GRK. dr Wojciech Palubicki

Transcription

1 GRK dr Wojciech Palubicki

2 Projekty (dwu-osobowe) Napisać program w OpenGLu w którym następujące cechy są zawarte: Scena pokazuje jakąś kreaturę wodną (Może to być np. ryba, roślina lub korala) Można w jakiś sposób wejść w interakcję ze sceną i scena jest animowana Wszystkie wielokąty są teksturowane Wszystkie wielokąty są oświetlone Napisać dokumentacje która opisuję program (około 30 wierszy) Przygotować 5 minutową prezentację projektu na ostatnich ćwiczeń Implementacja (50%), opis i prezentacja (50%)

3 Wybrać jakiś zaawansowany efekt graficzny Normal mapping Proceduralne tekstury: np. szum Perlina Hierarchia animacji Implementacja parallel transport frames Własne pomysły

4 Modeling with Transformations Create elementary geometric objects, then rotate, translate and scale them until you define a model

5 Modeling with Transformations But individual parts dont move in a constrained way to each other To introduce constraints and express kinematics we need to parametrize our model

6 Model to World Frame

7 Model World Position and orient the robot hammer in world space

8 Model World Each part of the object is transformed independently relative to the origin

9 Model World Alternatively transform every object relative to it s parent

10 Relative Transformations

11 Hierarchical Transforms

12 Making an Articulated Arm A minimal 2D jointed object: Two pieces, A( forearm ) and B( upper arm ) Attach point c on B to point a on A ( elbow ) Desired parameters: Shoulder position S (point at which b winds up) Shoulder angle β (A and B rotate together about b) Elbow angle α (A rotates about a = c)

13 Making an Arm: Step 1 Start with A and B in their untransformed configurations (B is hiding behind A) First apply a series of transformations to A.

14 Making an Arm: Step 2 Translate by a, bringing a to the origin

15 Making an Arm: Step 3 Next, rotate A by the elbow angle α

16 Making an Arm: Step 4 Translate A to form the elbow joint a c

17 Making an Arm: Step 5 Translate both objects by a bringing a to the origin (A nd B move together)

18 Making an Arm: Step 6 Next rotate by the shoulder angle β

19 Making an Arm: Last Step Finally, translate by the shoulder position S, bringing the arm to its final position

20 Parametrization S,α, β are parameters of the model a,b and c are structural constants

21 Hierarchical Transforms

22 Model Construction

23 Scene Graph

24 Scene Graph

25 Scene Graph Model e.g. as a stack Down the graph is a push operation Up the graph a pop operation

26 Scene Graph OpenGL 3.0+

27 Forward Kinematics

28 Inverse Kinematics

29 Inverse Kinematics Solution for more complex structures Non-linear optimization Keyframing with interpolation

30 Keyframing

31 Keyframing

32 Interpolation

33 Interpolation

34 Interpolation

35 Linear Interpolation

36 Linear Interpolation

37 Linear Interpolation: Limitations May need a large number of keyframes if motion is non-linear

38 Parametric Curves Define a continuous smooth curve f passing through the data points Explicit form y = f(x) Implicit form f(x, y) = 0 Parametric form x = f(t), y = g(t)

39 Równania parametryczne Parametr t jest używany żeby ustalić wartości zmiennych x t = 1 t x 0 + tx 1 y t = 1 t y 0 + ty 1 Gdzie 0 t 1. Niech P 0 = x 0, y 0, P 1 = (x 1, y 1 ) i P = (x, y), wtedy P(t) = (1 t) P 0 + tp 1

40 Równania parametryczne Parametr t jest używany żeby ustalić wartości zmiennych x t = 1 t x 0 + tx 1 y t = 1 t y 0 + ty 1 Gdzie 0 t 1. Niech P 0 = x 0, y 0, P 1 = (x 1, y 1 ) i P = (x, y), wtedy P(t) = (1 t) P 0 + tp 1 P 1 P 0 P(0.5)

41 Parametric Curve Example What curve does this represent?

42 Cubic Curves We can use a cubic function to represent a smooth curve in 3D Vector Form:

43 Cubic Curves We can use a cubic function to represent a smooth curve in 3D Vector Form:

44 Smooth Curves Controlling the shape of the curve

45 Constraints on the Cubics How many constraints do we need to determine a cubic curve?

46 Constraints on the Cubics How many constraints do we need to determine a cubic curve?

47 Constraints on the Cubics How many constraints do we need to determine a cubic curve?

48 Natural Cubic Curves

49 Natural Cubic Spline A spline is a curve that is piecewise-defined and is smooth at the places where the pieces connect

