Transformation CS 148: Summer 2016 Introduction of Graphics and Imaging Zahid Hossain

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1 Transformation CS 148: Summer 2016 Introduction of Graphics and Imaging Zahid Hossain

2 Placement of Objects Fisher et al. (2012) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 2

3 Placement of Objects Oriented Fisher et al. (2012) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 3

4 Placements of Objects Translated Fisher et al. (2012) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 4

5 Points and Vectors y p z Origin x CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 5

6 Points and Vectors y Origin x Vector Points z CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 6

7 Transformations What? Just functions acting on points: Why? Viewing: Convert between coordinates systems Virtual camera, e.g. perspective projections Modeling: Create objects in a convenient orientation Use multiple instances of a given shape Kinematics characters/robots 7

8 Linear Transformation CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 8

9 Linear Transformation CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 9

10 Linear Transformation CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 10

11 Linear Transformation Lines Map to Lines CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 11

12 Common Transformation Rotate Scale Shear 12

13 13

14 Composing Transformations First Rotate 45 CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 14

15 Composing Transformations Then Scale 2x along y CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 15

16 Composing Transformations Scale Rotation Order of transformations CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 16

17 Composing Transformation Not Commutative Rotate -> Scale Scale -> Rotate CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 17

18 Translation CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 18

19 Homogenous Coordinates For any non-zero c CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 19

20 Homogenous Coordinates CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 20

21 Homogenous Coordinates CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 21

22 Homogenous Coordinates 4D Homogenous Coordinate 3D Cartesian Coordinate CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 22

23 Homogenous Coordinates w-axis (x,y,w) w=1 (x/w,y/w,1) x,y-axis CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 23

24 Homogenous Coordinates z-axis Decrease w z=1 x,y-axis CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 24

25 Homogenous Coordinates z-axis Decrease w z=1 x,y-axis CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 25

26 Homogenous Coordinates CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 26

27 Homogenous Coordinates Represents a Vector! (Homogenous Coordinates express both Vectors and Points) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 27

28 Back to Translation Homogenous Coordinate Homogenous Coordinate CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 28

29 Some Exercise Convert 3D Cartesian Coordinate to Homogenous Coordinate (3,4,2) : CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 29

30 Some Exercise Convert 3D Cartesian Coordinate to Homogenous Coordinate (3,4,2) : (3,4,2,1) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 30

31 Some Exercise Convert 3D Cartesian Coordinate to Homogenous Coordinate (3,4,2) : (3,4,2,1) Convert Homogenous Coordinate to Cartesian Coordinate (3,4,2,2) : CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 31

32 Some Exercise Convert 3D Cartesian Coordinate to Homogenous Coordinate (3,4,2) : (3,4,2,1) Convert Homogenous Coordinate to Cartesian Coordinate (3,4,2,2) : (1.5,2,1) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 32

33 Some Exercise Rotate about the center of the box CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 33

34 Some Exercise Translate to center CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 34

35 Some Exercise Rotate CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 35

36 Some Exercise Translate back CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 36

37 Coordinate System CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 37

38 Coordinate System CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 38

39 Coordinate System CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 39

40 Coordinate System: Rotation CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 40

41 Coordinate System: Rotation CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 41

42 Coordinate System: Rotation CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 42

43 Coordinate System: Hierarchy CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 43

44 Coordinate System: Hierarchy CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 44

45 Coordinate System: Hierarchy CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 45

46 Interpreting Transformations CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 46

47 Two Interpretations With respect to Global Frame With respect to Local Frame CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 47

48 Coordinate System: Hierarchy CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 48

49 w.r.t Global Frame Translate(1,1) Rotate(45 ) Translate(0,1) Combined = Translate(0,1) Rotate(45 ) Translate(1,1) Order of Transforms CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 49

50 w.r.t Local Frame Translate(0,1) Rotate(45 ) Translate(1,1) Combined = Translate(0,1) Rotate(45 ) Translate(1,1) Order of Transform CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 50

51 w.r.t Local Frame Translate(0,1) Rotate(45 ) Translate(1,1) Combined = Translate(0,1) Rotate(45 ) Translate(1,1) Order of Transform Both interpretations are equivalent CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 51

52 Hierarchical Modeling y x translate(0,4) drawtorso() pushmatrix() translate(1.5,0) rotatex(lefthiprotate) drawthigh() pushmatrix() translate(0,-2) rotatex(leftkneerotate) drawleg()... popmatrix() popmatrix() pushmatrix() translate(-1.5,0) rotatex(righthiprotate) // Draw the right side Matrix Stack CurrentMatrix = Translate(0,4) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 52

