Midterm Review

1. Midterm Review CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2004

2. Project 3 • Load an .anim file and play back a keyframed animation on a skinned character • Extra Credit: • Display the channel curve (1 point) • Simple channel editor (2 points)

3. Anim File animation { range [time_start] [time_end] numchannels [num] channel { extrapolate [extrap_in] [extrap_out] keys [numkeys] { [time] [value] [tangent_in] [tangent_out] … } } channel … }

4. Project 3 • The first 3 channels will be the root translation (x,y,z) • After that, there will be 3 rotational channels for every joint (x,y,z) in the same order that the joints are listed in the .skel file

5. Project 3 • Suggested classes: • Keyframe: stores time, value, tangents, cubics… • Channel: stores an array (or list) of Keyframes • Animation: stores an array of Channels • Player: stores pointer to an animation & pointer to skeleton. Keeps track of time, accesses animation data & poses the skeleton. • Optional: • Rig: simple container for a skeleton, skin, and morphs • Pose: array of floats (or just use stl vector) • ChannelEditor: it’s always nice to separate editor classes from the data that they edit

6. Linear Algebra Review

7. Key Math Concepts • Vectors (dot & cross products…) • Matrices (& systems of equations) • Homogeneous vectors & matrices • Transformations, inverse transformations • Viewing, cameras • Euler angles • Quaternions • Summation (weighted sum, average, convex) • Cubic equations • Interpolation, splines, radial basis functions

8. Vector Arithmetic

9. Vector Magnitude • The magnitude (length) of a vector is: • Unit vector (magnitude=1.0)

10. Dot Product

11. Example: Angle Between Vectors • How do you find the angle θ between vectors A and B? b θ a

12. Example: Angle Between Vectors b θ a

13. Dot Products with Unit Vectors 0 <a·b < 1 a·b = 0 a·b = 1 b θ a -1 < a·b < 0 a·b a·b = -1

14. Dot Products with Non-Unit Vectors • If a and b are arbitrary (non-unit) vectors, then the following are still true: • If θ < 90º then a·b > 0 • If θ = 90º then a·b = 0 • If θ > 90º then a·b < 0

15. Dot Products with One Unit Vector • If |u|=1.0 then a·u is the length of the projection of a onto u a u a·u

16. • x n p • Example: Distance to Plane • A plane is described by a point p on the plane and a unit normal n. Find the distance from point x to the plane

17. • x n p • Example: Distance to Plane • The distance is the length of the projection of x-p onto n: x-p

18. Cross Product

19. Properties of the Cross Product area of parallelogram ab if a and b are parallel is perpendicular to both a and b, in the direction defined by the right hand rule

20. Example: Area of a Triangle • Find the area of the triangle defined by 3D points a, b, and c c b a

21. Example: Area of a Triangle c c-a b a b-a

22. Matrices

23. N a b M b d c c d a Example: Alignment to Target • A spaceship with a matrix M must target another ship with matrix N. Find a unit rotation axis u and an angle θ that will rotate M to be facing N

24. N a b M b d c c d a Example: Alignment to Target • As objects are defined looking down the -z axis in their local space, the -c axis of their world matrix will represent the object’s heading heading

25. N a b M b d c c d a Example: Alignment to Target • Therefore, we want to rotate matrix M so that it’s c axis lines up with the vector going from N to M, or (M.d - N.d) (M.d-N.d)

26. N a b M b d c c d a Example: Alignment to Target • We will rotate around the axis perpendicular to both M.c and (M.d - N.d) u (M.d-N.d)

27. N a b M b d c c d a Example: Alignment to Target • The rotation angle θ is just the angle between M.c and (M.d - N.d) u θ (M.d-N.d)

28. Orthonormality • If all row vectors and all column vectors of a matrix are unit length, that matrix is said to be orthonormal • This also implies that all vectors are perpendicular to each other • Orthonormal matrices have some useful mathematical properties, such as: • M-1 = MT

29. Orthonormality • If a 4x4 matrix represents a rigid transformation, then the upper 3x3 portion will be orthonormal

30. Matrix Dot Matrix

31. Camera Matrix • Think of the camera just like any other object. Just as a chair model has a matrix W that transforms it into world space, the camera matrix C would transform a camera model into world space. • We don’t want to transform the camera into world space. Instead, we want to transform the world into the camera’s space, so we use the inverse of C.

32. Example: Camera ‘Look At’ • Our eye is located at position e and we want to look at a target at position t. Generate an appropriate camera matrix M.

33. Example: Camera ‘Look At’ • Our eye is located at position e and we want to look at a target at position t. Generate an appropriate camera matrix M. • Two possible approaches include: • Measure angles and rotate matrix into place • Construct a,b,c, & d vectors of M directly

34. Example: Camera ‘Look At’ Matrix • The d vector is just the position of the camera, which is e: • The c vector is a unit length vector that points directly behind the viewer:

35. Example: Camera ‘Look At’ Matrix • The a vector is a unit length vector that points to the right of the viewer. It is perpendicular to the c axis. To keep the camera from rolling, we also want the a vector to lay flat in the xz-plane, perpendicular to the y-axis. • Note that a cross product with the y-axis can be optimized as follows:

36. Example: Camera ‘Look At’ Matrix • The b vector is a unit length vector that points up relative to the viewer. It is perpendicular to the both the c and a axes • Note that b does not need to be normalized because it is already unit length. This is because a and c are unit length vectors 90 degrees apart.

37. Example: Camera ‘Look At’ Matrix • Summary:

38. Inversion • If M transforms v into world space, then M-1 transforms v’ back into local space

39. Homogeneous Vectors • Technically, homogeneous vectors are 4D vectors that get projected into the 3D w=1 space

40. Homogeneous Vectors • Vectors representing a position in 3D space can just be written as: • Vectors representing direction are written: • The only time the w coordinate will be something other than 0 or 1 is in the projection phase of rendering, which is not our problem

41. Position Vector Dot Matrix

42. Position Vector Dot Matrix y v=(.5,.5,0,1) x (0,0,0) Local Space

43. Position Vector Dot Matrix b Matrix M y y d a v=(.5,.5,0,1) x x (0,0,0) (0,0,0) Local Space World Space

44. Position Vector Dot Matrix b v’ y y d a v=(.5,.5,0,1) x x (0,0,0) (0,0,0) Local Space World Space

45. Direction Vector Dot Matrix

46. Matrix Dot Matrix (4x4) • The row vectors of M’ are the row vectors of M transformed by matrix N • Notice that a, b, and c transform as direction vectors and d transforms as a position

47. Quaternions • In computer animation, quaternions are used as an alternative way to store orientations (other than matrices & Euler angles) • One of their biggest advantages is their ability to interpolate between arbitrary orientations nicely • They are also a compact, fast way to store & manipulate orientations • As we generally require matrices at some point, the ability to convert a quaternion to/from a matrix is very important • Once in quaternion format, we can interpolate smoothly using ‘slerp’ (spherical linear interpolation), or construct quaternion curves through several orientations

48. Weighted Blending & Averaging • Weighted sum: • Weighted average: • Convex average: • Additive blend:

49. Rigging & Animation Review

50. Rigging DOF Joint Skeleton Skin Morphing, PSD FFD, SOFFD Rig Animation Pose Keyframe Channel Animation Clip Player Blender State Machine Rigging & Animation Concepts