onts can also be easly comted: c c 6 6 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

45 Dervatve Crve The dervatve of a cbc Bezer crve s a qadratc Bezer crve c' Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

46 More Proertes Cbc Two crve sans are obtaned and both of them are standard Bezer crves throgh rearameterzaton c v v [ ] c c c [ ] [ ] The control onts for the left and the rght are l r v v [ ] Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

47 Hgh-Degree Crves Generalzng to hgh-degree crves Advantages: Easy to comte Infntely dfferentable Dsadvantages: x y z Comtatonally comlex ndlaton ndesred wggles How abot hgh-order Hermte? Not natral!!! n a b c Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

48 Bezer Slnes Bezer crves of degree n c n B n Control onts and bass fnctons Bernsten olynomals of degree n: B n n n n n!!! n Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

49 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Recrsve Comtaton... n n j j j c

50 Recrsve Comtaton N levels n n n n n c Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

51 Proertes Bass fnctons are non-negatve The smmaton of all bass fnctons s nty End-ont nterolaton c c n Bnomal exanson theorem n n Convex hll: the crve s bonded by the convex hll defned by control onts n n Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

52 More Proertes Recrsve sbdvson and evalaton Symmetry: c and c- are defned by the same set of ont onts bt dfferent orderng n n ; Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

53 Tangents and Dervatves End-ont tangents: c' n c' n I-th dervatves at two end-onts deend on n ; n n n Dervatves at non-end-onts nvolve all control onts Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

54 Other Advanced Tocs Effcent evalaton algorthm Dfferentaton and ntegraton Degree elevaton Use a olynomal of degree n to exress that of degree n Comoste crves Geometrc contnty Dslay of crve Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

55 Bezer Crve Renderng Use ts control olygon to aroxmate the crve Recrsve sbdvson tll the tolerance s satsfed Algorthm go here If the crrent control olygon s flat wth tolerance then ott the lne segments else sbdvde the crve at.5 Comte control onts for the left half and the rght half resectvely Recrsvely call the same rocedre for the left one and the rght one Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

56 Hgh-Degree Polynomals More degrees of freedom Easy to comte Infntely dfferentable Drawbacks: Hgh-order Global control Exensve to comte comlex ndlaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

57 Pecewse Polynomals Pecewse --- dfferent olynomals for dfferent arts of the crve Advantages --- flexble low-degree Dsadvantages --- how to ensre smoothness at the jonts contnty Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

58 Pecewse Crves Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

59 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

60 Pecewse Bezer Crves Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

61 Contnty One of the fndamental concets Commonly sed cases: C C C Consder two crves: a and b s n [] Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

62 Postonal Contnty a b Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

63 Dervatve Contnty a a b ' b ' Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

64 General Contnty Cn contnty: dervatves to n-th are the same at the jonng ont The ror defnton s for arametrc contnty Parametrc contnty deends of arameterzaton! Bt arameterzaton s not nqe! Dfferent arametrc reresentatons may exress the same geometry Re-arameterzaton can be easly mlemented Another tye of contnty: geometrc contnty or Gn a b... n Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

65 Geometrc Contnty G and G Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

66 Geometrc Contnty Deend on the crve geometry DO NOT deend on the nderlyng arameterzaton G: the same jont G: two crve tangents at the jont algn bt may or may not have the same magntde G: t s C after the rearameterzaton Whch condton s stronger??? Examles Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

67 Pecewse Hermte Crves How to bld an nteractve system to satsfy varos constrants C contnty a b C contnty G contnty a b a' b ' a b a ' α b ' Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

68 Pecewse Hermte Crves Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

69 Pecewse Bezer Crves Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

70 Pecewse Bezer Crves C contnty C contnty G contnty C contnty q Geometrc nterretaton G contnty q q q α q q q q q q q q q Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

71 Pecewse C Bezer Crves Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

72 Contnty Smmary C: straghtforward bt not enogh C: too constraned Pecewse crves wth Hermte and Bezer reresentatons satsfyng varos contnty condtons Interactve system for C nterolatng slnes sng ecewse Bezer crves Advantages and dsadvantages Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

