I haven't asked for this yet because there is one value I can't tell you how to calculate even though I know what it is. (I'm decades past my trig prime when I could have just stopped and derived the formulae, and I haven't put the work into figuring it out. So I've held off mentioning this. - But this is the thread where I should introduce the request or idea - and we can sort out the fine points later.
The horizon (H), meridian (M), and prime vertical (PV) are always at right angles to each other. That means that a planet on any one of these is always conjunct, opposite, or square another planet on one of these. This theoretical construct has proven itself hands down - no doubt about it - in ingresses, and probably is equally true in returns and nativities (but, due to the rarity and the math difficulties, hasn't been actually studied by me and, probably, by anyone else).
In a sense, it's just another mundane aspect. If this were demonstrated and accepted, there is little reason not to lump it in with other in mundo aspects (since that's what it is: just a larger view of the landscape), but at the moment I'm using different language to avoid confusing everybody (since using PV aspects alone hasn't become normalized in everybody's thinking yet).
Here is what I know and what defines the aspects as explored thus far:
- The Vertex-Antivertex axis (either ecliptically or mundanely) is worse than useless in measuring an angularity effect in ingresses (and IMHO in return charts), though it seems to have some expression in nativities.
- Regardless of that fact, the mundane conjunctions, oppositions, and squares appear definitely powerful between a planet on the prime vertical and another planet that is on the horizon, meridian, or prime vertical. These come in three flavors.
- Functionally (while exploring them thus far) my working method has been to start with a planet within 3° of azimuth 90° or 270°, i.e., the Antivertex (due east) or Vertex (due west).
- FLAVOR 1: PV to PV. These are easy: These are conjunctions or oppositions in azimuth.
- FLAVOR 2: PV to Meridian. These also are easy: They are squares in azimuth where the PV planet fits the above criterion and the meridian planet is foreground.
- FLAVOR 3: PV to Horizon. This is the one where I don't have an easy direct way to tell you how to do the math. I've been using a kludge as an approximation (hopefully a close approximation). What we want is a measurement along the meridian in the same way that azimuth is a measurement along the horizon and PVL is a measurement along the PV.
Here is how I've been faking it: I've the advantage that I'm only dealing with factors very close to the angle. When very close to the angle, several different ways of calculating will produce nearly identical results; e.g., for a planet within a few degrees of the horizon, both distance in PVL and in altitude will be similar and, often, identical. (My Moon is 3°15' off Dsc in PVL and 3°15' southern altitude.) Therefore, I use an alternative but similar pair of different measurements for PV-to-H squares: I take how far the first planet is past (plus) or before (minus) Vx/Av in azimuth, then how far the second planet is past (plus) or before (minus) the horizon in PV Amplitude (the latitude or declination analog to PVL) - a value I can squeeze out of Solar Fire. It's tedious and I probably miss some by inadvertence.
It would be better to figure out how to calculate this new value nobody else uses - meridian longitude - and make the measurement directly. I am prepared for the rude shock that the geometry of such a circle could be freakier than expected and produce surprise results but, if so, then it's because it's actually how things look.