That line used in martial arts as a guide in kata practice. Embusen is the line or pattern that a kata follows in its performance. It is the tool that teaches proper movement, stance, kamae, how stances and movement apply to the particular tactic/technique applied in any given situation or action. The embusen line is different for each kata. It has a start point and end point. It is a blueprint for kata practice.
演 - act, perform, play
武 - martial, manly, strong, mighty, brave, power of fighting
線 - a line, wire, a lane, a track, a figure, a level
The embusen is often thought of as a static line that does not move once the kata begins. In the shu stage of the shu-ha-ri that may be true. It is best if something static is provided for beginner/novice practitioners. Even the few who enter the "ha and ri" stages of practice still consider the embusen as a stationery or static line followed religiously - that is just not accurate.
As a practitioner enters into the "ha" and "ri" stages the embusen moves and is considered much like kamae, i.e. not a particular or static stance but something more fluid and transitional. This is true of the embusen line in kata as well.
Take a look at the following graphic of the standard then the 45 degree variations of the embusen for Naihanchi kata of Isshinryu. As you enter the "ha" and "ri" stages of training you have to visualize and implement the fact that once you start something you will invariably have to make adjustments in the heat of battle. This variation practice gets you thinking and working with such variations with kata and with drills derived from kihon and kata. Once you begin thinking and acting on such variations you will find interesting and beneficial results from kata, etc. practices.
The embusen is also meant to be subjected to the requirements of karate practices through things like the "shu-ha-ri" philosophies, etc. It is that which makes the difference between a physiokinetic practice and the full wholehearted practice we should achieve as would be insinuated by things like the gokui, shu-ha-ri and shin-gi-tai, etc.
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Think of this practice like drills, i.e. where uke or tori suddenly change the strategies and tactics. Once you begin with one direction be prepared to change it dynamically to achieve your goals and strategies.