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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