Three-Dimensional Object Representations Chapter 8

Transcription

1 Three-Dimensional Object Representations Chapter 8

2 3D Object Representation A surace can be analticall generated using its unction involving the coordinates. An object can be represented in terms o its vertices, edges and polgons. (Wire Frame, Polgonal Mesh etc.) Curves and suraces can also be designed using splines b speciing a set o ew control points. = (,z) z...

3 Solid Modeling - Polhedron A polhedron is a connected mesh o simple planar polgons that encloses a inite amount o space. A polhedron is a special case o a polgon mesh that satisies the ollowing properties: Ever edge is shared b eactl two aces. At least three edges meet at each verte. Faces do not interpenetrate. Faces at most touch along a common edge. Euler s ormula : I F, E, V represent the number o aces, vertices and edges o a polhedron, then V + F E =. 3

4 3D Object Representation The data or polgonal meshes can be represented with: Verte List, and Face List (Polgon List) Additionall, we oten keep Normal List, and/or Edge List 4

5 Vertices and Faces - E.g. Cube Verte Inde Face Inde 5

6 Data representation using verte, ace and normal lists: Verte List Normal List Polgon List

7 Data representation using verte, ace and edge lists: Verte List Edge List Polgon List [i] [i] z[i] L[j] M[j] P[k] Q[k] R[k] S[k]

8 Normal Vectors (OpenGL) Assigning a normal vector to a polgon: glbegin(gl_polygon); glnormal3(n,n,zn); glverte3(,,z); glverte3(,,z); glverte3(3,3,z3); glverte3(4,4,z4); glend(); Enabling automatic conversion o normal vectors to unit vectors: glenable(gl_normalize); 8

9 Regular Polhedra (Platonic Solids) I all the aces o a polhedron are identical, and each is a regular polgon, then the object is called a platonic solid. Onl ive such objects eist. 9

10 Wire-Frame Models I the object is deined onl b a set o nodes (vertices), and a set o lines connecting the nodes, then the resulting object representation is called a wire-rame model. Ver suitable or engineering applications. Simplest 3D Model - eas to construct. Eas to clip and manipulate. Not suitable or building realistic models.

11 Wire Frame Eamples - Glut Some Wirerame Models in OpenGL: Cube: glutwirecube(gldouble size); Sphere: glutwiresphere( ); Torus: glutwiretorus( ); Teapot: glutwireteapot(gldouble size); Cone: glutwirecone( ); ( ) Reer tet or list o arguments.

12 Wire Frame Model - Utah Teapot

13 Polgonal Mesh Three-dimensional suraces and solids can be approimated b a set o polgonal and line elements. Such suraces are called polgonal meshes. The set o polgons or aces, together orm the skin o the object. This method can be used to represent a broad class o solids/suraces in graphics. A polgonal mesh can be rendered using hidden surace removal algorithms. 3

14 Polgonal Mesh - Eample 4

15 Solid Modeling Polgonal meshes can be used in solid modeling. An object is considered solid i the polgons it together to enclose a space. In solid models, it is necessar to incorporate directional inormation on each ace b using the normal vector to the plane o the ace, and it is used in the shading process. 5

16 Solid Modeling - Eample 6

17 Solid Modeling Eamples - Glut Some predeined Solid Models in OpenGL: Cube: glutsolidcube(gldouble size); Sphere: glutsolidsphere( ); Torus: glutsolidtorus( ); Teapot: glutsolidteapot(gldouble size); Cone: glutsolidcone( ); ( ) Arguments same as wire-rame models. 7

18 Quadric Suraces Suraces, which are described with second-degree equations The include spheres, elipsoids, tori, paraboloids, and hiperboloids 8

19 Sphere In Cartesian coordinates In parametric orm r z / /, sin, sin cos, cos cos r z r r 9

20 Elipsoid In Cartesian coordinates In parametric orm z r z r r / /, sin, sin cos, cos cos z r z r r

21 Torus,generated b rotating a circle In Cartesian coordinates r aial z r In parametric orm ( r r cos)cos, ( r aial aial z r sin, r cos)sin,

22 Torus,generated b rotating an ellipse In Cartesian coordinates In parametric orm z aial r z r r, sin, )sin cos (, )cos cos ( z aial aial r z r r r r

23 Superquadrics-Superellipse s s r r, sin, cos s s r r 3

24 Superelipsoid In Cartesian coordinates In parametric orm s z s s s s r z r r / /, sin, sin cos, cos cos s z s s s s r z r r 4

25 Surace Modeling = (, z) Y X Z Man suraces can be represented b an eplicit unction o two independent variables, such as = (, z). 5

