Why only some eclipses are total. The Sun's diameter is about 400 times the Moon's, and it is also about 400 times farther away, so both discs are about half a degree wide. The Moon's orbit is an ellipse (perigee about 356,500 to 370,000 km, apogee about 404,000 to 406,700 km, mean 384,400 km), so the Moon's disc varies between about 29.4 and 33.5 arcminutes while the Sun's varies between 31.4 and 32.6. When the Moon's disc is the larger one, totality is possible; otherwise a ring of Sun (an annulus) remains. If the Moon is only partly in line, only a partial eclipse results.
theta = 2 atan(R / D) (apparent angular diameter)
L_umbra = D_sm R_m / (R_s - R_m) (Moon to umbra apex, similar triangles)
r_umbra(d) = R_m - d (R_s - R_m) / D_sm (negative: antumbra, annular)
r_penumbra(d) = R_m + d (R_s + R_m) / D_sm
total at distance d only if L_umbra > d
Obscuration. The Sun and Moon are treated as flat discs of angular radii a_s and a_m with angular separation s; the hidden fraction of the Sun's area is the overlap of two circles (the lens formula). The same formula runs for every pixel of the globe in a shader, so umbra, penumbra and the annular antumbra all come from the geometry. Magnitude is the Moon/Sun diameter ratio when total, otherwise the fraction of the Sun's diameter covered. Limb darkening and atmospheric scattering are ignored; the daylight dimming drawn is illustrative.
Totality duration. Duration = 2 r_umbra / (v_shadow - omega R cos(lat)). The Moon's transverse speed from angular momentum is v = h/r (about 1.09 km/s at 357,000 km); the ground moves east at omega R cos(lat) = 0.42 km/s near 25 N, so the shadow runs over the ground at about 0.67 km/s and a 258 km wide umbra takes about 383 s, 6 min 23 s. Near the ends of the path the shadow hits the surface at a slant and moves much faster, so totality there is shorter.
Where the shadow goes. The model keeps the shadow axis in the fundamental plane (through Earth's centre, perpendicular to the Sun-Moon line) with a velocity plus small fitted acceleration, and projects it onto a spherical Earth using the Sun's declination (17.8 degrees) and the Sun's hour angle. The fit uses a few published values: greatest eclipse near 25.5 N, 33.2 E at 10:07:50 UTC, gamma about 0.14, magnitude about 1.08, duration 6 min 23 s, and the path through the Strait of Gibraltar. Everything else follows from geometry, so times and widths elsewhere are approximate (minutes, tens of kilometres) and are no substitute for an official ephemeris.
Saros. After 6585.32 days (223 synodic months, 242 draconic months, 239 anomalistic months, about 18 years 11 days 8 hours) Sun, Moon and node return to nearly the same relative geometry, so a similar eclipse follows, shifted about 120 degrees west because of the extra 8 hours. The 2027 eclipse belongs to saros 136 (approximate catalogue figure).
Simplifications (honest list). Spherical Earth; circular discs; constant Sun distance and declination during the day; the Moon sits on the shadow axis at the chosen distance (the what-if slider moves it along that line); no refraction; the lunar limb profile in the observer view is an exaggerated illustration; corona and diamond ring are artistic, not predictions. Times are UTC; local mean solar time is UTC + longitude/15 h, official clocks use time zones.
Independent checks. The table compares the umbra radius from similar triangles with the radius found by searching numerically for the position where the Moon's disc exactly fits inside the Sun's, angular sizes with published ranges, the closed-form totality duration with the time between contacts found by bisection, and the saros interval with its three month lengths.
Educational model built from rounded public figures, not an official prediction. Independent educational model; check national almanacs before travelling.