GPS (Global Positioning System) determines a receiver's position by measuring the travel time of radio signals from multiple satellites, each of which broadcasts its precise position and the time of signal transmission. Since radio signals travel at the speed of light (c ≈ 3×10⁸ m/s), the distance from satellite to receiver equals c × time-of-flight. Each measured distance defines a sphere centered on the satellite; the receiver lies at the intersection of these spheres. This geometric process—finding the intersection of spheres—is called trilateration (distinct from triangulation, which uses angles).
Three satellites theoretically suffice to locate a receiver in 3D space (the intersection of three spheres gives two points, one of which is typically underground or in space and can be eliminated). However, receiver clocks are far less precise than atomic GPS satellite clocks (whose accuracy is ~1 nanosecond ≈ 30 cm), introducing a clock offset error that affects all distance measurements equally. A fourth satellite equation allows solving for both the 3D position and the clock offset simultaneously, making GPS practical without expensive atomic clocks in handsets. Modern receivers use 8–12 satellites for redundancy and accuracy improvement.
This simulator places satellites in orbit, lets you control their positions, and visualizes how the intersection of range spheres converges to the receiver position. You can introduce clock errors, atmospheric delays, and satellite geometry effects (dilution of precision, DOP) to observe how they degrade position accuracy—building intuition for why GPS works well in open sky but poorly in urban canyons, and how augmentation systems (WAAS, DGPS) correct systematic errors.
What is the difference between trilateration and triangulation?
Triangulation uses measured angles between the observer and reference points to determine position—a technique used in surveying with theodolites. Trilateration uses measured distances from known reference points. GPS is trilateration: satellites are the known reference points, and measured signal travel times converted to distances are the inputs. The distinction matters practically—GPS receivers measure time (converted to distance), not angles, so the correct term is trilateration. Both techniques require sufficient independent measurements (at least 3 in 2D, 4 in 3D) to uniquely determine position.
Why does GPS need four satellites instead of three?
Three satellite distances define three spheres whose intersection gives two candidate points in 3D space (usually one underground and one in the air—the latter is chosen automatically). However, receiver clocks have timing errors of microseconds, which translate to hundreds of meters of range error. A fourth satellite adds a fourth equation to the system: the four range equations have four unknowns (x, y, z, and clock offset), allowing simultaneous solution for both position and clock correction. With the clock corrected, all four range measurements agree, giving accurate 3D position without requiring an atomic clock in the receiver.
What is dilution of precision (DOP) and how does satellite geometry affect accuracy?
DOP quantifies how satellite geometry amplifies position errors. If satellites are clustered together in a small region of the sky, their range spheres intersect at a very shallow angle—a small error in any range shifts the intersection point dramatically. Well-spread satellites spanning the sky create range spheres that intersect nearly perpendicularly, so range errors cause much smaller position errors. PDOP (Position DOP) < 2 is excellent; PDOP > 6 significantly degrades accuracy. HDOP (horizontal) and VDOP (vertical) separately quantify horizontal and vertical accuracy. Urban canyons reduce visible sky, increasing DOP and degrading urban positioning.
GPS errors come from multiple sources: ionospheric delay (the ionosphere slows signals by 1–30 ns depending on solar activity and path length—a single-frequency receiver must use a model correction; dual-frequency receivers measure the differential delay directly); tropospheric delay (water vapor and air pressure add ~2.3 m of range error at zenith, correctable with meteorological models); satellite clock errors (~1 ns, broadcast in navigation message); ephemeris errors (satellite position uncertainty ~0.5 m); multipath (reflections from buildings add ghost signals); and receiver noise. Combined typical standalone accuracy is 3–5 m horizontal; with WAAS/SBAS corrections, 1–3 m; RTK achieves 1–2 cm.
A standard GPS receiver must download the satellite almanac (coarse orbital data for all satellites) and ephemeris (precise orbital data for visible satellites) from the GPS signal—a process taking 30 seconds to 12 minutes from cold start. A-GPS delivers this data via the cellular network or internet in milliseconds, dramatically reducing time-to-first-fix (TTFF). The phone already knows its approximate location from cell tower triangulation, which further narrows the satellite search space. A-GPS allows position fix in 1–3 seconds even with a weak GPS signal that would require several minutes to download navigation data directly.