Consider these two search patterns.
${\square}$ A square moves in straight lines forming what you might call a "square-spiral" pattern as it covers a much larger square space.
${\bigcirc}$ A circle spirals out as it searches a circular field with area proportional to the large square space.
The area of each "search window" is equal to 1 square unit. So, the circle would have a radius of $\frac{1}{\sqrt{\pi}}$
1) The $\square$ search window starts by moving up and then to the right one unit. From position (1,1) the square's behavior can be explained by a simple loop.
For every revolution $x$ the square moves $[2x]\Downarrow$ , $[2x]\Leftarrow$ , $[2x+1]\Uparrow$ , $[2x+1]\Rightarrow$.
Such that the entire area of the ground is covered with no point searched more than once.
During any interval the square moves some combination of straight movements over new ground. A rectangle, one unit wide, and the product of time and rate of change long, can represent the area covered at linear speed $s$ units per second as continuously $s$ square units per second.
2) For $\bigcirc$ we could use the Archimedean spiral to cover the ground to be searched, or should we use an involute curve instead ???
The search rate of $\square$, moving at speed $s$, is $s$ square units per second. How does that relate to the search rate of $\bigcirc$, moving at speed $s$, covering a circle with the same area ???
Based on the size of the diameter I'd approximate $\bigcirc$ is less than thirteen percent more efficient than $\square$.
