Direct interface tracking computes spray behavior based only on first principles. It is an advanced form of direct numerical simulation, but with the emphasis shifted from resolving details of turbulence to details of multiphase flow. The moving interface requires special treatment and advanced numerical methods. A code that is capable of accurate resolution of three-dimensional free-surface deformation has been constructed. The Navier-Stokes equations for the liquid phase are solved on a deforming unstructured mesh. This technique tracks the boundary precisely, similar to marker-and-cell methods. However, the adaptive mesh deforms with the interface. Furthermore, this new method avoids the surface reconstruction required in volume-of-fluid methods. The numerical scheme produces a positive-definite matrix that is solved using a conjugate gradient method. In order to maintain mesh quality, the mesh point connectivity and node locations are updated each time step. The results demonstrate the performance and accuracy of this technique. The method used for the surface tension force is shown to he second-order-accurate in space. By locally fitting the free surface to a parabola when evaluating curvature, problems with numerical noise in the solution are avoided. A time-sup criterion based on free-surface numerical stability is discussed. The results for a deforming drop and collapsing ligament are presented. The code is validated by comparison to the theoretical period for drop deformation.