Fractal Interpolation Curve In 3-dimensional Space And Its Box-Counting Dimension

In this paper,δ-Oscillation of curve in 3-dimensional space is defined and studied, then obtain dimension formulation. Fractal interpolation curve in 3-dimensional space is constructed. And itsδ-Oscillation properties are studied. On the base of these obtain the method of calculate Box-dimension of fractal interpolation curve in 3-dimensional space. Firstly, some typical fractal functions and their Box-dimension or Hausdorff dimension are proposed. Secondly, the fruits of functional digraph in 2-Dimensional space aboutδ-Oscillation of curve in 2-dimensional space and its dimension are summarized. On this base, 2D to 3D,δ-Oscillation of curve in 3-dimensional space is defined and studied, then obtain dimension formulation. According to this dimension theorem, the Box-counting dimension of the digraph of the sum of the two functions in 3-dimensional space is established. The maximum Box-counting dimension of the two functional digraphs plays a decisive part in the Box-counting dimension of the digraph of the sum of the two functions in 3-dimensional space. Thirdly, results of fractal interpolation function in 2-Dimensional space are introduced, including its basic concept, dimension, and properties. And then fractal interpolation curve in 3-dimensional space is constructed. Itsδ-Oscillation properties are studied. On the base of these obtain the method of calculate Box-dimension of fractal interpolation curve in 3-dimensional space.