The S−P gap tells one station how far away an earthquake was, but not in which direction. All it knows is that the epicenter (the point on the surface above the focus) lies somewhere on a circle around the station. With the distance formula, three circles pin it down.
Why it works
Two circles usually cross at two points, so two stations leave two possible epicenters. The two points are mirror images across the line joining the stations. A third station's circle passes through only one of them, and that is the epicenter. To check a candidate point by hand, compute its distance to the third station with $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ and compare it with that station's measured distance. Real networks use dozens of stations and solve for depth and start time too, but this is the core idea, called trilateration (GPS works the same way).
Key idea
Each station gives a circle of the right distance. Two circles cross at two points; the third circle picks the one that is the epicenter.
Worked examples
Worked example: Pick the right point
Station 3 is at $(100, 40)$ on a grid in km and is 50 km from the quake. The circles of stations 1 and 2 cross at $P(60, 10)$ and $Q(-20, 70)$. Which is the epicenter?
- Distance from station 3 to $P$: $\sqrt{(60 - 100)^2 + (10 - 40)^2} = \sqrt{1,600 + 900} = 50\,\text{km}$. It matches.
- Distance to $Q$: $\sqrt{(-120)^2 + 30^2} \approx 124\,\text{km}$. It doesn't.
- So the epicenter is $P(60, 10)$.
Answer: $P(60, 10)$.
Worked example: From gaps to circles
Three stations record S−P gaps of 10, 15 and 20 s. What are the radii of their circles (8.4 km per second)?
- $10 \times 8.4 = 84\,\text{km}$; $15 \times 8.4 = 126\,\text{km}$; $20 \times 8.4 = 168\,\text{km}$.
- The station with the shortest gap is closest to the quake.
Answer: 84, 126 and 168 km.
Common mistake
It is easy to think two stations are enough. Two circles usually cross at two points, and both fit the two stations equally well. Check by sketching: the third circle is what breaks the tie.