In this article, we discuss the mathematics on how to implement a continuous-time multi-agent crowd simulation. The article covers two agent models, circular, unorientable model, and three-circle, orientable model. We include extensive discussion about simulation objects, simulation mechanics, and implementation details and conclude by addressing the shortcoming and challenges of this type of simulation. We intended this article as a reference implementation for crowd simulation. We have experimented with the implementation in the crowddynamics library.