1. Lecture 2: linear algebra, animation basics
Computer Animation

2. Vector Arithmetic

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

4. Dot Product

5. Example: Angle Between Vectors
How do you find the angle θ between vectors a and b? b θ a

6. Example: Angle Between Vectorsθ a

7. 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

8. 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

9. 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

10. Cross Product

11. 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

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

13. Example: Area of a Trianglec
c-a b a
b-a

14. Example: Alignment to Target
An object is at position p with a unit length heading of h. We want to rotate it so that the heading is facing some target t. Find a unit axis a and an angle θ to rotate around. t • • p h

15. Example: Alignment to Target
t-p t • • θ p h

16. Trigonometry
cos2θ + sin2θ = 1
1.0
sin θ θ
cos θ

17. Laws of Sines and Cosines
Law of Sines:
Law of Cosines: b α γ c a β

18. Matrices Computer graphics apps commonly use 4x4 homogeneous matrices
A rigid 4x4 matrix transformation looks like this:
Where a, b, & c are orthogonal unit length vectors representing orientation, and d is a vector representing position
In mathematics, a rigid transformation or a Euclidean transformation is a transformation from a Euclidean space to itself that preserves distances between every pair of points (isometry). In particular, any object will have the same shape and size before and after a rigid transformation.

19. Matrices
The right hand column can cause a projection, which we won’t use in character animation, so we leave it as 0,0,0,1
Some books store their matrices in a transposed form. This is fine as long as you remember that: (A·B)T = BT·AT

20. 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

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

22. Determinants
The determinant of a 4x4 matrix with no projection is equal to the determinant of the upper 3x3 portion

23. Determinants
The determinant is a scalar value that represents the volume change that the transformation will cause
An orthonormal matrix will have a determinant of 1, but non-orthonormal volume preserving matrices will have a determinant of 1 also
A flattened or degenerate matrix has a determinant of 0
A matrix that has been mirrored will have a negative determinant

24. Transformations To transform a vector v by matrix M: v’=v·M
If we want to apply several transformations, we can just multiply by several matrices:
v’=(((v·M1)·M2)·M3)·M4…
Or we can concatenate the transformations into a single matrix:
Mtotal=M1·M2·M3·M4…
v’=v·Mtotal

25. Matrix Transformations
We usually transform vertices from some local space where they are defined into a world space
v’ = v·W
Once in world space, we can perform operations that require everything to be in the same space (collision detection, high quality lighting…)
Then, they are transformed into a camera’s space, and then projected into 2D
v’’ = v’·C-1·P
In simple situations, we can do this all in one step:
v’’ = v·W·C-1·P

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

27. Vector Dot Vector

28. Vector Dot Matrix

29. Matrix Dot Matrix

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

31. 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

32. Position Vector Dot Matrix

33. Position Vector Dot Matrixy
v=(.5,.5,0,1) x
(0,0,0)
Local Space

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

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

36. Direction Vector Dot Matrix

37. 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

38. Identity Take one more look at the identity matrix
It’s a axis lines up with x, b lines up with y, and c lines up with z
Position d is at the origin
Therefore, it represents a transformation with no rotation or translation

39. Animation basics

40. Animation Process Interactive vs. Non-Interactive
while (!finished) {
UpdateEverything();
DrawEverything(); }
Interactive vs. Non-Interactive
Real Time vs. Non-Real Time

41. Frame Rates Film 24 fps Imax 48 fps NTSC TV 30 fps (interlaced)
PAL TV 25 fps (interlaced)
HDTV fps
Computer >60 fps

42. Animation Tools Animation tools: Game/graphics engines: Maya
3D Studio Max
MotionBuilder
Blender
Game/graphics engines:
OGRE3D
Unity
Unreal

43. Virtual Characters Applications of animated characters Health
Analyzing walking styles
Analyzing muscle activations
Analyzing joint movement limits
Ergonomics
Usability of devices
Measuring comfort
Safety
Simulated worlds for crisis
management
Entertainment
Games, movies

44. Virtual Characters Different approaches: Keyframe animation
Motion Capture
Physics-based animation
Procedural animation

45. Virtual Characters
Different levels of character motion

46. Virtual Characters Representation: skeletal model
A VH is represented by a polyhedral model (or mesh)
An underlying skeleton deforms this mesh
Joints, connected by bones
A pose is defined by the rotations of the joints and the position of the root joint
Several standards
H-Anim

47. Kinematics Kinematics Forward Kinematics Inverse Kinematics
The analysis of motion independent of physical forces. Kinematics deals with position, velocity, acceleration, and their rotational counterparts, orientation, angular velocity, and angular acceleration.
Forward Kinematics
The process of computing world space geometric data from DOFs
Inverse Kinematics
The process of computing a set of DOFs that causes some world space goal to be met (I.e., place the hand on the door knob…)

48. Skeletons Skeleton Joint
A pose-able framework of joints arranged in a tree structure.
Joint
Allows relative movement within the skeleton.
Are essentially 4x4 matrix transformations
Can be rotational, translational, or other
Synonym: bone

49. DOFs Degree of Freedom (DOF)
A variable φ describing a particular axis or dimension of movement within a joint
Joints typically have around 1-6 DOFs (φ1…φN)
Changing the DOF values over time results in the animation of the skeleton
Note: in a mathematical sense, a free rigid body has 6 DOFs: 3 for position and 3 for rotation

50. Example Joint Hierarchy

51. Joints Core Joint Data Additional Data DOFs (N floats) Local matrix: L
World matrix: W
Additional Data
Joint offset vector: r
DOF limits (min & max value per DOF)
Type-specific data (rotation/translation axes, constants…)
Tree data (pointers to children, siblings, parent…)

52. Skeleton Posing Process
Specify all DOF values for the skeleton (done by higher level animation system)
Recursively traverse through the hierarchy starting at the root and use forward kinematics to compute the world matrices (done by skeleton system)
Use world matrices to deform skin & render (done by skin system)
Note: the matrices can also be used for other things such as collision detection, FX, etc.

53. Forward Kinematics
In the recursive tree traversal, each joint first computes its local matrix L based on the values of its DOFs and some formula representative of the joint type:
Local matrix L = Ljoint(φ1,φ2,…,φN)
Then, world matrix W is computed by concatenating L with the world matrix of the parent joint
World matrix W = L · Wparent

54. Joint Offsets
It is convenient to have a 3D offset vector r for every joint which represents its pivot point relative to its parent’s matrix

55. DOF Limits
It is nice to be able to limit a DOF to some range (for example, the elbow could be limited from 0º to 150º)
Usually, in a realistic character, all DOFs will be limited except the ones controlling the root

56. Poses
One can then adjust each of the DOFs to specify the pose of the skeleton
We can define a pose Φ more formally as a vector of N numbers that maps to a set of DOFs in the skeleton
Φ = [φ1 φ2 … φN]
A pose is a convenient unit that can be manipulated by a higher level animation system and then handed down to the skeleton
Usually, each joint will have around 1-6 DOFs, but an entire character might have 100+ DOFs in the skeleton

57. Joint Types

58. Joint Types Rotational Translational Compound Non-Rigid
Hinge: 1-DOF
Universal: 2-DOF
Ball & Socket: 3-DOF
Euler Angles
Quaternions
Translational
Prismatic: 1-DOF
Translational: 3-DOF (or any number)
Compound
Free
Screw
Constraint
Etc.
Non-Rigid
Scale
Shear
Design your own...

59. Hinge Joints (1-DOF Rotational)
Rotation around the x-axis:

60. Hinge Joints (1-DOF Rotational)
Rotation around the y-axis:

61. Hinge Joints (1-DOF Rotational)
Rotation around the z-axis:

62. Hinge Joints (1-DOF Rotational)
Rotation around an arbitrary axis a:

63. Universal Joints (2-DOF)
For a 2-DOF joint that first rotates around x and then around y:
Different matrices can be formed for different axis combinations

64. Ball & Socket (3-DOF)
For a 3-DOF joint that first rotates around x, y, then z:
Different matrices can be formed for different axis combinations

65. Quaternions

66. Prismatic Joints (1-DOF Translation)
1-DOF translation along an arbitrary axis a:

67. Translational Joints (3-DOF)
For a more general 3-DOF translation: