How to Find a Lost Bluetooth Earphone??
1. It's somewhere in the room, but I can't find it
Have you ever lost a single Bluetooth earphone in your room? It's still connected to your phone (music is playing), but you can't pinpoint its location.
Is it under the futon? Behind the shelf? Wandering around blindly or relying solely on sound is inefficient. Here, the knowledge of "Mathematics (Geometry)" learned in junior high school becomes surprisingly useful.
2. Mathematical Tool: Circumcenter of a Triangle
First, let's prepare a mathematical tool to solve the problem. Geometry offers the following important theorem:
"There is exactly one circle that passes through three points not on the same straight line."
This circle is called the Circumcircle, and its center is called the Circumcenter. The circumcenter is equidistant from the three points and can be found as the intersection of the perpendicular bisectors of each side.
Fig 1: The circumcenter (red dot) is the intersection of the perpendicular bisectors (green lines) of the sides.
3. Application to Physical Phenomena: Bluetooth Waves
How do we apply this to finding earphones?
Bluetooth radio waves spread in concentric circles if there are no obstacles. Although signal strength decays with distance, reading specific values (dBm) is difficult without specialized tools. However, there is a clear boundary that anyone can recognize.
That is the "boundary where the connection cuts off / connects."
This "connection limit" forms the circumference of a circle with a certain radius $R$ centered on the earphone. In other words, if you find 3 points in the room where the connection is lost, those 3 points lie on the same circumference, and the earphone is located at the center.
4. Practice: Circumcenter Search Algorithm
Play some music and follow these steps:
- Find Point A:
Start walking from the center of the room and place a marker exactly where the sound cuts off. - Find Point B:
Return to the connected area, walk in a different direction, and find another spot where the sound cuts off (Point B). - Find Point C:
Walk in yet another direction and find the third cutoff point (Point C). - Aim for the Intersection:
Visualize the perpendicular bisectors of the line segments AB and BC in your mind. Head towards their intersection; that is the location of your earphone.
Fig 2: Locating the target from 3 disconnection points (dotted triangle)
5. Dr.WataWata's Insight
In engineering, finding a "maximum value (peak)" is often difficult because it is susceptible to noise. On the other hand, detecting an "edge (boundary)" where a signal switches "on or off" is much easier and more reliable.
"If you don't know the center, fill in the outer moat (boundary) first." This is a mathematically and strategically logical approach common to edge detection in image processing and robot exploration algorithms.