73 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

74 C Interolatng Slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

75 Natral C Cbc Slnes A set of ecewse cbc olynomals x y z C contnty at each vertex c Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

76 Natral C Cbc Slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

77 Natral Slnes Interolate all control onts Eqvalent to a thn str of metal n a hyscal sense Forced to ass throgh a set of desred onts No local control global control N control onts N eces n- condtons We need two addtonal condtons Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

78 Natral Slnes Interactve desgn system Secfy dervatves at two end-onts Secfy the two nternal control onts that defne the frst crve san Natral end condtons: second-order dervatves at two end onts are defned to be zero Advantages: nterolaton C Dsadvantages: no local control f one ont s changed the entre crve wll move How to overcome ths drawback: B-Slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

79 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

80 B-Slnes Motvaton The goal s local control!!! B-slnes rovde local control Do not nterolate control onts C contnty Alternatvely Catmll-Rom Slnes Kee nterolatons Gve C contnty only C s acheved Wll be dscssed later!!! Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

81 C Aroxmatng Slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

82 From B-Slnes to Bezer Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

83 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Unform B-Slnes B-slne control onts: Pecewse Bezer crves wth C contnty at jonts Bezer control onts: B-slne control onts: Pecewse Bezer crves wth C contnty at jonts Bezer control onts: n v v v v v

84 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Unform B-Slnes In general I-th segment of B-slnes s determned by for consectve B-slne control onts In general I-th segment of B-slnes s determned by for consectve B-slne control onts v v v v

85 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Unform B-Slnes In matrx form Qeston: how many Bezer segments??? In matrx form Qeston: how many Bezer segments??? v v v v

86 B-Slne Proertes C contnty Aroxmaton Local control convex hll Each segment s determned by for control onts Qestons: what haens f we t more than one control onts n the same locaton??? Doble vertces trle vertces collnear vertces End condtons Doble endonts: crve wll be tangent to lne between frst dstnct onts Trle endont: crve nterolate endont start wth a lne segment B-slne dslay: transform t to Bezer crves Deartment of Comter Scence CSE5 Lectre Notes Srng 5 Center for Vsal Comtng

87 Catmll-Rom Slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

88 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Catmll-Rom Slnes Kee nterolaton Gve C contnty Control tangents locally Idea: Bezer crve between sccessve onts How to determne two nternal vertces Kee nterolaton Gve C contnty Control tangents locally Idea: Bezer crve between sccessve onts How to determne two nternal vertces ' ' v v v v c v v c v c v c

89 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Catmll-Rom Slnes In matrx form Problem: bondary condtons Proertes: C nterolaton local control nonconvex-hll In matrx form Problem: bondary condtons Proertes: C nterolaton local control nonconvex-hll v v v v

90 Cardnal Slnes For vertces defne end-onts and ther c v c v assocated tangents Secal case: Catmll-Rom slnes when More general case: Kochanek-Bartels slnes Tenson bas contnty arameters c c α v α v v v α Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

91 Cardnal Slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

92 Kochanek-Bartels Slnes For vertces to defne for condtons c c c v c v α β γ v α β γ v v v β γ v β γ v v v Tenson arameter: Bas arameter: Contnty arameter: α β γ Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

93 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

94 Pecewse B-Slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

95 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng B-Slne Bass Fnctons B B B otherwse B k k k k k k < <

96 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Bass Fnctons Lnear examles How does t look lke??? Lnear examles How does t look lke??? ] 4 [ 4 ] [ ] [ ] [ ] [ ] [ B B B

97 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Bass Fnctons Qadratc cases knot vector s [456] Cbc examle Qadratc cases knot vector s [456] Cbc examle < < < < < < < < < < < < B B B B

98 B-Slne Bass Fncton Image Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

99 B-Slnes n Mathematcs c B k Control onts and bass fnctons of degree k- Pecewse olynomals Bass fnctons are defned recrsvely We also have to ntrodce a knot seqence nk n a non-decreasng order... n k Note that the arametrc doman: [ k n ] Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

100 Bass Fnctons B B B B B B 4 B B B B 4 B B B B 4 B B 4 B B 4 4 B B 5 4 B B 5 6 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

