Some techical stuff about the Ascendant
I spent the weekend considering some technical issues about the Ascendant/Descendant axis.
The issue: Is the Descendant the true descending point?
Conclusion: Approximately yes.
Why: For it to be true 180 degrees of the ecliptic should always be over horizon, and 180 degrees below. We start at a point where the planes of the ecliptic and horizon cross. Using angular coordinates we start with the ecliptic passing above our head. We try to rotate the horizontal plane in one direction; now we still have 180 degrees of the ecliptic above us. Next we rotate along the other axis, now the ecliptic will tilt and come closer to the horizon, but we still have 180 degrees above.
Important point: However, there are two points where the ascendant is not defined: The places where the ecliptic and the horizon are parallel. However due to the axial tilt of the earth, these points move. They are the points in the polar region where the sun touches the horizon when it changes vertical direction.
Back to the question about 180 degrees above and below. Well, I stated it's true at the point where the horizon and ecliptic planes cross. However, due to earth having a radius, so most surface points are a little off the plane. However, since earth-sun distance >> earth radius, the difference in degrees are so small that it doesn't matter.
We should note that in the case that we are casting a chart on the moon, which has a orbit that take it off the ecliptic, we can't assume that asc. opp. desc.