CSE 167: Lecture #2: Coordinate Transformations. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011

Transcription

1 CSE 167: Introduction to Computer Graphics Lecture #2: Coordinate Transformations Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011

2 Announcements Homework #1 due Friday Sept 30, 1:30pm; presentation in lab 260 Don t save anything on the C: drive of the lab PCs in Windows. You will lose it when you log out! 2

3 Overview Linear Transformations Homogeneous Coordinates Affine Transformations Concatenating Transformations Change of Coordinates Common Coordinate Systems 3

4 Linear Transformations Scaling, shearing, rotation, reflection of vectors, and combinations thereof Implemented using matrix multiplications 4

5 Scaling Uniform scaling matrix in 2D Analogous in 3D 5

6 Scaling Nonuniform scaling matrix in 2D Analogous in 3D 6

7 Shearing Shearing along x-axis in 2D Analogous for y-axis, in 3D 7

8 Rotation in 2D Convention: positive angle rotates counterclockwise Rotation matrix 8

9 Rotation in 3D Rotation around coordinate axes 9

10 Rotation in 3D Concatenation of rotations around x, y, z axes are called Euler angles Result depends on matrix order! 10

11 Rotation in 3D Around arbitrary axis R(a,θ) = 1+ (1 cos(θ))(a 2 x 1) a z sin(θ) + (1 cos(θ))a x a y a y sin(θ) + (1 cos(θ))a x a z a z sin(θ) + (1 cos(θ))a y a x 1+ (1 cos(θ))(a 2 y 1) a x sin(θ) + (1 cos(θ))a y a z a y sin(θ) + (1 cos(θ))a z a x a x sin(θ) + (1 cos(θ))a z a y 1+ (1 cos(θ))(a 2 z 1) Rotation axis a a must be a unit vector: Right-hand rule applies for direction of rotation Counterclockwise rotation a = 1 11

12 Overview Linear Transformations Homogeneous Coordinates Affine Transformations Concatenating Transformations Change of Coordinates Common Coordinate Systems 12

13 Homogeneous Coordinates Generalization: homogeneous point Homogeneous coordinate Corresponding 3D point: divide by homogeneous coordinate 13

14 Homogeneous coordinates Usually for 3D points you choose For 3D vectors Benefit: same representation for vectors and points 14

15 Translation Using homogeneous coordinates 15

16 Translation Using homogeneous coordinates Matrix notation 16 Translation matrix

17 Transformations Add 4 th row/column to 3 x 3 transformation matrices Example: rotation 17

18 Transformations Concatenation of transformations: Arbitrary transformations (scale, shear, rotation, translation) Build chains of transformations Result depends on order 18

19 Overview Linear Transformations Homogeneous Coordinates Affine Transformations Concatenating Transformations Change of Coordinates Common Coordinate Systems 19

20 Affine transformations Generalization of linear transformations Scale, shear, rotation, reflection (linear) Translation Preserve straight lines, parallel lines Implementation using 4x4 matrices and homogeneous coordinates 20

21 Translation 21

22 Translation Inverse translation 22

23 Scaling Origin does not change 23

24 Scaling Inverse of scale: 24

25 Shear Pure shear if only one parameter is non-zero 25

26 Rotation around coordinate axis Origin does not change 26

27 Rotation around arbitrary axis Origin does not change Angle, unit axis a 27

28 Rotation matrices Orthonormal Rows, columns are unit length and orthogonal Inverse of rotation matrix: Its transpose 28

29 Overview Linear Transformations Homogeneous Coordinates Affine Transformations Concatenating Transformations Change of Coordinates Common Coordinate Systems 29

30 Rotating with pivot Rotation around origin Rotation with pivot 30

31 Rotating with pivot 1. Translation 2. Rotation 3. Translation 31

32 Concatenating transformations Arbitrary sequence of transformations Note: associativity 32

33 Overview Linear Transformations Homogeneous Coordinates Affine Transformations Concatenating Transformations Change of Coordinates Common Coordinate Systems 33

34 Change of coordinates Point with homogeneous coordinates Position in 3D given with respect to a coordinate system 34

35 Change of coordinates New uvwq coordinate system Goal: Find coordinates of with respect to new uvwq coordinate system 35

36 Change of coordinates Coordinates of xyzo frame w.r.t. uvwq frame 36

37 Change of coordinates Same point p in 3D, expressed in new uvwq frame 37

38 Change of coordinates 38

39 Change of coordinates Inverse transformation Given point w.r.t. frame Coordinates w.r.t. frame 39

40 Overview Linear Transformations Homogeneous Coordinates Affine Transformations Concatenating Transformations Change of Coordinates Typical Coordinate Systems 40

41 Typical Coordinate Systems Camera, world, object coordinates: Camera coordinates Object coordinates World coordinates

42 Object Coordinates Coordinates the object is defined with Often origin is in middle, base, or corner of object No right answer, whatever was convenient for the creator of the object Camera coordinates Object coordinates World coordinates 42

43 World Coordinates World space Common reference frame for all objects in the scene Chosen for convenience, no right answer If there is a ground plane, usually x/y is horizontal and z points up (height) In OpenGL x/y is screen plane, z comes out Camera coordinates Object coordinates World coordinates 43

44 World Coordinates Transformation from object to world space is different for each object Defines placement of object in scene Given by model matrix (model-to-world transform) M Camera coordinates Object coordinates World coordinates 44

45 Camera Coordinate System Camera space Origin defines center of projection of camera x-y plane is parallel to image plane z-axis is perpendicular to image plane Camera coordinates Object coordinates World coordinates 45

46 Camera Coordinate System The Camera Matrix defines the transformation from camera to world coordinates Placement of camera in world Transformation from object to camera coordinates Camera coordinates Object coordinates World coordinates 46

47 Camera Matrix Construct from center of projection e, look at d, upvector up: Camera coordinates 47 World coordinates

48 Camera Matrix Construct from center of projection e, look at d, upvector up: Camera coordinates 48 World coordinates

49 Camera Matrix z-axis x-axis y-axis 49

50 Inverse of Camera Matrix How to calculate the inverse of the camera matrix C -1? Generic matrix inversion is complex and computeintensive Observation: camera matrix consists of rotation and translation: R x T Inverse of rotation: R -1 = R T Inverse of translation: T(t) -1 = T(-t) Inverse of camera matrix: C -1 = T -1 x R -1 50

51 Objects in Camera Coordinates We have things lined up the way we like them on screen x to the right y up -z going into the screen Objects to look at are in front of us, i.e. have negative z values But objects are still in 3D Next step: project scene into 2D 51

52 Next Lecture Rendering Pipeline Perspective Projection 52