Two position-lines crossing at a right angle pin you down tightly; the same two lines crossing at a shallow angle leave you sliding along a long thin smear, even though each line is just as accurate. The GEOMETRY of the fix, not the measurements, sets how sharp it is — that’s why GPS needs satellites spread across the sky. Slide the crossing angle and watch the error blot stretch.
Even with equally precise measurements, the accuracy of a fix depends on GEOMETRY — the dilution of precision. Two lines of position crossing at angle α give a position error that scales as 1/|sinα|: at 90° the error blot is a tight circle (DOP = 1), but as the lines graze toward parallel the blot stretches into a long ellipse and the error blows up. This is why GPS receivers report GDOP and prefer satellites spread WIDE across the sky, not clustered. A fail-loud self-check throws unless a near-parallel crossing has several times the DOP of a perpendicular one. ◆ real navigation math, node-verified.
A two-line 1/sinα DOP model (the exact geometric factor); full GDOP is the covariance of an over-determined least-squares fix across all satellites — the perpendicular-is-best, grazing-is-worst behaviour is the exact core.