A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2020 Jan 19, 12:01 -0800
Using a Sumner approach to deal with this doesn't really change anything.
We can calculate celestial lines of position based on various different geometric approaches. By the usual intercept method we calculate azimuth and distance from some AP point. The so-called Sumner calculation generates an LOP from two points on the line given two latitudes (note: three points are not required). Whether you get your LOP by intercept+azimuth or by two-point or by various intermediate options (like calculate one point on the LOP and then calculate or look up azimuth), you end up with short segment of a line of position near where you think you are. It doesn't matter at all whether this is generated by the intercept method or the Sumner process. In either methodology, a pair of Sun sights can still give you LOP segments that are very far apart and can't be made to cross "on the page" (and which are also subject to inaccuracy in crossing due to the inaccuracy of using a flat piece of paper to represent a spherical globe over any distance greater than about a degree). Intercept method or "Sumner" method, you still need to start with a very good estimated position.
The celestial circle of position (with an error band) is the fundamental geometric entity that we get from any celestial altitude sight. It doesn't matter how it's generated, assuming we're using a mathematically-valid method. Any celestial circle of position reduces, of course, to a celestial line of position over short distances. Celestial LOPs generated by the two-point methodology (known historically as the "Sumner" method) are fundamentally identical to celestial LOPs generated by the intercept method (see PS). Historically the intercept method was slightly shorter in terms of computation but more work in plotting. Today computation effort is irrelevant. Either approach produces a celestial LOP, and both approaches need an iterative step or some other means of estimating position if we consider hypothetical scenarios where there is no reasonable DR position.
PS: For absolute technical accuracy (or nit-picking, depending on your point of view!), there is a very small difference between the LOPs generated by the two-point methodology and LOPs generated by the intercept method or other one-point techniques. If I pick off two points on a celestial LOP and draw a line through those, I am slicing across the celestial circle of position. The segment of the LOP between the two points is ever so slightly inside the proper circle of position. By contrast the line of position from the intercept method touches the proper circle of position at only one single point and is otherwise outside the circle. In terms of geometry on the plane, one LOP is a secant of the circle --a line that slices across a circle close to its circumference-- while the intercept LOP is a tangent of the circle --a line that just touches the circumference of a circle. The gap between the two is usually negligibly small in celestial navigation. Decades ago there was a fair amount of pointless debate over the merits of a secant of the circle of position compared to the tangent of a circle. As I say, "pointless debate". (N.B.: "secant" and "tangent" here refer to lines in relation to a circle, not the trig functions, though they do originally get their names from this geometry: https://en.wikipedia.org/wiki/Secant_line).