Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it’s difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number of Clifford algebra to make the singular ruled surfaces transform into the dual singular curves on the dual unit sphere. By using the research method on the singular curves, we give the definition of the dual evolute of the dual front in the dual unit sphere, we further provide the k-th dual evolute of the dual front. Moreover, we consider the ruled surface corresponding to the dual evolute and k-th dual evolute and provide the developable conditions of these ruled surfaces.

奇异的直纹表面是一个有趣的研究对象，并且是探索新问题的突破点。但是，由于奇异性，很难研究奇异直纹曲面的属性。在本文中，我们结合奇异性理论和Clifford代数来研究奇异的直纹曲面。我们利用Clifford代数的对偶数使奇异的直纹曲面转换成对偶单位球面上的对偶奇异曲线。通过对奇异曲线的研究，给出了双单位球面上双前沿的对偶演化的定义，进一步给出了双前沿的第k对偶演化。此外，我们考虑对应于对偶渐开线和k的直纹表面第二次进化，并提供这些直纹表面的可发展条件。