50 Natural Cubic Spline A spline is a curve that is piecewise-defined and is smooth at the places where the pieces connect 1st spline

51 Natural Cubic Spline A spline is a curve that is piecewise-defined and is smooth at the places where the pieces connect 2nd spline

52 Natural Cubic Spline A spline is a curve that is piecewise-defined and is smooth at the places where the pieces connect 3rd spline

53 Hermite Curves A Hermite curve is a cubic curve determined by Endpoints p 0 and p 1 Tangent vectors (velocities) v 0 and v 1 at endpoints

54 Example of Hermite Curves

55 Tangents (Derivatives)

56 Tangents (Derivatives)

57 Tangents (Derivatives)

58 Hermite Curves The value of the curve is Q(0)=p 0 at t=0 and Q(1)=p 1 at t=1 The derivative of the curve to be v 0 at t=0 and v 1 at t=1

59 Hermite Curves

60 Hermite Curves

61 Hermite Curves

62 Hermite Curves

63 Hermite Curves

64 Hermite Curves

65 Hermite Curves

66 Hermite Interpolation

67 Hermite Interpolation

68 Bezier Curves Variations of Hermite curves Indirectly specify tangent vectors, by specifying two intermediate points

69 Bezier Curve Formulation There exist a number of ways to formulate Bezier curves De Casteljau (recursive linear interpolations) Bernstein polynomials Cubic equations Matrix form most useful for CG purposes

70 Bezier Curves Find the curve Q(t) as a function of parameter t Endpoints p 0 and p 3 Tangents alinged with p 0 p 1 and p 3 p 2

71 De Casteljau

72 De Casteljau

73 De Casteljau

74 De Casteljau

75 De Casteljau

76 De Casteljau

77 De Casteljau

78 De Casteljau For four control points P 0,0, P 1,0, P 2,0 and P 3,0 : P 0,3 t = (1 t) 3 P 0,0 + 3t(1 t) 2 P 1,0 + 3t 2 (1 t)p 2,0 + t 3 P 3,0

79 Bezier Cubic Form For four control points P 0,0, P 1,0, P 2,0 and P 3,0 : P 0,3 t = (1 t) 3 P 0,0 + 3t(1 t) 2 P 1,0 + 3t 2 (1 t)p 2,0 + t 3 P 3,0

80 Bezier Matrix Form

81 Hermite vs. Bezier Curves

82 Hermite vs. Bezier Curves Bezier and Hermite curves can be related as follows:

83 Chaining Curve Segments B-spline Bézier 10th degree

84 Problems with Splines? Require explicit definition of derivatives at each point.

85 Catmull-Rom spline Tangent at point P i is half of the vector connecting P i-1 to P i+1

86 Catmull-Rom spline Tangent at point P i is half of the vector connecting P i-1 to P i+1

87 Catmull-Rom Example

88 Catmull-Rom Spline

89 Catmull-Rom Spline

90 Catmull-Rom Spline

91 Catmull-Rom Spline Curve piece between control points p i and p i+1 First and last curve segments are special cases

92 Catmull-Rom Spline Curve piece between control points p i and p i+1 First and last curve segments are special cases

93 Closed-loop Catmull-Rom

94 Parametric Curves and Animation of Objects

95 Parametric Curves and Animation of Objects Use parameter t of a curve Q(t) to represent time in animation

96 Curve Framing Defining orientation for every point of the curve Useful if curve is trajectory of a camera Or clone objects onto the curve and have their orientations vary smoothly

97 Curve Framing Have only tangent vector T

98 Parallel Transport Frames Hansen et al. 1995

99 Parametric Curve

100 Tangents

101 Torsion of a Curve

102 Torsion of a Curve

103 Torsion of a Curve

104 Smooth Variation of Torsion

105 Parallel Transport

106 Parallel Transport

107 Parallel Transport

108 Parallel Transport

109 Parallel Transport θ

110 Parallel Transport θ

111 Parallel Transport θ

112 Animation Forward Kinematics Scene Graph Keyframing Interpolation with parametric curves Curve Framing

113 Animation Forward Kinematics Scene Graph Keyframing Interpolation with parametric curves Curve Framing Physically-based Kinematics

114 Animation Forward Kinematics Scene Graph Keyframing Interpolation with parametric curves Curve Framing Physically-based Kinematics Motion Capture