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最近的研究表明，机械加工表面的微观形貌具有自仿射性和多尺度性，因而表面形貌的高度分布方差、斜率和曲率的方差不再是唯一的。基于分形几何理论，用Weierstrass－Mandelbrot分形函数来表征表面的微观形貌，所得到的参数是不依赖于测量尺度而变的“固有”参数。根据分形几何建立的表面轮廓的数学模型可对表面轮廓进行模拟，为建立三维表面形貌的评定参数提供理论依据。
Recent studies have shown that the micro-topography of a machined surface is self-affine and multi-scale, and therefore the variances in height distribution, slope and curvature of surface topography are no longer unique. Based on the fractal geometry theory, the micro-morphology of the surface is characterized by the Weierstrass-Mandelbrot fractal function. The obtained parameters are “intrinsic” parameters that do not depend on the measurement scale. According to the mathematical model of the surface contour established by fractal geometry, the surface contour can be simulated, which provides a theoretical basis for establishing the evaluation parameters of the three-dimensional surface topography.