A New Algorithm For Sierpinski Curves

The Sierpinski filling curve is a common fractal curve.Filling a square can be achieved by a certain iterative law,which leads to a kind of fractal structure---the space filling curve,and the Sierpinski filling curve is often applied to electrical appliances as a resistive component structure due to its own geometric properties.In this paper,we describe the Sierpinski filling curve from the perspective of the frame,and a new algorithm for studying the filling curve is given.The Sierpinski filling curve is a discrete curve essentially.The characterization of the geometric properties of discrete curves is generally based on the study of smooth curves fitted to them.In this paper,the discrete affine curvature value of the curve is defined by establishing the moving frame of the discrete curve.Combined with the programming,implementing the output of the Sierpinski filling curve.By studying the formation rule of Sierpinski filling curve,the moving frame is established based on the edge tangent vector of discrete points.Then studying the affine curvature sequence of the discrete points of the curve,the iterative law of the sequence is found,and simplifing the program by coding.Depicting the entire filling discrete curve by the geometric property of each discrete points.After completing the output of the curve,based on traditional Sierpinski filling curve,combined with that arbitrary polygons can be triangulated,and the purpose of extending the filling pattern from a square to an arbitrary polygon is realized,which provides more options for the the application of the curve structure of Sierpinski in real life.This method of studying discrete curves through moving frame is also applicable to other filling curves.It is also important to the study of discrete curves.