This paper presents a novel scheme for constructing bivariate spline surfaces over triangular meshes which are topologically equivalent to a disk. The core part of the scheme is a set of knot selection rules that define local configurations of a triangulation called the directed-one-ring-cycle (D1RC) configurations and bivariate splines defined over a D1RC configuration that are new non-tensor-product splines and possess many nice properties of a univariate B-spline. Using D1RC splines, we take an input triangular mesh as a control mesh and define a bivariate spline surface from the control mesh, which mimics the standard NURBS modeling. Moreover, we can introduce sharp features into the overall smooth spline surface by simply setting special D1RC configurations. As a result, the proposed scheme can define spline surfaces in a way similar to that of NURBS, but has less restriction on the connectivity of the input control mesh.