around the corner,, originally uploaded by ~Faiz.
Occasionally in my reading, I come across a particularly well-posed question that I feel is not only accessible to "lay" readers but also interesting to them. The problem of how to best look (or "peek") around a corner is just one of these questions.
As a warning, I will pose the question in this post and then solicit ideas and answers from the readers. Then next week I'll post a solution and hopefully an example.
First, the setting: You are Bruce Willis in Die Hard. You are pursuing and being pursued by bloodthirsty terrorists in a strange office building. You find yourself walking in a smoky room with your hand against a wall in order to not get lost. All of a sudden, you notice that your wall comes to an end exactly one meter in front of you. You pause and analyze how to best approach this corner in the wall in order to be able to see down the adjoining wall.
If you knew the angle that the adjoining wall intersected your wall (and this angle was acute), the quickest path would be one that departed the wall you were following at an angle of
90-\phi \sin\phi meters. This would take you directly to the "extension" of the line of the adjoining wall and would afford you an unobstructed view down that wall. Conversely, if the angle is obtuse, it would be quickest to just go directly along the wall to the corner. This concept is illustrated in the figure below (courtesy R. Dorrigiv and A. Lopez-Ortiz):
The tension between these two approaches (curl out into the room versus proceed directly to the corner) represent the crux of this path-planning problem.
Let me finally pose the question: Starting from one unit "south" of a corner with unknown acute angle (between 0 and 90 degrees, inclusive), what is the best approach path, and what does "best" mean to you?
A solution---and my take on it---coming soon.
