A blending surface is a surface that smoothly connects two given surfaces along two arbitrary curves, one on each surface. This is particularly useful in the modeling operations of filleting a sharp edge between joining surfaces or connecting disjoint surfaces. In this paper we derive a new surface formulation for representing surfaces which are blends of parametric surfaces. The formulation has the advantage over the traditional rational polynomial approach in that point and normal values have no gaps between the blending surface and the base surfaces. Shape control parameters that control the cross-sectional shape of the blending surface are also available. In addition, the base surfaces are not restricted to any particular type of surface representation as long as they are parametrically defined and have a well-defined and continuous normal vector at each point. The scheme is extensible to higher orders of geometric continuity, although we concentrate on G 1 .