有面积的曲线的性质及高维推广

The Properties of the Curve Having Area and its Generalizations of Higher Dimension

中文关键词:  分形  分形曲线  分维  胖分形指数

中文摘要:

      本文研究平面上有面积的曲线的分形性质及其在ｄ维欧氏空间Ｅ^ｄ中的推广．对瘦分形曲线，分维Ｄ_f＝ｌｎ２^ｄ／ｌｎ（２／ｋ^1/ｄ），而对于胖分形曲线，分形指教β＝－ｌｎｋ／ｌｎ２．α＝ｄ．ｄ＝２时的胖分形曲线包括经典的Ｐｃａｎｏ曲线为其特例。

英文摘要:

      This paper studied the fractal properties of the curve having area on the plane and itsgeneralization in the d-dimensional Euclid space E^ｄ.For thin fractal curves, the fractaldimension D_f= log2^ｄ/log(2/k^1/ｄ),for fat fractal curves,the fractal exponents β=-logk/log2,α=d. When d=2, the fat fractal curves include classical Peano’s curvedrawn by D.Hilbert.