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Thursday, December 27, 2007

Quadratic equations will help Dorchester!

Sounds unlikely, doesn’t it? How could quadratic equations possibly help Dorchester? Well, I should first note that we’re talking about quadratic relations —in particular, those represented by hyperbolas — not about quadratic functions in the familiar form of f(x)=ax2 + b+ c, represented by the almost equally familiar vertically oriented parabola.

That doesn’t help, I hear you say.

So listen up, and you’ll hear about a real-life application of quadratic relations with hyperbolic graphs, and you’ll see why it’s important to Dorchester. First of all, media coverage of Dorchester always seems to emphasize the number of shootings and stabbings here, almost always blurring the distinctions among neighborhoods... but that’s not what this post is about, so let’s not go off on that tangent. Anyway, the police report on a shooting yesterday included the following remarks:
This morning at approximately 1:21am, officers from District B-2 responded to the area of 206 Blue Hill Avenue based on the sound of gunshot detected by the ShotSpotter system.

On arrival, bystanders informed officers that there was a male lying on the ground suffering from a stab wound. The victim, a 22 year-old male, was found on the sidewalk in front of 203 Blue Hill Ave. suffering from what appeared to be stab wound. Officers immediately requested an ambulance to the scene to treat the victim. While being treated on-scene, officers learned that the victim was suffering from a single gunshot wound and not a stab wound.
Ignoring the misplaced modifier (surely the officers weren’t the ones who were treated on-scene), we note the reference to ShotSpotter. On Universal Hub you can read a brief discussion of the specific incident and the possible effectiveness of this technology, but here I just want to say something about the mathematics involved. For about three months, Boston has been using ShotSpotter to provide almost instantaneous detection of gunshots in high-crime neighborhoods. In addition to the interesting engineering going on in distinguishing gunshots from other sounds, ShotSpotter uses second-degree quadratic relations in the form of ax2 + bxy + cy2 + d + e+ f = 0; when b2–4ac > 0, the graph of such a relation turns out to be a hyperbola. By precisely timing how long it takes the sound of the shot to reach each of three detectors (and knowing the speed of sound during current weather conditions), a computer can calculate three different hyperbolic paths and can determine that the gunshot occurred at the unique intersection of all three. Then the nearest police officers can be immediately dispatched to the correct location. If this technology continues to fulfill its promise, we will have quicker responses to shootings in Dorchester and other Boston neighborhoods, sometimes quick enough to arrest a shooter — and quicker responses will presumably lead to reducing the number of such shootings. Stay tuned...

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