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Reasons to Suspect that
Echolocation Cannot Explain Sonic Vision
Hyperboloids and ellipsoids, and also spheres, are herein called conicoids. Each conicoid can be generated by the reflection of a planar conic section about an axis of that conic section. A sphere can be generated by spinning a circle about one of its diameters.
In “active” wavar, the time and place of generation of a wave are known and sensors sense reflections of that wave. From times-of-arrival of echoes at a few sensors, a point P might be computed as the intersection of some conicoids. Of each hyperboloid that is used, both foci will be on sensors. Of every ellipsoid used, one foci will be on a sensor and the other will be on a generator of clicks.
A method or act of computing, via an intersection of conicoids, a point P to represent a source of a wave is called triconication.
Triconication upon the outputs of sensors in a small (e.g. 3 or 4 member) set can be used, and probably is used in conscious computations by some people, in what some call “echo-location,” to locate sources of waves. Perhaps, for example, the submarine Scorpion was located, at least roughly, by triconication upon the sharp sounds sent across the Atlantic Ocean by its imploding hull.

Ecolocation Must Fail to Produce Proper Images?
While echolocation using triconication can locate isolated source-points of waves, there are two reasons to expect echolocation to fail as a basis of sonic imaging:
1) it produces too few points to properly depict an object, and
2) the points it produces are obscured by phantoms.

1) Points produced by echolocation are few for two reasons
A) There is Destructive Overlap: when two or more wave-reflectors are at nearly enough the same distance from a sensor, there can be overlap of waves that destroys time-of-arrival information on that sensor.
B) There is the Features-per-Duration Limitation. For example, in a computer-simulation of a “fish-finder” (animation of simulated fish-finder) a test fish, though it might be 24 inches long, “seen” sideways can easily span less than 6 inches from its nearest point to the Device to its farthest point from the Device. In active sonar, the differences in time-of-arrival from the farthest point and the time-of-arrival from the nearest point of the fish would be, in seconds, less than d = 6/Speed (Speed is about 60,000 inches per second). In that simulation, the duration of the one Feature being used, the dolphin-inspired "fang" ( described here) is p = 1/100,000 second. It seems unlikely that, on any one sensor, the number of fang-things from that “fish” could exceed d/p=10. So, in the given duration d, a sensor’s output would be limited to no more than 10 fangs resulting from some dolphin-like insonification of that “fish.” And, in the detection of representations of possible sources of waves, a triconicating set of sensors would detect no more than 10 points. Ten is not enough for a proper image of a fish. And, rather than 10, a more realistically, due to Destructive Overlap on each sensor, anticipated number of image-points on that fish per triconicating set, per click, might be just 2.

2) A second reason for failure of echolocation, in imaging objects rather than just locations of objects, can be briefly illustrated. Suppose that M1, M2, M3 are three detecting sensors of a detecting set S and that F denotes "fang" and F-toa means time-of-arrival of a fang. Suppose that representations of source-points of waves are detected via times-of-arrival, "toas," of wave-fangs on three (or more) sensors in a group. Suppose T1 is an F-toa ("toa"=time-of-arrival) on M1 and there are 100 F-toas on M2 and 100 F-toas on M3. There might be only one source of the wave that produced T1 but there might be 1*100*100 or 10,000 triples (T1,T2,T3) to consider for that T1. Of these, at least 9,900 would be “phantoms,” Given a ratio of 99 phantom points per good image-points, the true object would be obscured in a cloud of echoplasm.

Sometimes, to state a problem clearly is to solve it?

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