Transformations. Prof. Dr. Markus Gross

Transcription

1 Transformations Prof. Dr. Markus Gross

2 Transformations Transformations map geometry

3 Transformations Transformations in the graphics pipeline: Change position & orientation of objects Project objects to screen Animate objects

4 Notation Points and vectors are represented as Matrices are represented as A point is transformed as Transpose:

5 Linear maps Definitions Represented by matrices Affine maps

6 Translation 2D Transformations Scaling

7 2D Transformations Rotation by angle In matrix form

8 Homogeneous Coordinates Affine maps are linear maps in homogeneous coordinates

9 Homogeneous Coordinates Translation is represented as a matrix

10 Homogeneous Coordinates Rotation Scaling

11 Homogeneous Coordinates Shear along x- and y- axis

12 Homogeneous Coordinates A point has infinitely many homogeneous coordinates, for any

13 Homogeneous Coordinates Point as a line in 3D homogenous coordinates affine plane

14 Combining Transformations Combine via matrix multiplication Example: rotation followed by translation

15 Combining Transformations Commutativity Matrix Translation Rotation Scaling Scaling Matrix Translation Rotation Scaling Rotation Only for 2D!

16 3D Transformations Homogeneous coordinates: 4x4 matrices Project onto the hyperplane

17 3D Transformations Translation Scaling

18 3D Transformations Rotation around the x-, y-, z- axis

19 3D Transformations Rotation of angle around an axis

20 3D Transformations Shearing parallel to the principal planes

21 Coordinate Systems Represent a point/vector as a linear combination of orthonormal basis vectors

22 Coordinate Systems Change of coordinate systems t

23 Coordinate Systems Change of coordinate systems t

24 Coordinate Systems Change of coordinate systems t Rotation Translation

25 Coordinate Systems Change of coordinate systems t

26 Transforming Normal Vectors Surface normal

27 Transforming Normal Vectors Surface normal tangent plane

28 Transforming Normal Vectors How to transform a normal when Each on the plane satisfies Then the normal is given by

29 Transforming Normal Vectors How to transform a normal when Current normal Transformed normal Verify by some algebra! (Hint: the plane is given by )

30 Projection From 3D to 2D space Camera image plane

31 Projection From 3D to 2D space

32 Perspective Projection

33 Parallel vs. Perspective Projection Parallel Projection Perspective Projection

34 Parallel vs. Perspective Projection

35 Perspective Projection Vanishing points

36 Perspective Projection Mathematics of perspective projection

37 Perspective Projection Mathematics of perspective projection

38 Perspective Projection Mathematics of perspective projection 3D Coordinate

39 Parallel Projection Mathematics of parallel projection

40 Parallel projection Clipping Planes

41 Clipping Planes Parallel projection Perspective projection

42 Summary of Transformations Projective Rigid Affine Linear Translation Rotation Scaling Shear Perspective Parallel

43 Transformations in OpenGL Stages of transformations Vertex (x, y, z, 1) T Eye Coordinates Clip Coordinates Normalized Device Coordinates Window (Screen) Coordinates ModelView Transform Projection Perspective Division Viewport Transform

44 Transformations in OpenGL Stages of transformations Vertex (x, y, z, 1) T Eye Coordinates Clip Coordinates Normalized Device Coordinates Window (Screen) Coordinates ModelView Transform Projection Perspective Division Viewport Transform

45 Transformations in OpenGL ModelView Transform Stage 1: Model to world coordinates r 3 r 2 t Model Coordinates World Coordinates r 1

46 Transformations in OpenGL ModelView Transform Stage 2: World to camera coordinates Default in OpenGL: Eye (Camera) Coordinates World Coordinates

47 Transformations in OpenGL Stages of transformations Vertex (x, y, z, 1) T Eye Coordinates Clip Coordinates Normalized Device Coordinates Window (Screen) Coordinates ModelView Transform Projection Perspective Division Viewport Transform

48 Transformations in OpenGL Projection Option 1: Parallel projection right glortho(left, right, bottom, top, near, far);

49 Transformations in OpenGL Projection Option 1: Parallel projection

50 Transformations in OpenGL Projection Option 2: Perspective projection right glfrustum(left, right, bottom, top, near, far);

51 Transformations in OpenGL Projection Option 2: Perspective projection

52 Transformations in OpenGL Projection Clip the points by comparing, and with

53 Transformations in OpenGL Stages of transformations Vertex (x, y, z, 1) T Eye Coordinates Clip Coordinates Normalized Device Coordinates Window (Screen) Coordinates ModelView Transform Projection Perspective Division Viewport Transform

54 Transformations in OpenGL Perspective division Normalized Device Coordinates Determines coordinates on the screen Used for depth tests

55 Transformations in OpenGL Stages of transformations Vertex (x, y, z, 1) T Eye Coordinates Clip Coordinates Normalized Device Coordinates Window (Screen) Coordinates ModelView Transform Projection Perspective Division Viewport Transform

56 Transformations in OpenGL Viewport Transform Normalized Device Coordinates Screen Coordinates glviewport(o x, o y, w, h); gldepthrange(n, f);