Reverse Engineering (RE) requires representing with free forms (NURBS, Spline, Bézier) a real surface which has been pointsampled. To serve this purpose, we have implemented an algorithm that minimizes the accumulated distance between the free form and the (noisy) point sample. We use a dualdistance calculation point to / from surfaces, which discourages the forming of outliers and artifacts. This algorithm seeks a minimum in a function that represents the fitting error, by using as tuning variable the control polyhedron for the free form. The topology (rows, columns) and geometry of the control polyhedron are determined by alternative geodesicbased dimensionality reduction methods: (a) graph approximated geodesics (Isomap), or (b) PL orthogonal geodesic grids. We assume the existence of a triangular mesh of the point sample (a reasonable expectation in current RE). A bijective composition mapping allows to estimate a size of the control polyhedrons favorable to uniformspeed parameterizations. Our results show that orthogonal geodesic grids is a direct and intuitive pp 1-14