Street-Crossing For The Mathematically Disinclined

After the last few shopping trips I’ve taken, I am compelled to address an unfortunate consequence of the general public’s lack of mathematic skills. While you may think that high school Geometry class was useless, there is one area where an intuitive understanding of angles, lines and polynomials will greatly decrease your likelihood of being run over by a severely annoyed seat belt engineer (random example).

When crossing from a parking lot into a store, the general population choses from three possible techniques. These may be described thusly: Imagine two parallel lines each representing the boundary of the parking lot and the store entrance respectively. Pedestrian paths crossing this space can be then rendered graphically as shown below:

Technique 1: Line AB
Technique 2: Line AC
Technique 3: Curve DE

Now, it is intuitively obvious that the technique involving the shortest distance to cross the street is the least dangerous (less exposure to traffic). However based on my recent shopping experiences most people follow some version of Technique 3, which is undeniably incorrect of them. Allow me to prove it.

The points A, B, C, D & E are located as follows:
A (20, 0)
B (0, 0)
C (0, 7)
D (0, 30)
E (20, -5)

Based on this we can determine equations for each of the 3 pathways:


Distance from Point A to B:


Distance from Point A to C:


Distance from Point D to E:
(Since I could not discover via googling any way to directly calculate the length of a polynomial curve segment, I did an approximation by breaking up the curve into tiny little line segments.)
Note: I purposely chose a relatively “flat” equation to describe the path of the pedestrian. More wobbly paths intersecting the same points would be even longer. Two of these possibilities are:

Confirming the distances given by these equations is left as an exercise for the reader.

As the above calculations demonstrate, by choosing a curved path, you are exposing yourself and your companions to more danger of being run over by drivers who feel that while yielding to pedestrians is a good thing, that doesn’t apply to pedestrians too stupid to understand the difference between roads and sidewalks. So, please for your safety (and my sanity) always utilize Technique 1 when crossing streets.

Sidenote: The above analysis is equally applicable to people crossing aisles in parking lots. While obesity is becoming more common, those wide aisles lined with cars are not in fact sidewalks so please do not treat them as such. FYI.