Some of my students have expressed confusion after listening to various instructors (including me) talk about either "getting off the line" or "getting on the line." What is this magical line and what do we do with it? The confusion comes from the fact that there are actually several different lines that we refer to in different contexts.
the attack line
the parallel offbalance line
the perpendicular offbalance line
the line formed between our centers
The attack line is the path that uke's center travels down as he attacks tori. In the context of kata, this line is called embusen. It is important for tori to get her center off this line as the first action an any technique. The techniques for getting off the line are the first three actions of tegatana no kata. Any time that an instructor is telling you to "get off the line," he is referring to embusen.
Any time uke takes a step he is subject to at least two major weaknesses, or offbalances. It is easy to find the direction of these offbalances. If you draw an imaginary line between the balls of uke's feet and extend it forward this defines the first offbalance, the parallel offbalance. The third line is perpendicular to this line and extends forward and backward between uke's feet. If sensei is telling you to "step on/across/down the line," he is almost always talking about one of these lines, either the parallel or perpenducular offbalance lines.
The last line commonly discussed is the line between uke's and tori's centers. This line if often defined through metsuke (eye contact) and in some situations it is approximated by tori's and uke's extended arms. Many times in techniques that derive from wrist attacks, for instance, if you are about to switch from an ura condition (behind uke) to an omote condition (in front of uke) or vise versa, you will want to "run down the line," meaning to get as far from uke as he will let you before you switch. At the "end of the line" you will feel your connection to uke go taut and that will change your direction of travel.