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upstart writes in with an IRC submission for Bytram:

How drones can hear walls: Mathematicians show that sound can be used to locate flat surfaces:

One drone, four microphones and a loudspeaker: nothing more is needed to determine the position of walls and other flat surfaces within a room. This has been mathematically proved by Prof. Gregor Kemper of the Technical University of Munich and Prof. Mireille Boutin of Purdue University in Indiana, USA.

Can walls and flat surfaces be recognized using sound waves? Mathematicians have been studying this question from a theoretical standpoint for quite some time.

"The basic scenario is a room with flat walls, and maybe a ceiling and a floor," explains Prof. Gregor Kemper of the Chair of Algorithmic Algebra at TUM. The room is not assumed to be rectangular. It is also possible to measure the slope of the walls. Several microphones and a loudspeaker are contained in the room.

Previous studies have already mathematically proven that four microphones and a loudspeaker are sufficient to pinpoint the walls and also calculate their inclination. To prepare for this, the microphones have to be brought into the room at random positions, which will take quite some time and in some situations will be altogether impossible.

That is why Kemper and Boutin took the idea one step further. In their theoretical approach, they mounted the loudspeaker and four microphones on a drone -- making measurement much more practical, because the equipment does not have to be installed in the room.

[...] The question of how likely it is for such ghost walls to arise in the measurement process leads to the core statement of the paper: Kemper and Boutin have proved that the drone's freedom of motion is sufficient for the probability of placing it in a "good" position -- meaning a position where no ghost walls are detected -- to be equal to 1. In other words, such a placement is a near certainty.

"The six degrees of freedom of the drone are sufficient for the microphones to be almost certainly in an optimal position for the measurement," says Kemper. The only prerequisite is that the microphones are not arranged in a common plane on the drone.

Mireille Boutin, Gregor Kemper. A Drone Can Hear the Shape of a Room. SIAM Journal on Applied Algebra and Geometry, 2020; 4 (1): 123 DOI: 10.1137/19M1248534

Original Submission