The Research Of Curvatures About Two Kinds Of Discrete Fractal Curves

The two kinds of curves discussed in this paper belong to discrete fractal curves.Fractal graphics have the feature of self-similarity and iterative generation.In nature,physics,biology,art and other fields,fractal graphics have a wide range of applicatons which is a kind of curve that is worth to explore and research.In previous research,we have established the basic theory of discrete centroaffine curve:it is mainly given the definition of discrete curve,discrete tangent curve,planar and space centroaffine curve,and the definition and calculation formula of discrete first and second centroaffine curvatures in Section 3 and 4.Under an affine transformation,discrete first and second centroaffine curvatures are affine invariants.It has provided a theoretical basis for later research.Using the moving frame and invariants,any discrete curve in space could be uniquely identified by its centroaffine curvatures and torsions.Firstly,under affine transformation,by analyzing the iteration process of Peano's Space-Filling curve and using the calculation formula of discrete first and second centroaffine curvatures,we can deduce the iterative regularities about affine curvatures of each point and design an algorithm.Then we can directly generate an affine Peano's Space-Filling curve for any arbitrary step on Matlab.Secondly,under affine transformation,also by analyzing the iteration process of Moore's curve and using the calculation fomula of discrete first and second centroaffine curvatures,we can deduce from the 2k-1st step to the 2kth step and 2kth step to the 2k+1st step respectively,summarize the iterative regularities about affine curvatures of each point and design an algorithm.Then we can directly generate an affine Moore's curve for any arbitrary step on Matlab.Given three initial points which are non-collinear and affine curvature,we can directly generate the curve shape at any arbitrary step.Finally by the affine curvatures,Moore's curve can be quantified and ecoded accordingly to an iterative sequence of characters.