101 B-Slne Facts The crve s a lnear combnaton of control onts and ther assocated bass fnctons n control onts and bass fnctons resectvely Bass fnctons are ecewse olynomals defned recrsvely over a set of non-decreasng knots {... k... n... n k} The degree of bass fnctons s ndeendent of the nmber of control onts note that I s ndex k s the order k- s the degree The frst k and last k knots do NOT contrbte to the arametrc doman. Parametrc doman s only defned by a sbset of knots Deartment of Comter Scence CSE5 Lectre Notes Srng 5 Center for Vsal Comtng

102 B-Slne Proertes C: ecewse olynomal of degree k- Contnty at jonts: Ck- The nmber of control onts and bass fnctons: n One tycal bass fncton s defned over k sbntervals whch are secfed by k knots [kik] There are nk knots n total knot seqence dvdes the arametrc axs nto nk sb-ntervals There are n-k-n-k sb-ntervals wthn the arametrc doman [k-n] Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

103 B-Slne Proertes There are n-k ecewse olynomals Each crve san s nflenced by k control onts Each control onts at most affects k crve sans Local control!!! Convex hll The degree of B-slne olynomal can be ndeendent from the nmber of control onts Comare B-slne wth Bezer!!! Key comonents: control onts bass fnctons knots arametrc doman local vs. global control contnty Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

104 B-Slne Proertes Partton of nty ostvty and recrsve evalaton of bass fnctons Secal cases: Bezer slnes Effcent algorthms and tools Evalaton knot nserton degree elevaton dervatve ntegraton contnty Comoste Bezer crves for B-slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

105 Unform B-Slne Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

106 Another Formlaton Unform B-slne Parameter normalzaton s n [] End-ont ostons and tangents c c c c ' ' Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

107 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Another Formlaton Matrx reresentaton Bass matrx Matrx reresentaton Bass matrx c c c c c UM M UM UM h h ' ' ' M

108 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Bass Fnctons Note that s now n [] Note that s now n [] B B B B

109 B-Slne Renderng Transform t to a set of Bezer crves Convert the I-th san nto a Bezer reresentaton v v k Consder the entre B-slne crve... v... v v 4 n... v v 7... v k... v 4 n 4 n Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

110 Matrx Exresson v 4 n Μ n The matrx strctre and comonents of B? v Μ q Av AB B The matrx strctre and comonents of A? Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

111 B-Slne Dscretzaton Parametrc doman: [k-n] There are n-k crve sans eces Assmng m onts er san nform samlng Total samlng onts mn-kl B-slne dscretzaton wth corresondng q... q l arametrc vales: v q... v l c v j B n j j k v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

112 B-Slne Dscretzaton Matrx eqaton q B Μ q l B k k v Μ v l Λ Ο Λ B B n k n k v Μ Μ v l n A s lxn matrx n general l s mch larger than n so A s sarse The lnear dscretzaton for both modelng and renderng Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

113 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

114 From B-Slnes to NURBS What are NURBS??? Non Unform Ratonal B-Slnes NURBS Ratonal crve motvaton Polynomal-based slnes can not reresent commonlysed analytc shaes sch as conc sectons e.g. crcles ellses arabolas Ratonal slnes can acheve ths goal NURBS are a nfed reresentaton Polynomal conc secton etc. Indstry standard Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

115 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng From B-Slnes to NURBS B-slnes NURBS crve B-slnes NURBS crve B w w w w k n z y x c n k n k B w B w c

116 Geometrc NURBS Non-Unform Ratonal B-Slnes CAGD ndstry standard --- sefl roertes Degrees of freedom Control onts Weghts Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

117 Ratonal Bezer Crve Projectng a Bezer crve onto w lane Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

118 From B-Slnes to NURBS Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

119 NURBS Weghts Weght ncrease attracts the crve towards the assocated control ont Weght decrease shes away the crve from the assocated control ont Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

120 NURBS for Analytc Shaes Conc sectons Natral qadrcs Extrded srfaces Rled srfaces Srfaces of revolton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

