# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Two Bodies + Angle Between Them = Third LoP
From: David Iwancio
Date: 2020 Feb 19, 02:28 -0800

First off, terminology:  after dwelling on mouthfuls like "differences in azimuth," it occured to me that the simplest term for the angles I'm talking about is "cut angle," since it's literally the angle that two lines of position cross at.  This might be helpful in explaining the clearing of lunar distance measurements to someone already comfortable with lines of position ("The intercepts change but the cut angle remains the same.")

Setting aside my idea of drawing "curves of constant cut" for a moment, it seems combining a two-body fix with a measurement of the angle between the bodies may be a way to tease apart different sources of constant error in your sextant measurements.

A quick example:  two stars and the distance between them

• H1:  40°00.0'
• H2:  50°00.0'
• D:  30°00.0'

Cut angle:  40°38.6'

If there is a 1.0' error in vertical measurements (e.g. dip):

• H1:  40°01.0'
• H2:  50°01.0'
• D:  30°00.0'

Cut angle:  40°39.8'

If there is a 1.0' error in all measurements (e.g. index error)

• H1:  40°01.0'
• H2:  50°01.0'
• D:  30°01.0'

Cut angle:  40°41.0'

In practical terms, I suspect this is most sensitive when the cut angle is small (which is exactly what you're hoping to avoid in a two-body fix) and when you need to hold the sextant nearly horizontal to measure the distance between the stars, but it might be something to throw spare computer processor cycles at if you're curious.

As for drawing a curve of constant cut angle, without getting into geometric jargon, right now my hunch is that computing the shape of the curve looks like computing the shape of the moon's shadow during a solar eclipse:

https://eclipse.gsfc.nasa.gov/SEanimate/SEanimate2001/SE2024Apr08T.GIF

If ever anybody pondered navigation by cut angle before the Transistor Age (and their last name wasn't "Bessel"), this is probably where they put the idea back down again.

Until I wrap my head around the math, for the moment the best I can do for comparison purposes is computing the length of a perpendicular dropped from the cut angle to a line connecting the stars.  For the preceding examples:

• True triangle:  39°54.1'
• Dip error triangle:  39°53.1'
• Index error triangle:  39°53.2'

Looking at these numbers, I'm betting this won't be terribly useful at all, but I'm going to continue to gnaw at the math anyway.

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