53 Hierarchical Modeling y x translate(0,4) drawtorso() pushmatrix() translate(1.5,0) rotatex(lefthiprotate) drawthigh() pushmatrix() translate(0,-2) rotatex(leftkneerotate) drawleg()... popmatrix() popmatrix() pushmatrix() translate(-1.5,0) rotatex(righthiprotate) // Draw the right side Translate(0,4) Matrix Stack CurrentMatrix = Translate(0,4) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 53

54 Hierarchical Modeling y x translate(0,4) drawtorso() pushmatrix() translate(1.5,0) rotatex(lefthiprotate) drawthigh() pushmatrix() translate(0,-2) rotatex(leftkneerotate) drawleg()... popmatrix() popmatrix() pushmatrix() translate(-1.5,0) rotatex(righthiprotate) // Draw the right side translate(0,4) translate(1.5,0) rotatex(lefthiprotate) translate(0,4) Matrix Stack CurrentMatrix = translate(0,4) translate(1.5,0) rotatex(lefthiprotate) translate(0,-2) rotate(leftkneerotate) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 54

55 Hierarchical Modeling y x translate(0,4) drawtorso() pushmatrix() translate(1.5,0) rotatex(lefthiprotate) drawthigh() pushmatrix() translate(0,-2) rotatex(leftkneerotate) drawleg()... popmatrix() popmatrix() pushmatrix() translate(-1.5,0) rotatex(righthiprotate) // Draw the right side translate(0,4) Matrix Stack CurrentMatrix = translate(0,4) translate(1.5,0) rotatex(lefthiprotate) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 55

56 Hierarchical Modeling y x translate(0,4) drawtorso() pushmatrix() translate(1.5,0) rotatex(lefthiprotate) drawthigh() pushmatrix() translate(0,-2) rotatex(leftkneerotate) drawleg()... popmatrix() popmatrix() pushmatrix() translate(-1.5,0) rotatex(righthiprotate) // Draw the right side Matrix Stack CurrentMatrix = translate(0,4) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 56

57 Hierarchical Modeling y x translate(0,4) drawtorso() pushmatrix() translate(1.5,0) rotatex(lefthiprotate) drawthigh() pushmatrix() translate(0,-2) rotatex(leftkneerotate) drawleg()... popmatrix() popmatrix() pushmatrix() translate(-1.5,0) rotatex(righthiprotate) // Draw the right side translate(0,4) Matrix Stack CurrentMatrix = translate(0,4) translate(-1.5,0) rotatex(righthiprotate) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 57

58 Hierarchical Modeling translate(0,4) Torso pushmatrix() translate(1.5,0) rotatex(righthiprotate) RightThigh pushmatrix() y Left Side popmatrix() RightLeg translate(0,-2) rotatex(rightlegrotate) pushmatrix() x CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 58

59 Camera and Projection Matrices CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 59

60 3D to 2D Conversion The world is in 3D But our Screen is in 2D Imagine our screen is a camera looking out into the world. Tasks 1. Convert 3D world coordinates to Camera Coordinates 2. Project the Camera Coordinate on 2D screen CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 60

61 Camera Matrix CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 61

62 Camera Matrix Homework! CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 62

63 Projection Matrix (Basic) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 63

64 Projection Matrix (Basic) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 64

65 Normalized Device Coordinates NDC (Normalized Device Coordinate) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 65

66 OpenGL Projection Matrix Note the n and f (to be consistent with OpenGL) CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 66

67 OpenGL Projection Matrix CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 67

68 OpenGL Projection Matrix Similarly for y CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 68

69 OpenGL Projection Matrix z is a little tricky Solving for A and B CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 69

70 OpenGL Projection Matrix CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 70

71 Non-Linearity in Z Monotonic: values keeps increasing as z goes in ve direction 2. Resolution decreases as z decrease (along ve) or depth increases CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 71

72 Non-Linearity in Z For Later: Has interesting effects in Depth Buffering Monotonic: values keeps increasing as z goes in ve direction 2. Resolution decreases as z decrease (along ve) or depth increases CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 72

73 Start Learning OpenGL Now CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 73

74 Start Learning OpenGL Now Suggestions Only For OpenGL 3.3: For OpenGL < 2.0: Any GLUT or FreeGLUT tutorial CS 148: Introduction to Computer Graphics and Imaging (Summer 2016) Zahid Hossain 74

75 Transformation CS 148: Summer 2016 Introduction of Graphics and Imaging Zahid Hossain

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