121 NURBS Crcle ag abcdedg w knot [444] Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

122 NURBS Crve Geometrc comonents Control onts arametrc doman weghts knots Homogeneos reresentaton of B-slnes Geometrc meanng --- obtaned from rojecton Proertes of NURBS Reresent standard shaes nvarant nder ersectve rojecton B-slne s a secal case weghts as extra degrees of freedom common analytc shaes sch as crcles clear geometrc meanng of weghts Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

123 NURBS Proertes Generalzaton of B-slnes and Bezer slnes Unfed formlaton for free-form and analytc shae Weghts as extra DOFs Varos smoothness reqrements Powerfl geometrc toolkts Effcent and fast evalaton algorthm Invarance nder standard transformatons Comoste crves Contnty condtons Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

124 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

125 Geometrc Modelng Why geometrc modelng Fndamental for vsal comtng Grahcs vsalzaton Comter aded desgn and manfactrng Imagng Entertanment etc. Crtcal for vrtal engneerng Interacton Geometrc nformaton for decson makng Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

126 From Crve to Srface Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

127 Parameterzaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

128 Srfaces From crves to srfaces A smle crve examle Bezer c [] Consder each control ont now becomng a Bezer crve v [] j j B B j v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

129 Srfaces Then we have Matrx form s v UMPM s v j [ B B B B ] T V T j B j v B j B B j v j B B B B Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

130 Srfaces Frther generalze to degree of n and m along two arametrc drectons s v n j Qeston: whch control onts are nterolated? How abot B-slne srfaces??? m n m j B B j v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

131 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

132 Tensor Prodct Srfaces Where are they from? Monomal form Bezer srface s v a j j m n s v B B v j j j v j B-slne srface General case m n s v B B j l v j s F G v v v j j j k j Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

133 Tensor Prodct Srface Bezer Srface Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

134 B-Slnes B-slne crves c Tensor rodct B-slnes Qeston agan: whch control onts are nterolated??? Another qeston: can we get NURBS srface ths way??? Answer: NO!!! NURBS are not tensor-rodct srfaces Another qeston: can we have NURBS srface? YES!!! n B k m n s v B B j l v j j k Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

135 NURBS Srface NURBS srface mathematcs s v n m j n m j Understand ths geometrc constrcton Qeston: why s t not the tensor-rodct formlaton??? Comare t wth Bezer and B- slne constrcton j w w j j B B k k B B j l j l v v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

136 NURBS Srface Parametrc varables: and v Control onts and ther assocated weghts: mn Degrees of bass fnctons: k- and l- < <... < m k Knot seqence: v < v <... < v n l Parametrc doman: v k l < < v < < v m n Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

137 NURBS Srface The same rncle to generate crves va rojecton Idea: assocate weghts wth control onts Generalzaton of B-slne srface Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

138 Rectanglar Srface Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

139 Hermte Srfaces How abot Hermte srfaces??? Hermte Crve [ H H H H ] c c c c' c' C s not a crve sv whch s also a Hermte Crve: [ H v H v H v H v ] s v s s s v sv Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

140 Hermte Srfaces Smlarly c s now a crve sv whch s also a Hermte crve: The same are for c and c : s s [ H v H v H v H v ] s v v v H H v v s s s s s s s s v v v v s s s v sv Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

141 Hermte Srfaces It s tme to t them together! s s s s s s s v v T v H H v s s Contnty condtons for srfaces Bezer srfaces B-slnes NURBS Hermte srfaces C and G contnty s s s s v v v sv s sv sv Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

142 Hermte Srfaces Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

143 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

144 Srface Normal Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

145 Srface Renderng Parametrc grds []X[] as a set of rectangles Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

146 Srface Patch Renderng We se bcbc as an examle The smlest naïve: convert crved atches nto rmtves that we always know how to render From crved srfaces to olygon qadrlaterals nonlanar and/or trangles lanar Srface evalaton at grd onts Ths s straght forward bt neffcent becase t reqres many tmes of evalaton of sv The total nmber s δ δ v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

147 Srface Renderng Parametrc grds []X[] as a set of rectangles Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

148 Srface Renderng Better aroach: recomtaton v [ ] v s v M v M s constant throghot the entre atch. The followngs are the same along soarametrc lnes [ ] [ v v v ] Use one dmensonal array to comte and store evalaton only once Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

