By using blossom approach, four new cubic rational Bernstein-like basis functions, which include the cubic Bernstein basis functions and the cubic Said-Ball basis functions as special cases, are deduced. This new basis possesses two tension shape parameters and forms a normalized B-basis, based on which a new class of $C^2$ continuous cubic rational B-spline-like basis functions with two local tension shape parameters is constructed. The totally positive property of the cubic rational B-spline-like basis is proved. A new class of cubic rational Bernstein-B\'{e}zier-like basis functions over triangular domain with three tension shape parameters, which include the cubic Bernstein-B\'{e}zier basis functions over triangular domain and the cubic Said-Ball basis functions over triangular domain as special cases, are also constructed. The conditions for $G^1$ continuous smooth joining two cubic rational Bernstein-B\'{e}zier-like patches over triangular domain are given.