  In this paper, we construct a bijective mapping between a biquadratic spline
space over the hierarchical T-mesh and the piecewise constant space over the
corresponding crossing-vertex-relationship graph (CVR graph). We propose a
novel structure, by which we offer an effective and easy operative method for
constructing the basis functions of the biquadratic spline space. The mapping
we construct is an isomorphism. The basis functions of the biquadratic spline
space hold the properties such as linearly independent, completeness and the
property of partition of unity, which are the same with the properties for the
basis functions of piecewise constant space over the CVR graph. To demonstrate
that the new basis functions are efficient, we apply the basis functions to fit
some open surfaces.
