1. CSCE 552 Fall 2012 Animations By Jijun Tang

2. Animation terms  frame – an image that is displayed on the screen, usually as part of a sequence.  pose – an orientation of an objects or a hierarchy of objects that defines extreme or important motion.  keyframe – a special frame that contains a pose.  tween – the process of going “between” keyframes.  secondary motion – an object motion that is the result of its connection or relationship with another object.

3. Example

4. Homogeneous coordinates Four-dimensional space Combines 3  3 matrix and translation into one 4  4 matrix

5. Euler Angles Three rotations about three axes Intuitive meaning of values

6. Rotation Matrix

7. 3x3 Matrix Rotation Easy to use Moderately intuitive Large memory size - 9 values Interpolation is hard  Introduces scales and shears  Need to re-orthonormalize matrices after

8. Quaternions Quaternions are an interesting mathematical concept with a deep relationship with the foundations of algebra and number theory Invented by W.R.Hamilton in 1843 In practice, they are most useful to use as a means of representing orientations A quaternion has 4 components

9. Quaternions on Rotation Represents a rotation around an axis Four values is axis vector times sin(θ /2) w is cos(θ/2) Interpolation is fast

10. Illustration

11. Quaternions (Imaginary Space) Quaternions are actually an extension to complex numbers Of the 4 components, one is a ‘real’ scalar number, and the other 3 form a vector in imaginary ijk space!

12. Quaternions (Scalar/Vector) Sometimes, they are written as the combination of a scalar value s and a vector value v where

13. Unit Quaternions For convenience, we will use only unit length quaternions, as they will be sufficient for our purposes and make things a little easier These correspond to the set of vectors that form the ‘surface’ of a 4D hypersphere of radius 1 The ‘surface’ is actually a 3D volume in 4D space, but it can sometimes be visualized as an extension to the concept of a 2D surface on a 3D sphere

14. Quaternions as Rotations A quaternion can represent a rotation by an angle θ around a unit axis a: If a is unit length, then q will be also

15. Quaternions as Rotations

16. Quaternion to Matrix To convert a quaternion to a rotation matrix:

17. Pose

18. Triangles Fundamental primitive of pipelines  Everything else constructed from them  (except lines and point sprites) Three points define a plane Triangle plane is mapped with data  Textures  Colors “Rasterized” to find pixels to draw

19. Mesh

20. Vertices A vertex is a point in space Plus other attribute data  Colors  Surface normal  Texture coordinates  Whatever data shader programs need Triangles use three vertices  Vertices shared between adjacent triangles

21. Model Describes a single type of object Skeleton + rig One per object type Referenced by instances in a scene Usually also includes rendering data  Mesh, textures, materials, etc  Physics collision hulls, gameplay data, etc

22. Instance A single entity in the game world References a model Holds current states:  Position & orientation  Game play state – health, ammo, etc Has animations playing on it  Stores a list of animation controls  Need to be interpolated

23. Animation Control Links an animation and an instance  1 control = 1 anim playing on 1 instance Holds current data of animation  Current time  Speed  Weight  Masks  Looping state

24. Animation Storage The Problem Decomposition Keyframes and Linear Interpolation Higher-Order Interpolation The Bezier Curve Non-Uniform Curves Looping

25. Storage – The Problem 4x3 matrices, 60 per second is huge  200 bone character = 0.5Mb/sec Consoles have around 256-512Mb Animation system gets maybe 25% PC has more memory, but also higher quality requirements

26. Decomposition Decompose 4x3 into components  Translation (3 values)  Rotation (4 values - quaternion)  Scale (3 values)  Skew (3 values) Most bones never scale & shear Many only have constant translation But human characters may have higher requirement  Muscle move, smiling, etc.  Cloth under winds Don’t store constant values every frame, use index instead

27. Keyframes Motion is usually smooth Only store every n th frame (key frames) Interpolate between keyframes  Linear Interpolate  Inbetweening or “tweening” Different anims require different rates  Sleeping = low, running = high  Choose rate carefully

28. Linear Interpolation

29. Higher-Order Interpolation Tweening uses linear interpolation Natural motions are not very linear  Need lots of segments to approximate well  So lots of keyframes Use a smooth curve to approximate  Fewer segments for good approximation  Fewer control points B é zier curve is very simple curve

30. B é zier Curves (2D & 3D) B é zier curves can be thought of as a higher order extension of linear interpolation p0p0 p1p1 p0p0 p1p1 p2p2 p0p0 p1p1 p2p2 p3p3

31. The B é zier Curve (1-t) 3 F 1 +3t(1-t) 2 T 1 +3t 2 (1-t)T 2 +t 3 F 2 t=0.25 F1F1 T1T1 T2T2 F2F2 t=1.0 t=0.0

32. The B é zier Curve (2) Quick to calculate Precise control over end tangents Smooth  C0 and C1 continuity are easy to achieve  C2 also possible, but not required here Requires three control points per curve  (assume F2 is F1 of next segment) Far fewer segments than linear

33. C0/C1/C2 The curves meet the tangents are shared the"speed" is the same before and after

34. Catmull-Rom Curve Defined by 4 points. Curve passes through middle 2 points. P = C 3 t 3 + C 2 t 2 + C 1 t + C 0 C 3 = -0.5 * P 0 + 1.5 * P 1 - 1.5 * P 2 + 0.5 * P 3 C 2 = P 0 - 2.5 * P 1 + 2.0 * P 2 - 0.5 * P 3 C 1 = -0.5 * P 0 + 0.5 * P 2 C 0 = P 1