26 Surace Modeling - Eample 6

27 Representations o Curves Use a sequence o points Piecewise linear - does not accuratel model a smooth line Tedious to create list o points Epensive to manipulate curve because all points must be repositioned Instead, model curve as piecewisepolnomial = (t), = (t), z = z(t) where (), (), z() are polnomials 7

28 Maniold Splines, X. Gu, Y. He & H. Qin, Solid and Phsics Modeling 5. Spline Representations A spline is a smooth curve deined mathematicall using a set o constraints Splines have man uses: D illustration Fonts 3D Modelling Animation ACM 987 Principles o traditional animation applied to 3D computer animation 8

29 Big Idea User speciies control points Deines a smooth curve Curve Control Points Control Points 9

30 Images taken rom Hearn & Baker, Computer Graphics with OpenGL (4) Interpolation Vs Approimation A spline curve is speciied using a set o control points There are two was to it a curve to these points: Interpolation - the curve passes through all o the control points Approimation - the curve does not pass through all o the control points 3

31 Speciing Curves Control Points A set o points that inluence the curve s shape Knots Control points that lie on the curve Interpolating Splines Curves that pass through the control points (knots) Approimating Splines Control points merel inluence shape 3

32 Images taken rom Hearn & Baker, Computer Graphics with OpenGL (4) Conve Hulls The boundar ormed b the set o control points or a spline is known as a conve hull Think o an elastic band stretched around the control points 3

33 Images taken rom Hearn & Baker, Computer Graphics with OpenGL (4) Control Graphs A polline connecting the control points in order is known as a control graph Usuall displaed to help designers keep track o their splines 33

34 Cubic Polnomials (t) = a t 3 + b t + c t + d Similarl or (t) and z(t) Let t: ( <= t <= ) Let T = [t 3 t t ] Coeicient Matri C Curve: Q(t) = T*C z z z z d c d c d c b b b a a a t t t 3 34

35 Another nice eature o curves Derivatives Ver useul or lighting equations Useul or automatic eature detection 35

36 Piecewise Curve Segments One curve constructed b connecting man smaller segments end-to-end Must have rules or how the segments are joined Continuit describes the joint Parametric continuit Geometric continuit 36

37 Parametric Continuit C is tangent continuit (velocit) C is nd derivative continuit (acceleration) Matching direction and magnitude o d n / dt n C n continous 37

38 Geometric Continuit I positions match G geometric continuit I direction (but not necessaril magnitude) o tangent matches G geometric continuit The tangent value at the end o one curve is proportional to the tangent value o the beginning o the net curve 38

39 Parametric Cubic Curves In order to assure C continuit, curves must be o at least degree 3 Here is the parametric deinition o a cubic (degree 3) spline in two dimensions How do we etend it to three dimensions? 39

40 Parametric Cubic Splines Can represent this as a matri too 4

41 Coeicients So how do we select the coeicients? [a b c d ] and [a b c d ] must satis the constraints deined b the knots and the continuit conditions 4

42 Parametric Curves Diicult to conceptualize curve as (t) = a t 3 + b t + c t + d Instead, deine curve as weighted combination o 4 well-deined cubic polnomials Each curve tpe deines dierent cubic polnomials and weighting schemes 4

43 Parametric Curves Hermite two endpoints and two endpoint tangent vectors Bezier - two endpoints and two other points that deine the endpoint tangent vectors Splines our control points C and C continuit at the join points Come close to their control points, but not guaranteed to touch them 43

44 Hermite Cubic Splines An eample o knot and continuit constraints 44

45 Hermite Cubic Splines One cubic curve or each dimension A curve constrained to /-plane has two curves: d c b a t t t t d ct bt at t ) ( ) ( 3 3 h g e t t t t h gt t et t ) ( ) (

46 Hermite Cubic Splines A -D Hermite Cubic Spline is deined b eight parameters: a, b, c, d, e,, g, h How do we convert the intuitive endpoint constraints into these (relativel) unintuitive eight parameters? We know: (, ) position at t =, p (, ) position at t =, p (, ) derivative at t =, dp/dt (, ) derivative at t =, dp/dt 46

47 Hermite Cubic Spline We know: (, ) position at t =, p p d d c b a d c b a 3 3 () () () p h h g e h g e 3 3 () () () 47

48 Hermite Cubic Spline We know: (, ) position at t =, p p d c b a d c b a d c b a 3 3 () () () p h g e h g e h g e 3 3 () () () 48

49 Hermite Cubic Splines So ar we have our equations, but we have eight unknowns Use the derivatives d c b a t t t c bt at t d ct bt at t 3 ) ( 3 ) ( ) ( 3 h g e t t t g t et t h gt t et t 3 ) ( 3 ) ( ) ( 3 49

50 Hermite Cubic Spline We know: (, ) derivative at t =, dp/dt dt dp c d c b a c b a () 3 () 3 () dt dp g h g e g e () 3 () 3 () 5

