Animation Process

1. Animation Process while (not finished) { MoveEverything(); DrawEverything(); } • Interactive vs. Non-Interactive • Real Time vs. Non-Real Time

2. Character Rigging • Skeleton • Skin • Facial Expressions • Muscles • Secondary motion: fat, hair, clothing…

3. Character Animation • Keyframe Animation • Motion Capture • Inverse Kinematics • Locomotion • Procedural Animation • Artificial Intelligence

4. Character Animation

5. Physics Simulation • Particles • Rigid bodies • Collisions, contact, stacking, rolling, sliding • Articulated bodies • Hinges, constraints • Deformable bodies (solid mechanics) • Elasticity, plasticity, viscosity • Fracture • Cloth • Fluid dynamics • Fluid flow (liquids & gasses) • Combustion (fire, smoke, explosions…) • Phase changes (melting, freezing, boiling…) • Vehicle dynamics • Cars, boats, airplanes, helicopters, motorcycles… • Character dynamics • Body motion, skin & muscle, hair, clothing

6. Physics Simulation

7. Animation Tools • Maya • 3D Studio • Lightwave • Filmbox • Blender • Many more…

8. Animation Production • Conceptual Design • Production Design • Modeling • Materials & Shaders • Rigging • Blocking • Animation • Lighting • Effects • Rendering • Post-Production

9. Resolution & Frame Rates • Video: • NTSC: 720 x 480 @ 30 Hz (interlaced) • PAL: 720 x 576 @ 25 Hz (interlaced) • HDTV: • 720p: 1280 x 720 @ 60 Hz • 1080i: 1920 x 1080 @ 30 Hz (interlaced) • 1080p: 1920 x 1080 @ 60 Hz • Film: • 35mm: ~2000 x ~1500 @ 24 Hz • 70mm: ~4000 x ~2000 @ 24 Hz • IMAX: ~5000 x ~4000 @ 24-48 Hz • Note: Hz (Hertz) = frames per second (fps) • Note: Video standards with an i (such as 1080i) are interlaced, while standards with a p (1080p) are progressive scan

10. Interlacing • Older video formats (NTSC, PAL) and some HD formats (1080i) use a technique called interlacing • With this technique, the image is actually displayed twice, once showing the odd scanlines, and once showing the even scanlines (slightly offset) • This is a trick for achieving higher vertical resolution at the expense of frame rate (cuts effective frame rate in half) • The two different displayed images are called fields • NTSC video, for example, is 720 x 480 at 30 frames per second, but is really 720 x 240 at 60 fields per second • Interlacing is an important issue to consider when working with video, especially in animation as in TV effects and video games • Computer monitors are generally not interlaced

11. Rendering • There are many ways to design a 3D renderer • The two most common approaches are: • Traditional graphics pipeline • Ray-based rendering • With the traditional approach, primitives (usually triangles) are rendered into the image one at a time, and complex visual effects often involve a variety of different tricks • With ray-based approaches, the entire scene is stored and then rendered one pixel at a time. Ray based approaches can simulate light more accurately and offer the possibility of significant quality improvements, but with a large cost • In this class, we will not be very concerned with rendering, as we will focus mainly on how objects move rather than how they look

12. Coordinate Systems • Right handed coordinate system

13. 3D Models • Let’s say we have a 3D model that has an array of position vectors describing its shape • We will group all of the position vectors used to store the data in the model into a single array: vn where 0 ≤ n ≤NumVerts-1 • Each vector vn has components vnx vny vnz

14. Vector Review

15. Vector Arithmetic

16. Vector Magnitude • The magnitude (length) of a vector is: • A vector with length=1.0 is called a unit vector • We can also normalize a vector to make it a unit vector:

17. Dot Product

18. Dot Product

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

20. Example: Angle Between Vectors b θ a

21. Dot Products with General Vectors • The dot product is a scalar value that tells us something about the relationship between two vectors • If a·b > 0 then θ < 90º • If a·b < 0 then θ > 90º • If a·b = 0 then θ = 90º (or one or more of the vectors is degenerate (0,0,0))

