Projecting Between the Planes
Here is the page the rest of the category depends on. It has one idea in it, and the idea is a fence.
The fence
Picture a fence running across a field, with palings standing a degree apart. The palings are numbered and perpendicular to the base of the fence.
A planet hovers in the air above the fence — not sitting on it, floating some distance above. You want to know which degree it occupies, so you extend the palings upward until one of them reaches the planet. The number on that paling is the reading.
Now do the one thing this whole category turns on.
Leave the planet exactly where it is. Do not move it by a hair. Instead tilt the fence by 23.4°, hinged at one end. Extend the palings again.
The planet reads a different degree.
Nothing about the planet changed. What changed is the thing you were measuring it against — and since a coordinate is a statement about a body relative to a reference, moving the reference moves the coordinate.
That is projection between the two planes, entire. The ecliptic is one fence, the celestial equator is the other, and 23.4° is the angle between them.
The palings are the projection
The palings deserve a name, because they are doing the real work.
Each paling is the set of points that all share one reading — every position that projects down onto the same mark on the fence. In the sky they are great circles running from pole to pole of whichever plane you are using: hour circles through the celestial poles for the equatorial frame, and the corresponding circles through the ecliptic poles for the ecliptic frame.
And here is why they cannot be the same palings for both fences. As the obliquity page showed, the two planes have different poles, 23.4° apart. Palings run to the poles. Different poles, different palings. Tilt the fence and its whole family of palings tilts with it.
This is what "measure away from the plane" was hiding. Both frames measure away from their own plane towards their own pole, and those are two different directions.
Where the hinge is
The fence is hinged, and it matters enormously where.
The two planes cross at the vernal equinox and at the point opposite it. That crossing line is the hinge: the axis the tilt turns about. Along that line the two fences are in exactly the same place, because that is what crossing means.
So a body lying on the hinge reads the same in both frames. It has to. And everything else swings.
The two ways a body can be off the hinge
Everything in the next page follows from asking where a body sits relative to that hinge, and there are two independent ways to be away from it.
Along the fence. A body can be sitting right on the fence — flat on it, no height at all — but a long way from the hinge. Tilting the fence still moves it, because it swings up or down and its perpendicular meets the fence somewhere new. A body on the ecliptic at 45° of longitude reads about 42.5° of right ascension: the same point in the sky, two and a half degrees apart in the two frames. This effect is largest around 45°, 135°, 225° and 315°, and it drops to nothing at 0°, 90°, 180° and 270°.
Above the fence. A body can be hovering, at some celestial latitude, and this is the case the diagram draws. Now even standing directly over the hinge does not save it — because it is over the hinge, not on it. When the fence swings beneath it, the planet stays put and the palings move under it.
A body hovering a height h above the hinge ends up displaced along the fence by h × sin(23.4°). That factor is 0.398 — call it four-tenths. It is not a fudge; it is simply the sine of the tilt, and the fence is where you can see why it turns up.
The same thing on a sphere
The fence is flat and the sky is not, so here is the honest version.
One body, sitting perfectly still, carrying three different pairs of numbers. The two circles that concern this category are the equator and the ecliptic; the third is the horizon, included as a reminder that the same machinery built your local frame too.
None of the three readings is the correct one. They are readings against different fences.
Latitude and declination are not the same number
One consequence worth stating before the next page, because it catches people.
A body's celestial latitude is its height above the ecliptic. Its declination is its height above the celestial equator. Two fences, two heights, two genuinely different numbers.
The Sun is the cleanest demonstration. Its celestial latitude is essentially zero all year — it lies on the ecliptic by definition. Its declination swings from +23.4° to −23.4° and back. Same body, same motion, two frames, and one of them reports a constant while the other reports the entire mechanism of the seasons.
What comes next
A planet is not a fixed dot. It is moving along the fence — so if the two frames disagree about which degree it is at, they disagree about when it arrives at any given degree.
That is not a rounding difference. For some bodies it is years.