G1 scattered data interpolation with minimized sum of squares of principal curvatures

One of the main focus of scattered data interpolation is fitting a smooth surface to a set of non-uniformly distributed data points which extends to all positions in a prescribed domain. In this paper, given a set of scattered data V = {(x_i, y_i), i=1,...,n} ∈ R² over a polygonal domain and a corresponding set of real numbers {z_i}_i=1ⁿ, we wish to construct a surface S which has continuous varying tangent plane everywhere (G¹) such that S(x_i, y_i) = z_i. Specifically, the polynomial being considered belong to G¹ quartic Bezier functions over a triangulated domain. In order to construct the surface, we need to construct the triangular mesh spanning over the unorganized set of points, V which will then have to be covered with Bezier patches with coefficients satisfying the G¹ continuity between patches and the minimized sum of squares of principal curvatures. Examples are also presented to show the effectiveness of our proposed method.