22. 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

23. • 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

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

25. 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

26. Cross Product

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

28. Example: Normal of a Triangle • Find the unit length normal of the triangle defined by 3D points a, b, and c c b a

29. Example: Normal of a Triangle c c-a b a b-a

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

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

32. 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

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

34. Vector Class class Vector3 { public: Vector3() {x=0.0f; y=0.0f; z=0.0f;} Vector3(float x0,float y0,float z0) {x=x0; y=y0; z=z0;} void Set(float x0,float y0,float z0) {x=x0; y=y0; z=z0;} void Add(Vector3 &a) {x+=a.x; y+=a.y; z+=a.z;} void Add(Vector3 &a,Vector3 &b) {x=a.x+b.x; y=a.y+b.y; z=a.z+b.z;} void Subtract(Vector3 &a) {x-=a.x; y-=a.y; z-=a.z;} void Subtract(Vector3 &a,Vector3 &b) {x=a.x-b.x; y=a.y-b.y; z=a.z-b.z;} void Negate() {x=-x; y=-y; z=-z;} void Negate(Vector3 &a) {x=-a.x; y=-a.y; z=-a.z;} void Scale(float s) {x*=s; y*=s; z*=s;} void Scale(float s,Vector3 &a) {x=s*a.x; y=s*a.y; z=s*a.z;} float Dot(Vector3 &a) {return x*a.x+y*a.y+z*a.z;} void Cross(Vector3 &a,Vector3 &b) {x=a.y*b.z-a.z*b.y; y=a.z*b.x-a.x*b.z; z=a.x*b.y-a.y*b.x;} float Magnitude() {return sqrtf(x*x+y*y+z*z);} void Normalize() {Scale(1.0f/Magnitude());} float x,y,z; };

35. Translation • Let’s say that we want to move our 3D model from it’s current location to somewhere else… • In technical jargon, we call this a translation • We want to compute a new array of positions v′n representing the new location • Let’s say that vector d represents the relative offset that we want to move our object by • We can simply use: v′n = vn + d to get the new array of positions

36. Transformations v′n = vn + d • This translation represents a very simple example of an object transformation • The result is that the entire object gets moved or translated by d • From now on, we will drop the n subscript, and just write v′= v + d remembering that in practice, this is actually a loop over several differentvn vectors applying the same vector d every time

37. Transformations • Always remember that this compact equation can be expanded out into • Or into a system of linear equations:

38. Rotation • Now, let’s rotate the object in the xy plane by an angle θ, as if we were spinning it around the z axis • Note: a positive rotation will rotate the object counterclockwise when the rotation axis (z) is pointing towards the observer

39. Rotation • We can expand this to: • And rewrite it as a matrix equation: • Or just:

40. Rotation • We can represent a z-axis rotation transformation in matrix form as: or more compactly as: where

41. Rotation • We can also define rotation matrices for the x, y, and z axes:

42. Linear Transformations • Like translation, rotation is an example of a linear transformation • True, the rotation contains sin()’s and cos()’s, but those ultimately just end up as constants in the actual linear equation • We can generalize our matrix in the previous example to be:

43. Linear Equation • A general linear equation of 1 variable is: where a and d are constants • A general linear equation of 3 variables is: • Note: there are no nonlinear terms like vxvy, vx2, sin(vx)…

44. System of Linear Equations • Now let’s look at 3 linear equations of 3 variables vx, vy, and vz • Note that all of the an, bn, cn, and dn are constants (12 in total)

45. Matrix Notation

46. Translation • Let’s look at our translation transformation again: • If we really wanted to, we could rewrite our three translation equations as:

47. Identity • We can see that this is equal to a transformation by the identity matrix

48. Identity • Multiplication by the identity matrix does not affect the vector

49. Uniform Scaling • We can apply a uniform scale to our object with the following transformation • If s>1, then the object will grow by a factor of s in each dimension • If 0<s<1, the object will shrink • If s<0, the object will be reflected across all three dimensions, leading to an object that is ‘inside out’

50. Non-Uniform Scaling • We can also do a more general nonuniformscale, where each dimension has its own scale factor which leads to the equations: