This is the flat meridian-plane view of the same rings the 3D armillary sphere carries — the page itself is the plane of the observer's meridian, so the outer circle is the celestial sphere and the polar axis, equator and horizon all cut it edge-on as straight diameters.
north pole altitude = φ (your latitude)
equator tilts (90°−φ) from the horizon
tropics sit ±23.44° from the equator, toward each pole
sun's declination δ(day) = 23.44°·sin(2π(day−81)/365.25)
A circle of declination δ (the sun's actual daily path) is parallel to the equator and offset toward the pole, so it also cuts this plane as a straight chord. The chord's two ends are the only points of that daily circle that lie exactly in the meridian plane: local noon (upper end, marked by the sun) and local midnight (lower end). Their height above the horizon line is literally the sun's altitude at that moment — no extra formula needed, it's read straight off the diagram.
Day length still needs real spherical trig, since the visible arc length isn't linear in altitude: cos H₀ = −tan φ · tan δ, daylight = 2H₀·12/π hours (clipped to 0–24h for the polar day/night cases, where the chord never crosses the horizon at all).
- φ — observer's latitude; flips which celestial pole is above the horizon.
- day of year — moves the sun's declination between the tropics and back over 365 days.