Understanding flow in the three-dimensional phase space is challenging both to human experts and current computer science technology. To break through the barrier, we are building a program called PSX3 that can autonomously explore the flow in a three-dimensional phase space, by integrating AI and numerical techniques. In this paper, I point out that quasi-symbolic representation called flow mappings is effective as a means of capturing qualitative aspects of three-dimensional flow and present a method of generating flow mappings for a system of ordinary differential equations with three unknown functions. The method is based on a finding that geometric cues for generating a set of flow patterns can be classified into five categories. I demonstrate how knowledge about interaction of geometric cues is utilized for intelligently controlling numerical computation.