Moving least squares is a technique for deriving a smooth surface from a collection of scattered data points in space. We can't just fit a plane to scattered points because the surface the scattered points indicate may not be a simple plane. Instead, for a given point in space, we look for the nearest data points and fit a plane to them. Using a weighting function, e.g. the Gaussian theta(r) = exp(-r/-sigma), we want to minimize the weighted distance theta(|x - x_i_|) of an approximating plane P to the data points x_i_.

Marc's paper that introduced MLS to computer graphics (it was well known in approximation and engineering since the 70's) also describes how to use the local MLS-fit plane as a support to fit a higher-degree polynomial, but the plane gives a tangency fit which is smooth over the surface, so most applications are happy with just fitting the plane. Marc's paper performs a conjugate gradient minimization to find the best fit plane but this optimization ends up being approximate.

Marc's paper on ray tracing MLS surfaces frames them as an implicit surface function. Nina's paper uses this formulation to provide an exact method for fitting the locally optimal plane.