What is a spiral search?

Answer 1

We start with certain assumptions:

1) The searcher will be able to find the target are looking for it closely enough to it's location,

2) The target is somewhere near the initial search location though not close enough to find immediately, and

3) The searcher wants to minimize search time.

For example, if someone is calling for help it is possible to find them if you are close enough to hear. If you're close enough to hear them at first, no search is necessary.

The single curve that both covers the plane (assuming the search area to be somewhat flat) and minimizes travel distance is the Archimedean Spiral (https://en.wikipedia.org/wiki/Archimedean_spiral), a curve that rotates around a point (the initial search location) and maintains a constant distance between successive turns.

The searcher should start out in some direction and travel to twice the discovery distance, then turn either left or right and maintain twice the discovery distance from the previous area traveled.

So if you are looking for a pin in the carpet and believe you can find it in any particular inch, you should search for two inches in any direction, then turn (right or left) to start the search spiral. Maintain a distance of two inches from the previously searched turn; you will then search every piece of the carpet completely.

Modifications: It is possible to search with some overlap, or to adapt the spiral to obstacles such as furniture the pin could not have fallen under. While these will increase search time above the optimum, the search will still be complete.