149 Srface Renderng How abot many atches: the array s nchanged ts samlng rate s the same ths s more sefl How abot adatve samlng based on crvatre nformaton!!! How to comter normal at any grd ont aroxmaton s s v s v v δ v s v s v δv s v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

150 Reglar Srface Generated from a set of control onts. Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

151 Crve Network Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

152 Coons Patch s v s v s s Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

153 Coons Patch s v s v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

154 Coons Patch s s Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

155 Coons Patch s v s v s s Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

156 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Coons Patch Blnearly blended Coons atch Bcbcally blended Coons atch Blnearly blended Coons atch Bcbcally blended Coons atch v L v L P L v L v P P P P P P P P f f f f f f f f f v H v H v H v H P H v H v H v H v P v v f f f f f f f f f f

157 Coons Patch s v s s v s s s s s v v v v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

158 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Gordon Srfaces Generalzaton of Coons technqes A set of crves Boolean sm sng Lagrange olynomals Generalzaton of Coons technqes A set of crves Boolean sm sng Lagrange olynomals m j v n v j f f f f f f f f f P P P P P P P v L v P L v P m j m j j n n

159 Transfnte Methods Blnearly blended Coons atch Interolate for bondary crves Bcbcally blended Coons atch Interolate crves and ther dervatves Gordon srfaces Interolate a crve-network Tranglar extenson Interolate over trangles Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

160 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

161 Tranglar Srfaces Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

162 Recrsve Sbdvson Algorthm Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

163 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Crve Mathematcs Cbc Bezer crve Control onts and bass fnctons Image and roertes of bass fnctons Bezer crve Control onts and bass fnctons Image and roertes of bass fnctons B c B B B B

164 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Recrsve Evalaton Recrsve lnear nterolaton Recrsve lnear nterolaton c

165 Proertes Bass fnctons are non-negatve The smmaton of all bass fnctons s nty End-ont nterolaton c c n Bnomal exanson theorem n n Convex hll: the crve s bonded by the convex hll defned by control onts n n Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

166 Proertes Bass fnctons are non-negatve The smmaton of all bass fnctons s nty End-ont nterolaton c c n Bnomal exanson theorem n n Convex hll: the crve s bonded by the convex hll defned by control onts n n Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

167 Dervatves Tangent vectors can easly evalated at the endonts c' ;c' Second dervatves at end-onts can also be easly comted: c c 6 6 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

168 Dervatve Crve The dervatve of a cbc Bezer crve s a qadratc Bezer crve c' Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

169 More Proertes Cbc Two crve sans are obtaned and both of them are standard Bezer crves throgh rearameterzaton c v v [ ] c c c [ ] [ ] The control onts for the left and the rght are l r v v [ ] Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

170 Barycentrc Coordnates T r s t V rr ss tt tsr S T ; srt T S; rts S R V R S Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

171 Tranglar Bezer Patch Tranglar Bezer srface j k n j k > s v n j k B j k r s t Where rst and they are local barycentrc coordnates Bass fnctons are Bernsten olynomals of degree n B n j k r s t n! r! j! k! s j t k Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

172 Tranglar Bezer Patch How many control onts and bass fnctons: Partton of nty n j k > n B j k r s t n Postvty B n j k r s t > ; r s t [] Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

173 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Recrsve Evalaton n l k j l k j l k j l k j k j k j v k j l n k j t s r ; s >

174 Proertes Effcent algorthms Recrsve evalaton Drectonal dervatves Degree elevaton Sbdvson Comoste srfaces Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

175 Research Isses Contnty across adjacent atches Integral comtaton Tranglar slnes over reglar tranglaton Transform tranglar slnes to a set of ecewse tranglar Bezer atches Interolaton/aroxmaton sng tranglar slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

