I've been playing with Chipmunk lately. It is a C library that simulates physics in two dimensions, and is intended for use in games. It's quite easy to use, but there is a bit of overhead in setting up graphics since I have to do that on my own. I'm using OpenGL through the SDL library, and it works quite smoothly. Here is the result of today's dabbling:
There are two important fundamental concepts in Chipmunk: Bodies and collision shapes. What is the difference? A body has the usual properties like mass, position, velocity, etc. However, a body does not have an area (a shape). That is the purpose of collision shapes. A collision shape does not have any of the properties of a body, it merely defines a shape (such as a circle or a rectangle). The relationship between these two concepts is that multiple shapes can be attached to a body. Only shapes can collide; bodies do not collide by themselves.
The balls in the video above were constructed using one collision shape (a circle shape) for each of the three bodies. They were rendered using an 8x8 OpenGL 2D texture (loaded from a PNG image) on a quadrilateral (GL_QUADS). Easy as pie.
The rope was more difficult to make. For a long time, the rope would either disintegrate slowly (because it was being pulled apart by gravity), or explode (the line segments would vibrate quite violently).
When making the rope, we need another fundamental Chipmunk object: The joint. Joints are used as constraints in the simulation. A constraint is something that keeps an object from moving freely. There are different kinds of joints: Pin joints, slide joints, pivot joints, and groove joints. For example, the pin joint simply stitches together two bodies at certain offsets from the body's centre. When one body moves, the joint will cause the other to follow.
My rope uses slide joints. The only difference from a normal pin joint is that the joint allows a certain flexibility: The "stitch" can be given a minimum and a maximum length, so that the bodies can get closer together or further apart before the joint's constraint takes effect. We can easily imagine the maximum length property as that of a string connecting two objects. The string will keep the objects from getting too far apart, but it will also bend if the objects are closer together than the length of the string.
The slide joints connect the line segments of the rope. Each line segment consists of four collision shapes attached to a single body. The four collision shapes are all circles, laid out side by side. These four circles have fixed positions with regards to each other. They're not actually drawn as circles, but as solid rectangles (GL_QUADS). The rope has 30 such line segments, so that's 30 bodies and 120 collision shapes.
Finally, we have two fixed bodies at either end of the rope. These bodies have infinite mass, are not affected by gravity, and have no collision shapes. (Actually, they're not added to the Chipmunk space, so they are not affected by any forces at all.) This is what keeps the rope in place. And to my great surprise, the way in which these bodies are connected to the rope seems to be what determines the quality of the simulation. At first, I was using pin joints to connect these stationary bodies to the endpoints of the rope. It wasn't until I replaced them with slide joints that the rope actually got as smooth as it is in the video above.
I'm currently toying with the idea of putting a cart with wheels on the thread and controlling the wheels with the arrow keys. If that works, I think it could make for a nice game element as a variation on the paddle of e.g. Breakout.