35. Non-Uniform Curves Each segment stores a start time as well Time + control value(s) = “knot” Segments can be different durations Knots can be placed only where needed  Allows perfect discontinuities  Fewer knots in smooth parts of animation Add knots to guarantee curve values: Transition points between animations

36. Looping and Continuity Ensure C0 and C1 for smooth motion At loop points At transition points: walk cycle to run cycle C1 requires both animations are playing at the same speed: reasonable requirement for anim system

37. Playing Animations “Global time” is game-time Animation is stored in “local time”  Animation starts at local time zero Speed is the ratio between the two  Make sure animation system can change speed without changing current local time Usually stored in seconds  Or can be in “frames” - 12, 24, 30, 60 per second

38. Scrubbing Sample an animation at any local time Important ability for games  Footstep planting  Motion prediction  AI action planning  Starting a synchronized animation Walk to run transitions at any time Avoid delta-compression storage methods  Very hard to scrub or play at variable speed

39. Delta Compression Delta compression is a way of storing or transmitting data in the form of differences between sequential data rather than complete files. The differences are recorded in discrete files called deltas or diffs. Because changes are often small (only 2% total size on average), it can greatly reduce data redundancy. Collections of unique deltas are substantially more space-efficient than their non-encoded equivalents.

40. Animation Blending The animation blending system allows a model to play more than one animation sequence at a time, while seamlessly blending the sequences Used to create sophisticated, life-like behavior  Walking and smiling  Running and shooting

41. Blending Animations The Lerp Quaternion Blending Methods Multi-way Blending Bone Masks The Masked Lerp Hierarchical Blending

42. The Lerp Foundation of all blending “Lerp”=Linear interpolation Blends A, B together by a scalar weight  lerp (A, B, i) = iA + (1-i)B  i is blend weight and usually goes from 0 to 1 Translation, scale, shear lerp are obvious  Componentwise lerp Rotations are trickier – normalized quaternions is usually the best method.

43. Quaternion Blending Normalizing lerp (nlerp)  Lerp each component  Normalize (can often be approximated)  Follows shortest path  Not constant velocity  Multi-way-lerp is easy to do  Very simple and fast Many others:  Spherical lerp (slerp)  Log-quaternion lerp (exp map)

44. Which is the Best No perfect solution! Each missing one of the features All look identical for small interpolations  This is the 99% case  Blending very different animations looks bad whichever method you use Multi-way lerping is important So use cheapest - nlerp

45. Multi-way Blending Can use nested lerps  lerp (lerp (A, B, i), C, j)  But n-1 weights - counterintuitive  Order-dependent Weighted sum associates nicely  (iA + jB + kC + …) / (i + j + k + … )  But no i value can result in 100% A More complex methods  Less predictable and intuitive  Can be expensive

46. Bone Masks Some animations only affect some bones  Wave animation only affects arm  Walk affects legs strongly, arms weakly Arms swing unless waving or holding something Bone mask stores weight for each bone  Multiplied by animation’s overall weight  Each bone has a different effective weight  Each bone must be blended separately Bone weights are usually static  Overall weight changes as character changes animations

47. The Masked Lerp Two-way lerp using weights from a mask  Each bone can be lerped differently Mask value of 1 means bone is 100% A Mask value of 0 means bone is 100% B Solves weighted-sum problem  (no weight can give 100% A) No simple multi-way equivalent  Just a single bone mask, but two animations

48. Hierarchical Blending Combines all styles of blending A tree or directed graph of nodes Each leaf is an animation Each node is a style of blend  Blends results of child nodes Construct programmatically at load time  Evaluate with identical code each frame  Avoids object-specific blending code  Nodes with weights of zero not evaluated

49. Triangles Fundamental primitive of pipelines  Everything else constructed from them  (except lines and point sprites) Three points define a plane Triangle plane is mapped with data  Textures  Colors “Rasterized” to find pixels to draw

50. Mesh

51. Vertices A vertex is a point in space Plus other attribute data  Colors  Surface normal  Texture coordinates  Whatever data shader programs need Triangles use three vertices  Vertices shared between adjacent triangles

52. Textures Array of texels  Same as pixel, but for a texture  Nominally R,G,B,A but can mean anything 1D, 2D, 3D and “cube map” arrays  2D is by far the most common  Basically just a 2D image bitmap  Often square and power-of-2 in size Cube map - six 2D arrays makes hollow cube  Approximates a hollow sphere of texels  For environmental

53. Cube Map

54. Texture Example

55. High-Level Organization Gameplay and Rendering Render Objects Render Instances Meshes Skeletons Volume Partitioning

56. Gameplay and Rendering Rendering speed varies according to scene  Some scenes more complex than others  Typically 15-60 frames per second Gameplay is constant speed  Camera view should not change game  In multiplayer, each person has a different view, but there is only one shared game  1 update per second (RTS) to thousands (FPS) Keep the two as separate as possible!

57. Render Objects Description of renderable object type Mesh data (triangles, vertices) Material data (shaders, textures, etc) Skeleton (+rig) for animation Shared by multiple instances

58. Render Instances A single entity in a world References a render object  Decides what the object looks like Position and orientation Lighting state Animation state

59. Meshes Triangles Vertices Single material “Atomic unit of rendering”  Not quite atomic, depending on hardware Single object may have multiple meshes  Each with different shaders, textures, etc  Level-Of-Distance (LOD)

60. LOD Objects have different mesh for different distance from the player The mesh should be simpler if object is faraway Many games have LOD, for example, Microsoft Train Simulator

61. Example (40m, 11370 poly)

62. Example (120m, 3900 poly)