176 Tranglar Bezer Srface Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

177 Recrsve Evalaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

178 Control onts Cbc Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

179 Bass Fnctons Cbc sss sst rss stt 6rst rrs ttt rtt rrt rrr Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

180 Tranglar Patch Sbdvson Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

181 Tranglar Doman Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

182 Tranglar Coons-Gordon Srface r ; f s t t ; f r s s ; f r t Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

183 Tranglar Coons-Gordon Srface s const. r const. t const. Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

184 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Tranglar Interolaton s r r L t r L t s P t r r L t s L s r P t s s L s r L t r P α γ γ α β β α α α f f f f f f f f f

185 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

186 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Tranglar Interolaton The Boolean sm of any two oerators reslts the same! Use cbc blendng fnctons for C nterolaton! The Boolean sm of any two oerators reslts the same! Use cbc blendng fnctons for C nterolaton! f f f f f f P P P P P P P P P f f f f f f f Q Q H s r H s r D H t r D H t r Q α α α α α α

187 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Gregory s Method Convex combnaton Generalze to entagonal atch! Convex combnaton Generalze to entagonal atch! a a t s r s a T a T a T a T T T T T T T T T T T T t r D t r T f f f f f f f f f f f f f α α

188 Tranglar B-slnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

189 Srface Proertes Inhert from ther crve generators More! Effcent algorthms Contnty across bondares Interolaton and aroxmaton tools Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

190 Shercal Parameterzaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

191 Shercal Parameterzaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

192 Possble Alcatons Smooth srface fttng Shae classfcaton Medcal regstraton Solvng PDEs on srfaces Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

193 Shae Morhng Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

194 Morhng Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

195 Mltresolton Mang Mltresolton morhng Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

196 Featre Mang Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

197 Textre Mang Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

198 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

199 Sold Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

200 Parametrc Solds Trcbc sold Bezer sold B-slne sold v w v w [ ] j k v w B B v B w j k j k v w jkb I Bj J v Bk K w a jk jk v j w j k k NURBS sold v w j k j k jk q q jk jk B B I I B B j J j J v B v B k K k K w w Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

201 Parametrc Solds Trcbc Hermte sold In general x v w v w y v w z v w v w [ ] Also known as hyeratch Parametrc solds reresent both exteror and nteror Examles A rectanglar sold a trlnear sold Bondary elements 8 corner onts crved edges and 6 crved faces Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

202 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Crves Srfaces and Solds Isoarametrc crves for srfaces Isoarametrc crves for solds Isoarametrc srfaces for solds Isoarametrc crves for srfaces Isoarametrc crves for solds Isoarametrc srfaces for solds..; const v const v v v j j s s s k j k j w v w v w v w v s s s s k j w v w v w v w v s s s s

203 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Crves Srfaces and Solds Non-soarametrc crves for srfaces Non-soarametrc crves for solds Non-soarametrc srfaces for solds Non-soarametrc crves for srfaces Non-soarametrc crves for solds Non-soarametrc srfaces for solds t v t t v t t v s c s t w t v t t w t v t t w v s c s b a w b a v b a w v s s

204 CSE5- Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

205 Srfaces of Revolton z x y Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

206 Srfaces of Revolton Geometrc constrcton Secfy a lanar crve rofle on y-z lane Rotate ths rofle wth resect to z-axs Procedre-based model What knds of shae can we model? Revew: three dmensonal rotaton w.r.t. z-axs x' cos θ sn θ x y' sn θ cos θ y z' z Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

207 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Srfaces of Revolton Mathematcs: srfaces of revolton Mathematcs: srfaces of revolton cos sn z v y v y v z y s c

208 Frenet Frames Motvaton: attach a smoothly-varyng coordnate system to any locaton of a crve Three ndeendent drecton vectors for a D coordnate system: tangent; b-normal; normal t normalze b normalze c c c n normalze b t Frenet coordnate system frame tbn vares smoothly as we move along the crve c Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

209 Frenet Coordnate System b c c c n t Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

210 Sweeng Srface y z x y x Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

211 General Sweeng Srfaces Srface of revolton s a secal case of a sweeng srface Idea: a rofle crve and a trajectory crve c c v Move a rofle crve along a trajectory crve to generate a sweeng srface Qeston: how to orent the rofle crve as t moves along the trajectory crve? Answer: varos otons Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

