"...is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows. The hyperbola is one of the four kinds of conic section, formed by the intersection of a plane and a cone."So "hyperbolic" not hyperbole. I could have said that a hyperbola is the locus of points on the plane whose difference of distances to two fixed points, foci, are constant. But then no one except one or two math majors would have a damned idea what I was saying. Anyway I'm dissembling, let's get back to radio. the Decca Navigator system consisted of sets of land-based radio beacons arranged in chains. Each chain consisted of a master station and two or three slave stations. These were positioned at the vertices of an equilateral triangle with the master at the centre.
The system used low frequencies from 70 to 129 kHz. Each master transmitted on a base frequency, and it's slaves on it's harmonics so if the master was on 85 kHz, the first slave would be on 70.833, then the second on 113.333, and third on 127.50. Each station transmitted CW and a navigator could compare the phase difference of the signals from the master and slave stations. Phase comparison could then identify hyperbolic lines of position. But even that is not a unique location. Then a "Decometer" was used. This device counted the hyperbolic "lanes" the ship or aircraft crossed. Then knowing the point of departure a unique position could be calculated. GPS it was not. Later they replaced the Decometer with a multi-pulse system that was a bit tidier.
The system was developed over time beginning with the work of William J. O'Brien on location-finding via the phase comparison of continuous wave transmissions. He tried to sell the military on it but they didn't like the complexity. But O'Brien had a friend named Harvey F. Schwarz. He was the chief engineer at Decca Records. Schwarz was impressed and pitched it to the British. Demonstrations were done in 1942. It was deemed superior to the existing Gee system. Decca even tried using it in cars!
After WWII Decca went commercial with the system and sold it in the USA, Japan, South Africa, Canada Iran.. etc. The downside was that the receivers were all leased making the system very expensive for users though very profitable for Decca. In the 1980s they dropped that requirement as competitors were making superior receivers. the company floundered the the UK had to take over operating the system. Later in court, The European Union actually forced the UK to end it's subsidies of the system which let to it's shut down in 2000.
After being shut down in the spring of 2000, it has been superseded by systems such as the American GPS and the planned European GALILEO positioning system.