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I haven't spent much time for stunding the parabola but i think that this is true only when your own course is a linear one.In this case the parabola is tangent to all bearings and the axis of parabola is ,indead,parallel to DRM.
In the other case that our course is not linear ,as in this video, the parabola is tangent to all Spiess Lines produced by the three bearings and i haven't searched if the axis of this parabola is still parallel to DRM (99,9% it is,i just haven't study it).You can't 'imagine' if this axis is parallel in the video as it is impossible to draw this parabola (only one Spiess Line is showing).
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I think you are right, Spiess Lines are equivalent to future bearings as if your course *was* linear. So axis of the parabola surely needs to be paralell to target's course. When own course is not linear, however, the only way to make the parabola tangent to bearing lines is to achieve such a position of your u-boat, that bearing line in time of observation will be drawn exactly on previously computed Spiess line (this is called singularity).
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B_K
Did you study what conditions (proportions of paralell and perpendicular speed components) should be met to make all bearing lines intersect in exactly one point (what are the conditions for speed and course of both vessels to be in pure lag LOS?)
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No, i haven't searched about it.Such case looks to me like having more a threoritical interest than a practical. I mean that ,even if you manage to make the three bearings intersect to one point (which practically will be extremelly hard to achieve), what would be the advantage ?
If i understand you right , you are asking to solve a problem with knowing only the first two bearings (neither target course or speed or range) and with (after the second bearing) proper own speed adjustment to have ,at the time of third observation, a third bearing which pass through the intersection point of the two previous two bearings. right ? if yes, i think that such a problem is not solvable.
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In Dangerous Waters often all bearing lines crossed in one point, at least it looked like this on TMA screen. If we could discover speed component proportions (conditions of pure lag LOS) to achieve this, TMA could be based on two real bearings, one assumed bearing (in fact Spiess line beginning in assumed u-boat future position and crossing the common crossing point) and 4th bearing achieved by triangulation. With some aproximation and to minimize error - preferably for far distances - even not a single point but some surrounding of that point should be enough to assume all bearing lines intersect there and to conduct simplified calculations.