The path of a solar eclipse: where on the Earth it is total or annular, as the central line and the northern and southern limits of the umbra, or the antumbra, and where it is seen at all, as the limits of the penumbra and the curves on the horizon where it begins or ends at sunrise or sunset, through the time the shadow is on the Earth.

The Besselian elements (see Astro.Solar.Eclipse) put the shadow on the fundamental plane, through the centre of the Earth and perpendicular to the shadow axis. On it the Earth is an ellipse, and a point (ξ, η) inside it is on the surface at a height ζ above the plane. Following the Explanatory Supplement, the auxiliary declinations d1 and d2 turn the ellipsoid into a unit sphere, η1 = η / ρ1 and ζ1 = √(1 − ξ² − η1²), and then

• ζ = ρ2 (ζ1 cos(d1 − d2) − η1 sin(d1 − d2)),

• sin φ1 = η1 cos d1 + ζ1 sin d1, the latitude on the sphere, and tan φ = tan φ1 / √(1 − e²) the geodetic latitude, and

• H = atan2(ξ, ζ1 cos d1 − η1 sin d1), the hour angle of the shadow axis there, and λ = H − μ the east longitude.

The central line

The central line is where the shadow axis meets the surface, ξ = x and η = y. It begins and ends at sunrise and sunset, where the axis grazes the Earth. Along it:

• The eclipse is total where the umbra's radius at the surface, L2 = l2 − ζ tan f2, is negative, and annular where it is positive. A hybrid eclipse changes from one to the other.

• The shadow moves over the observer at (a, b) on the plane, its own motion (x′, y′) less the observer's, which the Earth's rotation carries at ξ′ = μ′ (ζ cos d − η sin d) and η′ = μ′ ξ sin d − ζ d′. The duration of totality or annularity is the time to cross the shadow, 2 |L2| / √(a² + b²).

• The path width is the width of the band whose observers come within |L2| of the axis: 2 |L2| across the motion on the plane, lengthened where the ground is tilted to the plane, so 2 |L2| / √(1 − (q · n)²), where q is the unit vector across the motion and n the vertical at the central line.

• The Sun's altitude is that of the shadow axis, sin h = sin d sin φ + cos d cos φ cos H, geometric.

The limits

A limit is where the shadow's edge just grazes an observer: at the moment the observer is nearest the axis, m = |L2|. The observers nearest the axis at a moment lie on the line through the axis across the shadow's motion (a, b), turned slightly along it as the cone's radius changes, u a + v b = L2 L2′ for (u, v) the axis less the observer. On that line the Earth is a chord, every point of which is on the surface, and the limit is where the chord leaves the shadow: where the distance from the axis overtakes the cone's radius at that point's height, found by bisection. The motion and L2′ depend a little on where the observer is, so the limit is found again from where it was. The northern limit is to the left of the shadow's motion, which is always eastward, and the southern limit to the right.

Where the shadow reaches past the end of the chord, the edge of the path is the horizon: the place on the limb having its greatest eclipse at sunrise or sunset, the curve NASA's path tables give as the second limit of an eclipse whose axis misses the Earth. So near the ends of a path the limits follow the horizon, and they begin together where the shadow first touches the Earth and end together where it leaves. The cone's radius changes with the height above the plane, which changes quickly near the limb, so the shadow first touches the Earth a little inside the limb, where the chord is deepest in the shadow. There the ground is so tilted to the shadow that a hundred metres on the plane is tens of kilometres on the Earth, so at those two instants the limits are found from the line straight across the motion at the axis, the same for both, and they meet.

The limits are those of the umbra of the mean lunar limb, k2. The mountains and valleys of the limb move a limit by a kilometre or two and are not included, as NASA's path tables do not include them.

The region of partial eclipse

The partial eclipse is seen wherever the penumbra passes while the Sun is up. Its region is bounded by:

• The penumbral limits, north and south, found as the umbral limits are, from the penumbra's radius L1 = l1 − ζ tan f1. Where a limit would lie beyond the horizon it is left out: the region is bounded there by the horizon instead.

• The sunrise limit, where the eclipse ends just as the Sun rises, and the sunset limit, where it begins just as the Sun sets. At a moment, the penumbra's edge meets the Earth's limb, (cos θ, ρ1 sin θ) on the plane, at up to two points, found by Newton's method from where a circle of the penumbra's radius meets the unit circle. A point where the shadow is leaving, u a + v b < 0 for (u, v) the point less the axis, and the ground is turning into the day, ζ′ = −μ′ ξ cos d + d′ η > 0, sees the eclipse end at sunrise; one where the shadow is arriving and the ground turning into the night sees it begin at sunset. Places a little inside these curves see some of the eclipse with the Sun up.

Within the region, the curves of maximum eclipse at sunrise and at sunset are the places having their greatest eclipse on the horizon: the ends of the chord across the shadow's motion through the axis, where they are within the penumbra. First and last contact are the moments the penumbra first touches the Earth's limb and last leaves it, at the point of the limb nearest the axis, where the Sun is on the horizon.

A curve on the horizon can have two branches at the same moment, as when the penumbra overlaps the limb on one side of the Earth and the eclipse ends at sunrise both north and south of the axis, so each of those curves is a list of lines. The horizon is geometric: the Sun's centre at zero altitude, without refraction.

Because μ is the Greenwich hour angle at the moment, longitudes need no correction for ΔT. NASA's path tables use an ephemeris hour angle with ΔT applied to it, so they agree when they use the same ΔT.