复有理曲线是近期出现在几何造型领域的一种平面曲线表示形式，该表示形式在分析与计算上有着独特的优势。为了将复有理曲线推广到三维空间，我们拟基于四元数系统的特点，给出四元数有理曲线曲面的定义，并分析该表示的特点与优势。我们还将讨论四元数有理曲线曲面的μ基计算与应用；如何判别一个一般有理曲线曲面能否表示成四元数有理曲线曲面；canal曲面、cyclide曲面等常见曲面的四元数有理表示。上述研究结果将为几何造型中曲线曲面的表示提供一种新的解决方法，进而促进Clifford代数与Lie代数在几何造型中的应用。

Complex rational curve is a special kind of representation for planar rational curve which emerged recently in the field of geometric modeling. Complex rational representation has several advantages over classical rational representation. To generalize complex rational curve to 3-dimension cases, we are going to introduce the definition of quaternion rational curves and surfaces. Properties and advantages of quaternion rational curves and surfaces will be analyzed. We will also discuss μ-bases computation for quaternion rational curves and surfaces and their application, conditions for determining a rational curve or surface to be a quaternion rational curve or surface, how to represent canal surfaces and cyclides by quaternion rational surfaces. Thus, we can give a novel method to represent curves and surfaces which will facilitate their applications. Moreover, Clifford algebra and Lie algebra will be further introduced into the field of geometric modeling.