51 Hermite Cubic Spline We know: (, ) derivative at t =, dp/dt dt dp c b a d c b a c b a 3 () 3 () 3 () dt dp g e h g e g e 3 () 3 () 3 () 5

52 Hermite Speciication Matri equation or Hermite Curve p p r p r p t 3 t t t 3 a b c d e g h p p dp dp dt dt dp dp p p dt dt t = t = t = t = 5

53 Solve Hermite Matri h g e d c b a dt dp dt dp p p dt dp dt dp p p 3 53

54 Spline and Geometr Matrices p p a e p 3 3 p b dp dp dt dt c g dp d h dp dt dt M Hermite G Hermite 54

55 Resulting Hermite Spline Equation 55

56 Sample Hermite Curves 56

57 Blending Functions B multipling irst two matrices in lowerlet equation, ou have our unctions o t that blend the our control parameters These are blending unctions 57

58 Hermite Blending Functions I ou plot the blending unctions on the parameter t 58

59 Hermite Blending Functions Remember, each blending unction relects inluence o P, P, DP, DP on spline s shape 59

60 Bézier Spline Curves A spline approimation method developed b the French engineer Pierre Bézier or use in the design o Renault car bodies A Bézier curve can be itted to an number o control points although usuall 4 are used 6

61 Bézier Spline Curves (cont ) Consider the case o n+ control points denoted as p k =( k, k, z k ) where k varies rom to n The coordinate positions are blended to produce the position vector P(u) which describes the path o the Bézier polnomial unction between p and p n P( u) n k p k BEZ k, n( u), u 6

62 Bézier Spline Curves (cont ) The Bézier blending unctions BEZ k,n (u) are the Bernstein polnomials BEZ k, n ( u) k nk C( n, k) u ( u) where parameters C(n,k) are the binomial coeicients n! C( n, k) k!( n k)! 6

63 Bézier Spline Curves (cont ) So, the individual curve coordinates can be given as ollows n k n k k u BEZ u, ) ( ) ( n k n k k u BEZ z u z, ) ( ) ( n k n k k u BEZ u, ) ( ) ( 63

64 Images taken rom Hearn & Baker, Computer Graphics with OpenGL (4) Bézier Spline Curves (cont ) 64

65 Important Properties o Bézier Curves The irst and last control points are the irst and last point on the curve P() = p P() = p n The curve lies within the conve hull as the Bézier blending unctions are all positive and sum to n k BEZ k, n( u) 65

66 Cubic Bézier Curve Man graphics packages restrict Bézier curves to have onl 4 control points (i.e. n = 3) The blending unctions when n = 3 are simpliied as ollows: BEZ BEZ BEZ BEZ,3,3,3 3,3 ( u) 3u u 3 3 3u( u) ( u) 66

67 Cubic Bézier Blending Functions 67

68 Bézier Spline Curve Eercise (3, 7) (, 5) (, 4) (7, ) 68

69 Properties o Bezier Curve Bezier curve is a polnomial o degree one less than the number o control points p p p 3 p p p p Quadratic Curve Cubic Curve 69

70 Properties o Bezier Curve (cont.) Bezier curve alwas passes through the irst and last points; i.e., and () () () () m m 7

71 Properties o Bezier Curve (cont) The slope at the beginning o the curve is along the line joining the irst two control points, and the slope at the end o the curve is along the line joining the last two points. p p p

72 Properties o Bezier Curve (cont) Bezier blending unctions are all positive and the sum is alwas. m i Bez m, i ( u) This means that the curve is the weighted sum o the control points. 7

73 Design Technique using Bezier Curves: A closed Bezier curve can be generated b speciing the irst and last control points at the same location p 3 p p 4 p p =p 5 73

74 Design Technique (Cont) A Bezier curve can be made to pass closer to a given coordinate position b assigning multiple control points to that position. p = p p 3 p p 4 74

75 A Bezier curve can be ormed b piecing several Bezier section with lower degree. p p = p p 3 p p p 75

76 Bezier Suraces m r( u, v) Bez ( u) Bez ( v) r, i j m i l, j i j u, v where r( u, v) ( ( u, v), ( u, v), z( u, v)) r (,, z ) ij ij ij ij l 76

77 Bezier Patch A set o 6 control points The Bezier Patch 77

78 Bezier Patch Utah Teapot Deined Using Control Points 78

79 Bezier Patch Utah Teapot Generated Using Bezier Patches 79

80 B-Spline Curves Most widel used Commonl available in CAD sstems and graphics packages Like Bezier splines, B-Splines are generated b approimating a set o control points advantages over Bezier splines degree o the polnomial is (generall) independent o number o control points allows local shape control Disadvantage: more comple 8