  We present a fully Lagrangian particle level-set method based on high-order
polynomial regression. This enables closest-point redistancing without
requiring a regular Cartesian mesh, relaxing the need for particle-mesh
interpolation. Instead, we perform level-set redistancing directly on
irregularly distributed particles by polynomial regression in a Newton-Lagrange
basis on a set of unisolvent nodes. We demonstrate that the resulting particle
closest-point (PCP) redistancing achieves high-order accuracy for 2D and 3D
geometries discretized on highly irregular particle distributions and has
better robustness against particle distortion than regression in a monomial
basis. Further, we show convergence in a classic level-set benchmark case
involving ill-conditioned particle distributions, and we present an application
to an oscillating droplet simulation in multi-phase flow.
