The Moon crossing Earth's shadow, drawn in the plane perpendicular to the shadow axis at the Moon's distance: umbra, penumbra, contact times and magnitude.
The science
Earth's shadow is a cone. With solar radius Rs = 696,000 km, Earth radius Re = 6378 km and Sun distance a = 149.6 million km, the umbra half-angle is sin(au) = (Rs - Re)/a and the penumbra half-angle is sin(ap) = (Rs + Re)/a. At the Moon's distance D the shadow radii are ru = Re - D tan(au) and rp = Re + D tan(ap). Air around Earth adds a little: following Danjon the umbra is enlarged by about 2 % (slider).
The Moon of radius r = 1737.4 km moves across this plane with relative speed v = vm - w D (its orbital speed minus the apparent motion of the shadow axis, about 0.0172 rad/day). If the track passes a distance b from the axis, contact happens when the separation equals ru + r (U1, U4), total phase begins when it equals ru - r (U2, U3), and the penumbral limits use rp + r (P1, P4). The umbral magnitude is (ru + r - d)/(2r), and totality lasts 2 sqrt((ru - r)^2 - b^2)/v.
Simplifications: orbital speed scaled with sqrt(1/D), a straight track, no parallax variation and a uniform dark red tint instead of the real Danjon brightness scale. Checks below: the umbra radius against the textbook 4.6 thousand km at mean distance, and a time-stepped track against the analytic totality duration. This is educational geometry only.