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Mastering Particle Systems in Computer Graphics

Explore the fundamentals and applications of particle systems for modeling natural phenomena like clouds, terrain, plants, and crowd scenes. Learn Newtonian particle equations, force vectors, solution methods, spring forces, and more. Implement spring-mass systems, Hooke’s Law, damping, attraction, and repulsion forces. Discover spatial subdivision techniques with boxes, linked lists, and field calculations. Understand numerical methods like Euler’s and improved Euler’s for ODE solutions.

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Mastering Particle Systems in Computer Graphics

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  1. Particle Systems Mohan Sridharan Based on slides created by Edward Angel CS4395: Computer Graphics

  2. Introduction • Most important of procedural methods: • Used to model: • Natural phenomena: • Clouds. • Terrain. • Plants. • Crowd Scenes. • Real physical processes. CS4395: Computer Graphics

  3. Newtonian Particle • Particle system is a set of particles. • Each particle is an ideal point mass. • Six degrees of freedom: • Position. • Velocity. • Each particle obeys Newton's law: f = ma CS4395: Computer Graphics

  4. Particle Equations • Define position and velocity: pi = (xi, yi zi) vi = dpi /dt = pi‘= (dxi /dt, dyi /dt , zi /dt) m vi‘= fi • Hard part is defining force vector. CS4395: Computer Graphics

  5. Force Vector • Independent Particles : • Gravity. • Wind forces. • O(n) calculation. • Coupled Particles O(n): • Meshes. • Spring-Mass Systems. • Coupled Particles O(n2): • Attractive and repulsive forces. CS4395: Computer Graphics

  6. Solution of Particle Systems float time, delta state[6n], force[3n]; state = initial_state(); for(time = t0; time<final_time,time+=delta>){ force = force_function(state, time); state = ode(force, state, time, delta); render(state, time) } CS4395: Computer Graphics

  7. Simple Forces • Consider force on particle i: fi= fi(pi, vi) • Gravityfi= g. gi= (0, -g, 0) • Wind forces. • Drag. pi(t0), vi(t0) CS4395: Computer Graphics

  8. Meshes • Connect each particle to its closest neighbors. • O(n) force calculation. • Use spring-mass system: CS4395: Computer Graphics

  9. Spring Forces • Assume each particle has unit mass and is connected to its neighbor(s) by a spring. • Hooke’s law: force proportional to distance (d = ||p – q||) between the points. CS4395: Computer Graphics

  10. Hooke’s Law • Let s be the distance when there is no force: f = -ks(|d| - s) d/|d| ks is the spring constant. d/|d| is a unit vector pointed from p to q. • Each interior point in the mesh has four forces applied on it. CS4395: Computer Graphics

  11. Spring Damping • A pure spring-mass will oscillate forever! • Must add a damping term. f = -(ks(|d| - s) + kdd·d/|d|)d/|d| • Must project velocity: · CS4395: Computer Graphics

  12. Attraction and Repulsion • Inverse square law: f = -krd/|d|3 • General case requires O(n2)calculation. • In most problems, not many particles contribute to the forces on any given particle. • Sorting problem: is it O(n log n)? CS4395: Computer Graphics

  13. Boxes • Spatial subdivision technique. • Divide space into boxes. • Particle can only interact with particles in its box or the neighboring boxes. • Must update which box a particle belongs to after each time step. CS4395: Computer Graphics

  14. Linked Lists • Each particle maintains a linked list of its neighbors. • Update data structure at each time step. • Must amortize cost of building the data structures initially. CS4395: Computer Graphics

  15. Particle Field Calculations • Consider simple gravity. • We do not compute forces due to sun, moon, and other large bodies  • Rather we use the gravitational field. • Usually we can group particles into equivalent point masses. CS4395: Computer Graphics

  16. Solution of ODEs • Particle system has 6n ordinary differential equations. • Write set as du/dt= g(u,t) • Solve by approximations : Taylor’s theorem. CS4395: Computer Graphics

  17. Euler’s Method u(t + h) ≈u(t) + h du/dt = u(t) + hg(u, t) • Per step error is O(h2). • Require one force evaluation per time step • Problem is numerical instability: depends on step size. CS4395: Computer Graphics

  18. Improved Euler u(t + h) ≈u(t) + h/2(g(u, t) + g(u, t+h)) • Per step error is O(h3). • Also allows for larger step sizes. • But requires two function evaluations per step. • Also known as Runge-Kutta method of order 2 CS4395: Computer Graphics

  19. Contraints • Easy in computer graphics to ignore physical reality. • Surfaces are virtual. • Must detect collisions separately if we want exact solution. • Can approximate with repulsive forces. CS4395: Computer Graphics

  20. Collisions • Once we detect a collision, we can calculate new path. • Use coefficient of restitution. • Reflect vertical component. • May have to use partial time step. CS4395: Computer Graphics

  21. Contact Forces CS4395: Computer Graphics

  22. Other Topics • Language-based models: Section 11.7. • Recursive methods and fractals: Section 11.8. • Procedural noise: Section 11.9. • Examples in Appendix (A.18). CS4395: Computer Graphics

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