The math behind the arcs
Stars appear to rotate around the celestial pole at 360° per sidereal day (1,436.07 minutes). For any exposure of t minutes, the trail angle is t / 1436.07 × 360°. On the sensor, that angle becomes an arc whose radius in pixels depends on your focal length and sensor size.
Stars near the pole trace tiny tight circles. Stars near the celestial equator (declination 0°) trace the longest arcs for a given exposure. At the zenith, all stars rotate around a point directly overhead.
The 500 Rule gives the maximum exposure before trails become visible as streaks rather than points: divide 500 by your effective focal length. The NPF Rule is more precise and accounts for aperture and pixel pitch. This simulator shows you the crossover point.