Jinxiang Chai CSCE441: Computer Graphics Coordinate & Composite Transformations 0

Outline 2D/3D Coordinate transformation 2D/3D Composite transformation Required readings: HB 7-8, 9-6 1

Image space Coordinate Transform: 3D Geometry Pipeline 2 Normalized project space View space World spaceObject space Aspect ratio & resolution Focal length Rotate and translate the camera

Coordinate Transformation: 3D Modeling/Design Coordinate transformation from one reference frame to another 3

Coordinate Transformation: Animation/Robotics How to model 2D movement of animated characters or robots? 4 Click herehere

Coordinate Transformation Coordinate transformation from one reference frame to another 5

Coordinate Transformation Coordinate transformation from one reference frame to another 6 Local reference frame

Coordinate Transformation Coordinate transformation from one reference frame to another 7 Local reference frame Global reference frame

Coordinate Transformation Coordinate transformation from one reference frame to another ? 8 Local reference frame Global reference frame

Review – Vector Operations Dot Product 9

Review – Vector Operations Dot Product: measuring similarity between two vectors 10

Review – Vector Operations Dot Product: measuring similarity between two vectors 11

Review – Vector Operations Dot Product: measuring similarity between two vectors Unit vector: 12

Review – Vector Operations Dot Product: measuring similarity between two vectors 13

Review – Vector Operations Dot Product: measuring similarity between two vectors 14

Review – Vector Operations Cross Product: measuring the area determined by two vectors 15

Review – Vector Operations Cross Product: measuring the area determined by two vectors 16

2D Coordinates 2D Cartesian coordinate system: 17

2D Coordinate Transformation 2D Cartesian coordinate system: 18

2D Coordinate Transformation 2D Cartesian coordinate system: any 2D vector can be represented as 19

2D Coordinate Transformation 2D Cartesian coordinate system: P: (x,y) 20

2D Coordinate Transformation 2D Cartesian coordinate system: P: (x,y) 21

2D Coordinate Transformation Transform object description from to p 22

2D Coordinate Transformation Transform object description from to p 23 Given the coordinates (x’,y’) in i’j’ - how to compute the coordinates (x,y) in ij?

2D Coordinate Transformation Transform object description from to p 24 Given the coordinates (x’,y’) in i’j’ - how to compute the coordinates (x,y) in ij?

2D Coordinate Transformation Transform object description from to p 25 Given the coordinates (x’,y’) in i’j’ - how to compute the coordinates (x,y) in ij?

2D Coordinate Transformation Transform object description from to p 26

2D Coordinate Transformation Transform object description from to p 27

2D Coordinate Transformation Transform object description from to p 28

2D Coordinate Transformation Transform object description from to p 29

2D Coordinate Transformation Transform object description from to p 30

2D Coordinate Transformation Transform object description from to p 31

2D Coordinate Transformation Transform object description from to p 32

2D Coordinate Transformation Transform object description from to p 33

2D Coordinate Transformation Transform object description from to p 34

2D Coordinate Transformation Transform object description from to p 35

2D Coordinate Transformation Transform object description from to p 36

2D Coordinate Transformation Transform object description from to p 37

2D Coordinate Transformation Transform object description from to p 38

2D Coordinate Transformation p 39 What does this column vector mean?

2D Coordinate Transformation Transform object description from to p 40 What does this column vector mean? Vector i’ in the new reference system

2D Coordinate Transformation Transform object description from to p 41 What does this column vector mean?

2D Coordinate Transformation Transform object description from to p 42 What does this column vector mean? Vector j’ in the new reference system

2D Coordinate Transformation Transform object description from to p 43 What does this column vector mean?

2D Coordinate Transformation Transform object description from to p 44 What does this column vector mean? The old origin in the new reference system

2D Coordinate Transformation 2D translation p 45

2D Coordinate Transformation 2D translation p ? ? ? ? 46

2D Coordinate Transformation 2D translation p

2D Coordinate Transformation 2D translation&rotation p ? 48

2D Coordinate Transformation 2D translation&rotation p ? 49

2D Coordinate Transformation 2D translation&rotation p 50

2D Coordinate Transformation 2D translation&rotation p ? 51

2D Coordinate Transformation 2D translation&rotation p 52

2D Coordinate Transformation 2D translation&rotation p ? 53

2D Coordinate Transformation 2D translation&rotation p 54

2D Coordinate Transformation An alternative way to look at the problem P=[x,y] 55 - set up a transformation that superimposes the x’y’ axes onto the xy axis

2D Coordinate Transformation An alternative way to look at the problem P=[x,y] 56 - set up a transformation that superimposes the x’y’ axes onto the xy axis

2D Coordinate Transformation An alternative way to look at the problem P=[x,y] 57 - set up a transformation that superimposes the x’y’ axes onto the xy axis

2D Coordinate Transformation An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) p 58

2D Coordinate Transformation An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to (x,y)? p 59

2D Coordinate Transformation An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p 60

2D Coordinate Transformation An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p 61

2D Coordinate Transformation An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p 62

2D Coordinate Transformation An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p 63

2D Coordinate Transformation An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’) How to transform the point from (x’,y’) to (x,y)? Invert the matrix! p 64

2D Coordinate Transformation Same results! p 65

2D Coordinate Transformation 2D translation&rotation p 66

2D Coordinate Transformation 2D translation&rotation p 67

2D Coordinate Transformation 2D translation&rotation p 68

2D Coordinate Transformation 2D translation&rotation p 69

2D Coordinate Transformation 2D translation&rotation p 70

2D Coordinate Transformation 2D translation&rotation p 71

3D Coordinate Transformation Transform object description from to p 72

2D Coordinate Transformation Transform object description from to p 73

3D Coordinate Transformation Transform object description from to p 74

3D Coordinate Transformation Transform object description from to p 75

3D Coordinate Transformation Transform object description from to 76 p

3D Coordinate Transformation Transform object description from to 77 x y z

Composite 2D Transformation How to model 2D movement of characters or robots? 78 Click herehere

Composite 2D Transformation A 2D lamp character 79

Composite 2D Transformation A 2D lamp character – skeleton size 80

Composite 2D Transformation How can we draw the character given the pose ? 81

Composite 2D Transformation How can we draw the character given the pose ? 82 - This requires computing the global coordinates for any point on the character. - But we only have local coordinates of points. - So how can we map the local coordinates to the global coordinates?

Articulated Character Local reference frames with a default pose (0,0,0,0,0,0)

Composite 2D Transformation What’s the pose? 84

Composite 2D Transformation What’s the pose? 85

Composite 2D Transformation A 2D lamp character Given,, how to compute the global position of a point (e.g., A) based on its local coordinates? ? 86

Composite 2D Transformation What’s local coordinate ? ? 87

Composite 2D Transformation What’s local coordinate ? ? 88

Composite 2D Transformation What’s the current coordinate A ? ? 89

Composite 2D Transformation What’s the current coordinate A ? ? 90

Composite 2D Transformation What’s the current coordinate A ? ? 91

Composite 2D Transformation What’s the current coordinate A ? ? 92

Composite 2D Transformation What’s the current coordinate A ? ? 93

Composite 2D Transformation What’s the current coordinate A ? 94

How to Animate the Character? A 2D lamp character 95

How to Animate the Character? Keyframe animation - Manually pose the character by choosing appropriate values for - Linearly interpolate the inbetween poses. - Works for any types of articulated characters! 96

Composite 3D Transformation Similarly, we can easily extend composite transformation from 2D to 3D 97

Composite 3D Transformation 98