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Computer Animation Algorithms and Techniques

Computer Animation Algorithms and Techniques. Special Models. L-Systems. L-Systems. Branching Structures Botany Display geometric substitution turtle graphics Animating plants, animals. Plant examples http://algorithmicbotany.org/papers/#abop. As a Formal Grammar.

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Computer Animation Algorithms and Techniques

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  1. Computer AnimationAlgorithms and Techniques Special Models

  2. L-Systems

  3. L-Systems Branching Structures Botany Display geometric substitution turtle graphics Animating plants, animals

  4. Plant examples http://algorithmicbotany.org/papers/#abop

  5. As a Formal Grammar Related to fractals recursive branching structure often self-similar under scale Grammar parallel rewriting system context-free (in basic version)

  6. Historical development Aristid Lindenmayer botanist the ‘L’ in L-systems Przemyslaw Prusinkiewicz U. of Calgary introduced L-systems to graphics The Algorithmic Beauty of Plants

  7. DOL-systems Basic version Deterministic Context-free words rules

  8. Geometric interpretation Geometric substitution symbol -> geometric element Turtle graphics symbol -> drawing command

  9. Geometric substitution

  10. Turtle graphics F move forward w/ drawing f move forward w/o drawing + turn left - Turn right rules words S -> ABA A -> FF B -> TT T->-F++F- S ABA FFTTFF FF-F++F--F++F-FF

  11. Turtle graphics FF-F++F--F++F-FF

  12. Botany: terms Stems, roots, buds, leaves, flowers Nodes, internodes Herbaceous v. woody Dichotomous, monopodial Lateral bud Leavs from buds: alternate, opposite whorled Cell influence: lineage, tropisms, obstacles Discrete components: apices, internodes, leaves, flowers Finite number of components Components represented by symbols

  13. Bracketed L-Systems Also add non-determinism database amplification procedural models S->FAF A->[+FBF] A->F B->[-FBF] B->F Brackets -> branch

  14. Example S->FAF A->[+FBF] A->F B->[-FBF] B->F

  15. More examples S->FAF A->[+FBF] A->F B->[-FBF] B->F

  16. Stochastic L-System Add probabilities to non-deterministic L-systems Controls average termination level

  17. Context-sensitive Better control of rule application

  18. Animating plant growth Changes in topology Elongation of existing structures Changing angles, lengths

  19. Animating branches

  20. Parametric L-systems

  21. Context-sensitive, timed w/ conditions

  22. Open L-systems environmental interaction Plant Reception of information from environment Transport and processing of info inside plant Response in form of growth changes Environment Perception of plants actions Simulate processes of environment (e.g. light propagation) Present modified environment to plant

  23. Open L-systems environmental interaction Add construct to L-systems ?E(x1,x2,…,xm) (a bit simply) Query appears in production - sends message to environment which then returns value to production See paper by Mech and Prusinkiewicz for details

  24. Implicit Surfaces

  25. Implicit Surfaces Surface is only *implicitly* defined explicit parametric

  26. Implicit Surfaces Basic formulation Animation Collision detection Deforming implicits Level set methods

  27. Implicit Surfaces Usually define so: surface = 0 inside <0 outside > 0

  28. Displaying Implicit Surfaces Ray Tracing Search along ray to find zero point Marching cubes embed in 3D volume of cells intersect cell edges with surface define polygonal pieces from cell intercepts see examples a few slides later

  29. Metaball spherical, distance-based implicit d T In isolation: a sphere

  30. Multiple Implicits Sum overlapped implicits - with weights

  31. Implicit Surfaces

  32. Topology smoothly changes http://local.wasp.uwa.edu.au/~pbourke/modelling_rendering/implicitsurf/

  33. Signed-distance-based prmitives • From • Point • Edge • Face • Polyhedron

  34. Implicit Surfaces

  35. Testing - good for collision detection ?

  36. Polyhedra embedded in implicits

  37. Implicit Surfaces

  38. Colliding Implicit Surfaces

  39. Colliding Implicit Surfaces

  40. Level Sets

  41. Level Set Methods Adds dynamics to implicit surfaces Usually operate on signed distance function Isosurface updated according to velocity field defined over interface Tracing particles on curve is problematic

  42. Fundamental Idea Instead of evolving curve C(t) = 0 Evolve surface, U, that curve is a level set of Common surface used is signed distance function U(x,y) = distance to nearest point in C If U evolves according to Ut, C will evolve by Ct

  43. Level Set - Fundamental idea

  44. Front Propogation Isosurface advects Normally in direction of gradient Can have constant magnitude Or use magnitude of curvature

  45. Level Set Methods

  46. Front Propogation More generally • Velocity can depend on: • Local properties (e.g. curvature) • Global properties (e.g. position of front) • Properites Independent of shape (e.g. transport function)

  47. Level Set Equation V - velocity field Convection equation

  48. Level Set Equation • F can be: • Constant • Function of gradient • Function of curvature

  49. Implementation Use grid to hold distance function http://www.cs.cornell.edu/Courses/cs664/2005fa/Lectures/lecture24.pdf

  50. Narrow band

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