  I present a fast algorithm to find apparent horizons. This algorithm uses an
explicit representation of the horizon surface, allowing for arbitrary horizon
resolutions and, in principle, shapes. Novel in this approach is that the
tensor quantities describing the horizon live directly on the horizon surface,
yet are represented using Cartesian coordinate components. This eliminates
coordinate singularities, and leads to an efficient implementation. The
apparent horizon equation is then solved as a nonlinear elliptic equation with
standard methods. I explain in detail the coordinate systems used to store and
represent the tensor components of the intermediate quantities, and describe
the grid boundary conditions and the treatment of the polar coordinate
singularities. Last I give as examples apparent horizons for single and
multiple black hole configurations.
