Most work in computer simulation has focused on off-line algorithms where each step can take arbitrarily long to compute. This thesis focuses instead on interactive simulations that run in real-time. Besides running quickly, such simulations must allow interaction, remain stable under unexpected user input, and remain "correct" (in some sense) for unbounded periods of time. Further, biomechanical simulation requires near-instantaneous planning without simulating the future. These requirements motivate the development of fundamentally new simulation algorithms.

This thesis approaches the interactive simulation problem by developing new, low-dimensional representations of the phenomena being simulated. These representations perform several functions at once. First, they can correlate many degrees of freedom of the underlying phenomenon, allowing us to represent the system with fewer variables. Second, the reduction is constructed so as to allow rapid simulation or control. Finally, the representations allow us to express correctness constraints.

Three examples of such simulations are presented, covering fluids, crowds, and human animation. The fluid model enables large, real-time, detailed flows with continuous user interaction, and can handle moving objects immersed in the flow. The crowd model is based on a fluid-like continuum representation, and naturally exhibits emergent phenomena that have been observed in real crowds. Finally, the human model can automatically compute near-optimal human animations using a low-dimensional basis representation of the planning space.