Linear and Affine Transformations Coordinate Systems

Transcription

1 Linear and Affine Transformations Coordinate Systems

2 Recall A transformation T is linear if

3 Recall A transformation T is linear if Every linear transformation can be represented as matrix

4 Linear Transformation Examples Uniform Scaling Non-uniform Scaling Rotations Reflections Orthogonal Projections Translations?

5 Problem with Translation Translation by not linear! Would like a unified framework for handling all transformations

6 Homogeneous Coordinates Main idea: add a dummy 4 th dimension points: vectors:

7 In Homogeneous Coordinates

8 In Homogeneous Coordinates

9 In Homogeneous Coordinates

10 In Homogeneous Coordinates

11 Homogeneous Coordinates Main idea: add a dummy 4 th dimension points: vectors: Now translation is matrix multiplication! 4 x 4 matrix transformations called affine

12 Linear Transformation Zoo Translation:

13 Linear Transformation Zoo Translation:

14 Linear Transformation Zoo Rotation:

15 Linear Transformation Zoo Rotation:

16 Linear Transformation Zoo Rotation: what about in homogeneous coordinates?

17 Linear Transformation Zoo Rotation:

18 Linear Transformation Zoo Rotation:

19 Linear Transformation Zoo Uniform scaling:

20 Linear Transformation Zoo Uniform scaling:

21 Linear Transformation Zoo Scaling:

22 What About Non-Axis-Aligned?

23 What About Non-Axis-Aligned? compose transformations!

24 What About Non-Axis-Aligned? compose transformations!

25 Linear Transformation Zoo Reflection:

26 Linear Transformation Zoo Reflection: axis to reflect

27 Linear Transformation Zoo Reflection:

28 Linear Transformation Zoo Shear:

29 Linear Transformation Zoo Shear:

30 Linear Transformation Zoo Shear: shear y-axis in x-axis direction

31 Linear Transformation Zoo Shear:

32 Combining Transformations matrix multiplication does not commute

33 Example: Rotate About Point

34 Example: Rotate About Point

35 Transforming Normals The problem:

36 Transforming Normals The problem:

37 Transforming Normals The problem: Points and vectors: Normals:

38 What is a Coordinate System? 1. an origin 2. a frame of vectors spanning space

39 What is a Coordinate System? 1. an origin 2. a frame of vectors spanning space usually orthonormal usually right-handed

40 What is a Coordinate System? 1. an origin 2. a frame of vectors spanning space usually orthonormal usually right-handed How represented?

41 What is a Coordinate System? 1. an origin 2. a frame of vectors spanning space usually orthonormal usually right-handed How represented? in other coordinates (turtles all the way down?)

42 Cartesian World Coordinates Canonical root coordinate system Usually y points up, x and z horizontal But this is arbitrary

43 Transforming Coordinate Systems Can define coordinate system in terms of world coordinates

44 Transforming Coordinate Systems Can define coordinate system in terms of world coordinates Given in world coords

45 Transforming Coordinate Systems Can define coordinate system in terms of world coordinates Given in world coords

46 Change of Coordinates Matrix Maps from local to world coordinates

47 Change of Coordinates Matrix Maps from local to world coordinates How to map back?

48 More Coordinates Systems world

49 More Coordinates Systems world

50 Coordinate Systems in Graphics world camera

51 Coordinate Systems in Graphics world view matrix (also called look at ) camera

52 Building the View Matrix Three axes: tangent, up, look

53 Building the View Matrix Three axes: tangent, up, look Note: camera looks down negative look direction for extra confusion

54 tangent up -look eye Building the View Matrix Three axes: tangent, up, look Note: camera looks down negative look direction for extra confusion

55 tangent up -look eye Building the View Matrix Three axes: tangent, up, look Note: camera looks down negative look direction for extra confusion

56 tangent up -look eye -R eye Building the View Matrix R tangent up -look

57 Coordinate Systems in Graphics world object view matrix camera

58 Why Use Object Coordinates?

59 Why Use Object Coordinates? Easier to work with / animate

60 Why Use Object Coordinates? Easier to work with / animate Instancing

61 Coordinate Systems in Graphics model matrix world object view matrix camera

62 Transformations Every transformation creates child coordinate system

63 Two Interpretations of Backwards: transforms applied right to left in original coordinate system

64 Two Interpretations of Forwards: transforms applied left to right in new coordinate systems

65 Two Interpretations of Same answer either way, but both interpretations useful

66 Scene Graph Represents hierarchy of transformations Assignment 3: bones in character body