The Variation And Minkowski Dimension Of Self-reflecting Fractal Interpolation Surface

Variation and dimension are the very important factors to the surface roughness.In this dissertation,the self-reflecting and the generalized self-reflecting fractal interpolation surface on the rectangle are discussed. The fractal interpolation function is a bivariate continuous function,so the oscillation and variation and their properties of the bivariate continuous function are studied.For a kind of bivariate fractal interpolation function whose graph is the fractal interpolation surface,the value of its variation is estimated.By deducing the relation between the Minkowski dimension of the graph of continuous function and its variation,the exact value of the Minkowski dimension of the fractal interpolation surface is obtained. Equal-interval section of the rectangle is usually discussed,but actually inequal-interval section is rather comman.In Chapter 5,the author generalized this method based on the condition of equal-interval section and a iterated function system and a fractal interpolation surface can be constructed on R~3 by the reflecting mappings.Then the author give the proof of existence,unique proposition and continuity of this generalized self-reflecting fractal interpolation surface.Furthermore based on the study of its variation,the exact value of the Minkowski dimension of the generalized self-reflecting fractal interpolation surface is obtainedIn Chapter 1,we briefly review the fractal naissance,the development of fractal geometry and the current situation of this project. Furthermore the major content of the project is summarized.In Chapter 2,we mainly study the fundamental theory of fractal geometry including two definitions of dimension,iterated function system (IFS) and the theory of fractal interpolation. In Chapter 3,we study the oscillation and variation and their properties of the continuous function.In Chapter 4 and Chapter 5,we discuss the self-reflecting and generalized self-reflecting fractal interpolation surface on the rectangle. Their variation and the properties are studied and the exact value of the Minkowski dimension of them is obtained.