212 General Sweeng Srfaces Fxed orentaton smle translaton of the coordnate system of the rofle crve along the trajectory crve Rotaton: f the trajectory crve s a crcle Move sng the Frenet Frame of the trajectory crve smoothly varyng orentaton Examle: srface of revolton Dfferental geometry fndamentals: Frenet frame Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

213 Frenet Swet Srfaces Orent the rofle Crve C sng the Frenet frame of Cv Pt C on the normal lane nb Place the orgnal of C on Cv Algn the x-axs of C wth n Algn the y-axs of C wth b Examle: f Cv s a crcle Varaton generalzaton Scale C as t moves Morh C nto C as t moves Use yor own magnaton! Deartment of Comter Scence CSE5 Lectre Notes Srng 5 Center for Vsal Comtng

214 Rled Srfaces Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

215 Rled Srfaces Move one straght lne along a crve Examle: lane cone cylnder Cylndrcal srface s v v a Srface eqaton s v v s Isoarametrc lnes More examles s v vs vq vb Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

216 Develoable Srfaces Deform a srface to lanar shae wthot length/area changes Unroll a srface to a lane wthot stretchng/dstortng Examle: cone cylnder Develoable srfaces vs. Rled srfaces More examles??? Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

217 Develoable Srface Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

218 Smmary Parametrc crves and srfaces Polynomals and ratonal olynomals Free-form crves and srfaces Other commonly-sed geometrc rmtves e.g. shere ellsod tors serqadrcs blobby etc. Motvaton: Fewer degrees of freedom More geometrc coverage Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

219 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

220 Straght Lne x y 4 x y 4> x y 4< Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

221 Straght Lne Mathematcs ax by c α ax by c α ax y c Examle x y 4 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

222 Crcle x y > x y < x y Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

223 Conc Sectons Mathematcs Examles Ellse Hyerbola Parabola Emty set Pont Par of lnes Parallel lnes Reeated lnes ax bxy cy dx ey f x x x x x x x x y 5 y 5 y y y y 7 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

224 Concs Parametrc eqatons of concs Generalzaton to hgher-degree crves How abot non-lanar satal crves Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

225 Plane x y z Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

226 Plane and Intersecton n c a b Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

227 Plane Examle x y z General lane eqaton Normal of the lane Arbtrary ont on the lane a n ax a a a x y z a b c by cz y Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

228 Plane Plane eqaton dervaton x a x ax by a y a cz a b z a Parametrc reresentaton gven three onts on the lane and they are non-collnear! x y a a y b c a c v v a b a c a z z Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

229 Plane Exlct exresson f c s non-zero z ax by d c Lne-Plane ntersecton l n n n n d lane lane lane Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

230 Crcle Imlct eqaton x Parametrc fncton y c θ < cos θ sn θ θ < π Parametrc reresentaton sng ratonal olynomals the frst qadrant x y [ ] Parametrc reresentaton s not nqe! Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

231 CSE5- Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

232 z f x y f x y Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

233 Imlct Eqatons for Crves Descrbe an mlct relatonsh Planar crve ont set x y f x y The mlct fncton s not nqe { } { x y αf x y } { x y αf x y } Comarson wth arametrc reresentaton x y Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

234 Imlct Eqatons for Crves Imlct fncton s a level-set Examles straght lne and conc sectons z z ax by ax f c x y bxy cy dx ey f Other examles Parabola two arallel lnes ellse hyerbola two ntersecton lnes Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

235 Imlct Fnctons for Crves Parametrc eqatons of concs Generalzaton to hgher-degree crves How abot non-lanar satal crves Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

236 Imlct Eqatons for Srfaces { x y z f x y z } Srface mathematcs Agan the mlct fncton for srfaces s not nqe { x y z αf x y z } { x y z αf x y z } Comarson wth arametrc reresentaton x v v y v z v Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

237 Imlct Eqatons for Srfaces Srface defned by mlct fncton s a level-set Examles w w Plane qadrc srfaces tor serqadrcs blobby objects Parametrc reresentaton of qadrc srfaces Generalzaton to hgher-degree srfaces f x y z Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

