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Introduction to Procedural Methods, Particles

Introduction to Procedural Methods, Particles. Glenn G. Chappell CHAPPELLG@member.ams.org U. of Alaska Fairbanks CS 481/681 Lecture Notes Monday, March 22, 2004. Where Are We? [1/2]. So far, we have covered: Organizing a 3-D Scene Drawable objects Tree representations Advanced HSR

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Introduction to Procedural Methods, Particles

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  1. Introduction toProcedural Methods, Particles Glenn G. ChappellCHAPPELLG@member.ams.org U. of Alaska Fairbanks CS 481/681 Lecture Notes Monday, March 22, 2004

  2. Where Are We? [1/2] • So far, we have covered: • Organizing a 3-D Scene • Drawable objects • Tree representations • Advanced HSR • Image-space methods • Object-space methods • Buffer-Based Effects • Accumulation, including jittering • Stenciling • Virtual Reality • Representing transformations • VR Juggler programming • User-interface issues • Describing Objects • Explicit vs. implicit • Splines • Implicit representations & marching cubes • Rapid prototyping & file formats CS 481/681

  3. Where Are We? [2/2] • Next: • Procedural Methods • Modeling particles • Particle systems • Fractals • Non-Pipeline-Based Rendering • Ray tracing, etc. • As well as whatever VR-related topics can be made to work … • Also, the take-home midterm exam is due on Wednesday (3/24) at the start of class. • Do not turn it in via e-mail. Give me stapled sheets of paper. • The midterm is available on the web page. • Complete instructions are on the exam. CS 481/681

  4. Procedural Methods:Introduction [1/2] • We have considered various ways to represent a scene and objects within a scene. • Sometimes objects are complex enough (in appearance or behavior) that we think primarily about the algorithm that generates the object, rather than any static representation. • Such algorithms are generally known as procedural methods. • Perlin’s noise-based texture generation technique (from CS 381, fall 2003) is a good example of a procedural method. CS 481/681

  5. Procedural Methods:Introduction [2/2] • We will consider two main categories of procedural methods: • Particle systems • Independently moving particles, obeying some set of laws. • Used for: • Smoke. • Sparks (from welding or whatever). • Explosions (not too different from sparks). • Semi-rigid solids (particles connected by stiff springs). • Cloth (particles connected by things that act like fibers). • Crowd scenes (each particle is a person). • Flocks of birds & schools of fish. • Etc. • Fractals • Definitions later … • Used for: • Clouds. • Terrain. • Tree bark. • Pretty pictures.  CS 481/681

  6. Particles:Definition • A particle is an object that has a time-dependent position. • That’s essentially all it has. • So it can be somewhere, and it can move. • In CG, particles are used in many types of modeling. • Particles may interact with each other in complex ways. • We can render a particle however we want. • Particles are most interesting when they form groups in which each particle moves (somewhat) independently. • This is called a particle system. CS 481/681

  7. Particles:Position & Motion [1/2] • We look at how to model a single particle. • Suppose our particle obeys Newton’s Laws of Motion. • Newton’s First Law says that a particle’s velocity stays constant unless it is subjected to an external force. • So the state of a very simple particle can be stored as a position and a velocity. • In TRANSF, this might be a pos and a vec. • So a simple particle can be represented by a two-element struct: struct Particle { pos position; vec velocity; }; CS 481/681

  8. Particles:Position & Motion [2/2] • Code for a particle, with no force acting on it, might look like this: • Global: Particle p; • In the idle function: static double previous_time = get_the_time(); double current_time = get_the_time(); double elapsed_time = current_time – previous_time; p.position += p.velocity * elapsed_time; glutPostRedisplay(); previous_time = current_time; • In the display function: render(p); CS 481/681

  9. Particles:Force • Newton’s Second Law says that when a force acts on a particle, the resulting acceleration is proportional to the force and inversely proportional to the mass of the particle: • So when we deal with forces, we might want to store a mass for our particle as well. • The simplest force is gravity, which, at human scales, can be considered as giving a constant acceleration in the downward direction. • How do we find the position of a moving particle whose velocity is changing? • This question leads to the very deep topic of differential equations and their solutions. We will not cover this in detail here, but we can say a little bit … CS 481/681

  10. Particles:Simple Approximation • Suppose the force on a particle is small. • Then its velocity will not change much between two frames. • And we can approximate its changing velocity by its current velocity. vec force = compute_force(p.position); p.position += p.velocity * elapsed_time; p.velocity += force / p.mass * elapsed_time; • This is based on a simple method for solving differential equations, called Euler’s Method. • This method has two problems: • Accuracy • The velocity is not constant, after all. • Stability • Small initial errors in position can quickly turn into large errors. CS 481/681

  11. Particles:Better Approximation • We can do better if, instead of using the current velocity of the particle to update its position, we use the average of its current velocity and its next velocity. • Except we do not necessarily know what the latter is.  • So, we approximate it, the same way we have been: vec force = compute_force(p.position); vec next_velocity = p.velocity + force / p.mass * elapsed_time; p.position += (p.velocity + next_velocity) / 2. * elapsed_time; p.velocity = next_velocity; • This is based on a method for solving differential equations, called the Order 2 Runge-Kutta Method. • It has both better accuracy and stability. • When the acceleration is constant (e.g., gravity), it is exactly right. • I mean mathematically, of course. Floating-point computations are always wrong. • When the acceleration is not constant, we should do better than the above code. CS 481/681

  12. Particles:Other Forces • Other common forces: • Planetary-Scale Gravity • Attractive force between two particles. • Proportional to the product of their masses. • Inversely proportional to the distance between them. • Drag/Friction/Air Resistance • Force is opposite to the direction of motion. • Speed and mass may have effects. • Spring Forces • Pulls an object toward the at-rest point of the spring. • Force is proportional to the distance of the object from this point. • Usually combined with drag to give a realistic simulation. • Can be used in a “grid” to simulate a semi-rigid object. • Buoyancy • In air (for smoke!) or water. • Elastic (or Inelastic) Collisions CS 481/681

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