The multivariate quartic NURBS surfaces

In this paper, we construct a kind of multivariate quartic nonuniform rational B-spline (NURBS) surfaces by using bivariate quartic B-spline bases in the multivariate spline space S42(△mn(2)), and discuss some properties of this kind of NURBS surfaces with multiple knots on the type-2 triangulation. Compared with the bicubic (rational) Bézier surfaces, the new multivariate NURBS surfaces on the knot vectors of the form U={0,0,0,0,1,1,1,1} and V={0,0,0,0,1,1,1,1} have similar properties at the four edges of the surfaces. Several examples show that our multivariate B-spline surfaces are better than the corresponding bicubic Bézier surfaces.