NavList:

Yes, it's an interesting concept, and, for a while, there were sextants on spacecraft to take advantage of this geometry.

I do every lunars workshop a little differently [upcoming lunars workshops in March and April of this year]. Occasionally I have enough time to get to the concept of cones of position, but not usually. This is the fundamental geometry behind traditional space navigation by sextant, which was really only applied once --by Jim Lovell on Apollo 8 in December 1968. Practice sights were taken in later Apollo missions, but primarily after Apollo 8, the sextant was employed as a sighting instrument for spacecraft orientation --in effect, as an astro-compass. And after Apollo, automated star finders replaced that last role for a space sextant.

A circle of position on the Earth's surface is really the intersection between the Earth's surface and a three-dimensional cone extending from the Earth's center outward. Without information about observer altitude, a given astronomical zenith distance places an observer somewhere on the surface of the cone. When we are at great range from the Earth or Moon, we can still measure angles from the Earth or Moon limb to a bright star. That limb is just the horizon at great distance (and if the Earth is being used, that limb could literally be the sea horizon, just as in standard marine navigation). Each sight places the observer on a cone in space. This constrains our position and can be fed into an analysis of a spacecraft's trajectory. In the purest form, we can take three cones of position (two from the Moon's limb, one from the Earth's limb, or some similar combination) and cross them to get a three-dimensional position fix.

From the perspective of simple altitude observations, consider this scenario:
I see the Moon due South and a bright star directly above it, also due South. I measure the angle between them. This is a "lunar distance" but not in the historical sense for finding GMT since this star would be well off the ecliptic necessarily, and it would not be useful for determining time.

I measure this angle with better than average accuracy. Then I go to "clear" the lunar distance as observed for parallax (and a little refraction but that's much less important usually). Parallax depends on the Moon's altitude so I need to measure the Moon's altitude to continue with a normal lunars clearing process. But turn it around. If I know reasonably well the geocentric angular distance between the star and the Moon, which I would at known GMT, then the difference between my observed lunar distance and the geocentric computed distance is exactly the same as the Moon'a altitude correction... which depends on the Moon's altitude! So my lunar observation allows me to deduce the Moon's altitude without observing it. Maybe it's foggy... Maybe there are mountains on the horizon towards the Moon. Regardless of the reason, I can take the Moon's deduced altitude and use it now to get latitude, treating this as a "common" latitude sight ("common" but it's a Moon altitude so there are some other issues to deal with).

Latitude from a lunar distance... There's a heresy! But it works just fine (though demanding high accuracy from our sextant sights), and if we extend it, it leads directly to the idea of cones of position from lunars and three-dimensional position fixes in space.

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA