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Well, I guess I can't resist weighing in on this old topic again.  There is nothing complicated about this issue.  A single sextant altitude detremines an LOP, without any other info, such as a DR position, or that confusing AP.  As we all know, the LOP is a circle with radius of the co-altitude, centered at the observed body's GP.  A picture of this LOP requires plotting:  One simple, and obvious, way is to use the familiar equation for altitude, solved for LHA:  Cos LHA = (sin H -sin L sin d )/cos L cos d).  Enter with the body's declination, its observed altitiude, and the latitude at which you which to know the longitude of the point on the LOP, then get that longitude from the LHA & GHA of the body.  Compute as many points on the LOP as you wish.  It has nothing to do with an assumed, or a DR, position.  (This is exactly what Thomas Sumner did in 1837).

Similar plotting can be done with the St. Hilaire method.  As I've said several times before, there's no assumption made in the St. Hilaire method (not considering the LOP straight-line approximation here).  The AP is an Assigned Poisition that specifies where the unique point is on the LOP that is nearest to this designated point.  By using different APs (just as different latitudes above), calculate as many ponts on the circlular LOP as you wish.  That's plotting the LOP (exactly).

Maybe I've misunderstood the issue on the pevious posts (and I'm not addressing historical comments here), but this is my take on the subject "LOP's without DR Position."

J Karl