Abstract: Based on the analysis of the convexity for planar discrete data points,a type of local,interpolatory and convexity-preserving discrete subdivision scheme for the generation of smooth curves is presented,and the schemes proposed in1 and2 are the special cases.When the suitable shape parameters are chosen,the 4-point interpolatory scheme proposed by Dyn can be reconstructed.A modified scheme avoiding self-circle is derived also.In addition,some useful and interesting properties are discussed,and finally,some numerical examples are given.