The wheel is drawn as seen facing south: east, where signs rise, is on the left; the half above the horizon line is the visible sky. Signs run anticlockwise.
When each rashi starts and stops rising on the eastern horizon, from local midnight to midnight.
The lagna (लग्न, ascendant) is the point of the ecliptic — the Sun's yearly path through the zodiac — that is rising on the eastern horizon. The midheaven (MC, Sanskrit madhya lagna or dashama) is the point of the ecliptic crossing the meridian, due south for northern observers. Both turn once around the zodiac in a sidereal day, so the lagna changes rashi roughly every two hours.
Both follow from the local sidereal time θ (the right ascension on the meridian), the latitude φ and the obliquity of the ecliptic ε:
lagna = atan2( cos θ, −(sin θ cos ε + tan φ sin ε) ), MC = atan2( sin θ, cos θ cos ε ).
These give tropical longitudes; the sidereal values subtract the Lahiri ayanamsa (23°15′00.658″ on 21 March 1956, carried forward with the IAU 2006 precession, plus nutation). Sidereal time and the true obliquity come from the astronomy-engine library.
The ecliptic is tilted 23.4° to the equator, and the equator, not the ecliptic, turns steadily past the horizon. Away from the equator the ecliptic meets the eastern horizon at an angle that changes through the day. When it rises steeply, 30° of it clears the horizon quickly; when it lies nearly flat along the horizon, the same 30° takes much longer. In the northern hemisphere the tropical signs from Capricorn to Gemini (signs of "short ascension") rise fast and those from Cancer to Sagittarius rise slowly; the difference grows with latitude, and the pattern is reversed south of the equator. At the equator signs opposite each other take equal times. Each rashi's rising time is the difference of the oblique ascensions of its two ends, α − asin(tan φ tan δ), converted from sidereal degrees to clock minutes.
The ascendant formula was checked against the Swiss Ephemeris house routine (agreement better than 2″ for 1990–2050) and against an independent brute-force search for the ecliptic point on the eastern horizon. Sign change times are found to the second; the real limit is how exactly the place and clock time are known. Beyond latitude 66° parts of the ecliptic never rise, and the lagna jumps; the rising-time chart stops at 60°.