1. Procedural Animation and Physics Engine Yingcai Xiao

2. Outline What is it? How to design it in a game? How to design it in a game engine? How to implement it in Unity3D?

3. Yingcai Xiao Procedural Animation Dynamically generated by a procedure (function) at runtime. Flexible: can use any algorithms, can simulate real- world phenomenon (Good for building physics engines). Precise: no key frames, no lerping, can have smooth animation of any order in time and space.

4. Yingcai Xiao Procedural Animation of Dynamics

5. . Motion Dynamics Change of GO’s Geometry: location size orientation shape https://en.wikipedia.org/wiki/Procedural_animation Change of Camera: location orientation FOV (field of view) focal point

6. . Update Dynamics Change of GO’s Attributes: color texture lighting shading/reflection (via normal, material properties)

7. Yingcai Xiao Procedural Animation of Motion Dynamics (Wave Example)

8. . GO: Swimming Pool Animation: Wave Motion https://en.wikipedia.org/wiki/Procedural_animation Procedure: Physics Equations Data Structure: Elevation Grid

9. Data Structure Data Structure Design Criterion Compact (save space) Efficient (fast retrieval) Map-able (easy to convert) Minimal Coverage (small foot prints) Simple (easy to use)

10. Uniform Grid  Commonly used in game engines as Elevation Grid for terrain data structure.  Compact: No need to store x, z values,; they are calculated at runtime.  Parametric space (i,j)  x = i*d x + x 0; 0<= i < n x  z = j*d z + z 0; 0<= j < n z  Data array (i, j), loop i first, then j.  Speedy retrieval of data in a 2D array.  Simple  Not flexible

11. Elevation Grid  Elevation Grid is a special case of Uniform Grid.  IK space  x = i*d x + x 0; 0<= i < n x  z = j*d z + z 0; 0<= j < n z  Elevation values stored in as data at every grid note.  The stored y value along with the computed x, z values define the coordinates of each grid note.

12. Yingcai Xiao Elevation Grid origin: 0,0 spacing: 2, 2 dimension: 3, 3 elevation: 10,20,30, 11,21,31, 21, 22, 33

13. Yingcai Xiao Physics of Waves

14. A snapshot of a decaying cosine wave V(r) = e -r cos(10r);

15. . 1D Cosine Wave: y = COS(x);

19. Yingcai Xiao Procedural Animation of Update Dynamics (Wave Example)

20. Change the shades to simulate waves  Shades are produced by reflection.  Reflection at each location depends on the surface normal at the location.  If we change the normal by following the wave motion pattern, it could appear as a real wave.

21. Shading Shading: determining light reflection from objects at each pixel. Basic Reflection Model: Phong Reflection Model (most commonly used) I= k a I a + k d I d (l · n) + k s I s (v · r ) α I : reflected-light intensity

23. Procedural Animation in Unity3D Learn how to use scripts to change the geometry of game objects at run-time. Example: change the geometry of a plane to a simulate a wave.

24. Example in Unity3D To change the geometry of a plane to simulate a wave. Create a new project with Character Controller package. Add a plane (GameObject->CreateOhters->Plane) Scale it in the Inspector to 10X10 (in x and z). Uncheck the Mesh Collider (in the Inspector) Add a light (GameObject->CreateOhters->DirectionalLight) Add a FPC (Project->Assets->StandardAssets- >CharacterControllers->FirstPersonController, drag-and- drop it to Hierarchy.) Move it up. (Change its y position in the Inspector to 11). Add script to morph the plane (Plane->Inspector- >AddComponent->NewScript->JavaScript) name it Wave. Add the code on the next page to the Wave.js in Mono. Build->Build All in Mono to make sure there is no compilation errors.

25. Demo in Unity3D (Wave) #pragma strict function Start () { var mesh: Mesh = this.GetComponent(MeshFilter).mesh; var verts: Vector3[] = mesh.vertices; for (var v = 0; v < verts.Length; v++) { verts[v].y = Random.Range(0,10); } mesh.vertices = verts; mesh.RecalculateBounds(); mesh.RecalculateNormals(); this.gameObject.AddComponent(MeshCollider); } function Update () { }

26. Cosine Wave Formula y(r,t) = A e -r-at cos(2 π (r-Vt) / λ ); r = sqrt ((x-x 0 )*(x-x 0 )+ (z-z 0 )*(z-z 0 )) P 0 (x 0, y 0, z 0 ) : center of the wave A: amplitude of the wave V: velocity of the wave λ : wave length of the wave a: speed of decaying t = current time – time of impact (t 0 )