Admissible Location of Sample Points of Interpolation by Bivariate C 1 Quadratic Splines

In this paper, we will discuss the admissible location of sample points of C 1 quadratic spline interpolation in $$S\frac{1}{2}\left( {\Delta _{MN}^{\left( 2 \right)}} \right).$$ . General and simple poisedness conditions have been obtained by using smooth and conformality conditions expressed in terms of Bézier coefficients, the techniques of univariate spline interpolation, and theory of multivariate polynomial interpolations. First, we construct poised sets on an arbitrary rectangular cell $${R_{ij}} = \left[ {{x_i}{x_i} + 1} \right] \otimes \left[ {{y_j}{y_j} + 1} \right],$$ , that is, to find the location of sample points which admits unique Lagrange interpolation in $$S\frac{1}{2}\left( {\Delta _{11}^{\left( 2 \right)},{R_{ij}}} \right).$$ . Then, by means of the so-called flows from a source cell $$\mathop R\nolimits_{io,jo} $$ in which the poised set of sample points has been obtained, we can arrange the location of remaining sample points in other cells R ij along the flows.