How Astrocartography Lines Are Actually Calculated
Most explanations of astrocartography stop at "this is where the planet was rising". That's true, but it skips the part that makes the lines the shape they are. The whole system falls out of one idea.
Start with the sub-planet point
At any instant, each planet is directly overhead at exactly one point on Earth. Astronomers call it the sub-planet point (you'll know the solar version: the sub-solar point, which is why it's noon somewhere right now).
Finding it takes two numbers. The planet's declination gives you the latitude directly. Its right ascension, compared against Greenwich sidereal time at that instant, gives you the longitude. Sidereal time is the awkward part — it tracks Earth's rotation relative to the stars rather than the Sun, gaining about four minutes a day on the clock.
Once you have that single point, every line is defined relative to it.
Two of the lines are trivial
The planet culminates — reaches its highest point — everywhere along the meridian running through the sub-planet point. That's the MC line, and it's straight because meridians are straight. The IC line is the meridian exactly opposite it.
The other two are a great circle
Here's the part usually left out. The planet is on the horizon wherever it sits 90° away from straight up. So the set of all points where a planet is exactly rising or setting is the set of all points exactly 90° from the sub-planet point — and that is a great circle, the same kind of shape as the equator.
The rising half of that circle is the AC line; the setting half is the DC line. They meet at the two extreme-latitude points of the circle.
This matters practically. The textbook formula for horizon lines is cos H = -tan(lat) × tan(dec), solved latitude by latitude. It works, but it's undefined wherever the latitude exceeds 90° minus the planet's declination — which is exactly where the line does its most dramatic bending — and it needs special handling near the poles. Treating the whole thing as a great circle has no such gaps. It's always a clean closed loop, at every latitude, for every declination.
Why flat maps make this look harder than it is
On a Mercator projection, a tilted great circle becomes a long asymmetric squiggle that appears to wander off one edge of the map and reappear on the other. Generations of astrologers have read those squiggles and reasonably concluded the maths must be gnarly.
It isn't. The squiggle is an artefact of flattening a sphere. Put the same line back on a globe and it's a circle — you can spin the planet and watch it close on itself.
Where precision actually matters
Planetary positions are the easy part; modern ephemerides are accurate to a fraction of an arcminute, which is far finer than any interpretive claim could justify. The error that actually bites is much more mundane: the birth time. Earth turns 15° of longitude every hour, so an hour of uncertainty slides every line about 1,100 km sideways.
Getting the timezone right is fiddlier than it sounds, too — it means knowing the daylight-saving rules in force at that place on that date, which have changed more often than most people expect. Britain, for instance, stayed on UTC+1 right through the winters of 1968 to 1971.