In this study, we aim to construct a sequence of towel folding operations by a robot arm. In this paper, we propose the methods for spreading a towel and detecting the grasp positions. For the easy spreading of the towel, we propose how to put the towel on the table. The proposed method lifts one vertex of the towel, and detects the shadow part by using a camera image for recognizing the overlapped condition of the towel. For the folding motion of the towel, we set the vertices of the towel edges as grasping positions. So, we propose the way to detect the vertices of the towel correctly by generating a rectangle which includes all feature points of the towel by using image processing.