Introduction to Procedural Methods, Particles Glenn G. Chappell U. of Alaska Fairbanks CS 481/681 Lecture Notes Monday, March 22, 2004

22 Mar 2004CS 481/6812 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

22 Mar 2004CS 481/6813 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 . Give me stapled sheets of paper. The midterm is available on the web page. Complete instructions are on the exam.

22 Mar 2004CS 481/6814 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.

22 Mar 2004CS 481/6815 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.

22 Mar 2004CS 481/6816 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.

22 Mar 2004CS 481/6817 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; };

22 Mar 2004CS 481/6818 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);

22 Mar 2004CS 481/6819 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 …

22 Mar 2004CS 481/68110 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.

22 Mar 2004CS 481/68111 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.

22 Mar 2004CS 481/68112 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