Application Of Affine Curvature On Gosper Curve

The Gosper curves are space filling curves,that is,we can use one curve to fill a Gosper island if the curve is long enough.This kind of curves can be applied in mathematical programming,sparse multidimensional database indexing,electronics,biology and internet maps.Therefore,the Gosper curves are worth studying.In the previous research process,the basic theory of discrete affine curves has been perfected:the definitions of discrete curves,tangent lines,affine mappings and the first and second affine curvatures,and calculation formulas and geometric meanings of the first and second affine curvatures are given.Under affine transformation,the first and second affine curvatures of the plane discrete curve are affine invariants,which provide a theoretical basis for the study of Gosper curve by affine curvature.The Peano-Gosper curve is a fractal curve with self-similarity.Based on the moving frame and by means of affine curvature,a new Peano-Gosper space filling curve algorithm can be derived.The specific steps are as follows:firstly,the generation process of the Peano-Gosper curve is reviewed;secondly,under the affine transformation,using the first and second affine curvature calculation formulas,the affine curvature values of each point on the curve are calculated,and the iterative law is derived;again,as long as giving three initial points of non-collinearity and affine curvature values of each point,for any positive integer n,we can draw the Peano-Gosper curve in the nth step;finally,observing affine curvature values of the points on the Peano-Gosper curve,we can find that the affine curvature of the curve can be expressed as a regular iterative sequence of letters,a new algorithm is obtained,and the algorithm is realized by programming with MATLAB.The new algorithm reduces effectively the number of iterations and draws the Peano-Gosper curve more efficiently.The paper also analyze the Node-Gosper curve,calculate the affine curvature values of each point on the Node-Gosper curve by the first and second affine curvature calculation formulas,and find out the general law,design the corresponding algorithm.Each vertex of the Node-Gosper curve is the center of some regular hexagons,which boundary curve of these regular hexagons is the Gosper island.According to the above characteristics,a new algorithm can be used to draw the Gosper island accompanying the corresponding Node-Gosper curve with MATLAB.