238 Qadrc Srfaces Imlct fnctons Examles ax Shere Cylnder Cone Parabolod Ellsod Hyerbolod More by cz dxy exz fyz gx hy jz k x x x x x x y y y y y y z 4 z 4 5 Two arallel lanes two ntersectng lanes sngle lane lne ont z z z Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

239 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Qadrcs: Parametrc Re. Shere Ellsod Geometrc meanng of these arameters Shere Ellsod Geometrc meanng of these arameters ] [ ]; [ sn sn cos cos cos π π β π π α α β α β α r z r y r x r z y x ] [ ]; [ sn sn cos cos cos π π β π π α α β α β α c z b y a x c z b y a x

240 Generalzaton Hgher-degree olynomals j k jk x y z j k a Non olynomals Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

241 Srng 5 CSE5 Lectre Notes Deartment of Comter Scence Center for Vsal Comtng Serqadrcs Geometry generalzaton of qadrcs Serellse Serellsod Parametrc reresentaton What s the meanng of these control arameters? Geometry generalzaton of qadrcs Serellse Serellsod Parametrc reresentaton What s the meanng of these control arameters? s s a y a x a z a y a x s s s s [ ]; [ sn sn cos sn cos π π β π π α α β α β α s s s s s a a a z y x

242 Algebrac Fncton Parametrc reresentaton s olar bt Formlaton Proertes j k jk x y z j k a Powerfl bt lack of modelng tools Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

243 Algebrac Patch Tetrahedron Control ont weght Algebrac atch Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

244 Algebrac Patch A tetrahedron wth non-lanar vertces v n v n v n v Trvarate barycentrc coordnate rst for r v n s v r s t n tv n v A reglar lattce of control onts and weghts jkl v jv j k l > ; j kv n n k l n l n n n n v n Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

245 Algebrac Patch There are nnn/6 control onts. A weght wijkl s also assgned to each control ont Algebrac atch formlaton Proertes j k l n j k Meanngfl control local control bondary nterolaton gradent control self-ntersecton avodance contnty condton across the bondares sbdvson w jkl n! r s! j! k! l! j t k l Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

246 Satal Crves Intersecton of two srfaces f g x x y y z z Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

247 Algebrac Sold Half sace { x { x y y z z f f x x y y z z < > }; or } Usefl for comlex objects refer to notes on sold modelng f x y z f f x y z x y z f x y z Λ Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

248 Volme Datasets Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

249 Isosrface Renderng Isovale Isovale Deartment of Comter Scence Center for Vsal Comtng Isovale Isovale CSE5 Lectre Notes Srng 5 Isovale Isovale STzNY BRzzK

250 Drect Volme Renderng Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

251 Imlct Fnctons Long hstory: classcal algebrac geometry Imlct and arametrc forms Advantages Dsadvantages Crves srfaces solds n hgher-dmenson Intersecton comtaton Pont classfcaton Larger than arameter-based modelng Unbonded geometry Object traversal Evalaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

252 Imlct Fnctons Effcent algorthms toolktssoftware Comter-based shae modelng and desgn Geometrc degeneracy and anomaly Algebrac and geometrc oeratons are often closed Mathematcs: algebrac geometry Symbolc comtaton Deformaton and transformaton Shae edtng renderng and control Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

253 Imlct Fnctons Converson between arametrc and mlct forms Imlctzaton vs. arameterzaton Strategy: ntegraton of both technqes Aroxmaton sng arametrc models Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

254 Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

255 Free-Form Deformaton Free-Form Deformaton Examle Orgnal Model Deformed Mesh Sold Mesh Reslt Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

256 Free-Form Deformaton Free-Form Deformaton Examle Comlex >> 49 faces Orgnal Model Sold Mesh Deformed Reslts n both srface rendered and wreframe Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

257 Free-Form Deformaton Free-Form Deformaton Examle Non-trval toology Orgnal Model Deformed Mesh Sold Mesh wth a hole Reslt no change n central cylnder Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

258 Free-Form Deformaton Free-Form Deformaton Examle Localzed Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

259 Shae Modelng Drect Modelng / Manlaton Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5

260 Materal Modelng Materal Reresentaton Non-homogeneos Deartment of Comter Scence Center for Vsal Comtng CSE5 Lectre Notes Srng 5