Friday, May 2, 2014

Best Position Estimate for a Number of Intersecting Great Circles

  If one increases the scale of distance for finding positions the curvature of the Earth needs to be considered. One then has to find the intersection of great circles instead of straight lines on a plane. If all the circles don't meet at a point then a best estimate of the position needs to be made. Since the estimated positions should be closely grouped together a simple average correct to produce a unit vector will do. Consider the following intersection problem in which the intersections of pairs of great circles are used to make rough estimates of the position and an improved estimate found by combining the estimates. One computes the binormals, the angles along the great circle and the intersection points as before.

A plot helps visualize the situation.

At greater magnification we can distinguish the points of intersection and the improved estimate. The average position is close to that found by deviations from the great circles and much easier to calculate.

A simple procedure for finding the angular distance, φ, of a point, q, from a great circle and the arc length, θ, for the closest point along the circle is shown below. It was found by using vector analysis.

The point of intersection is where φ = 0 and q is on the line p. At this point q·b = 0 and one can see how the binormal b can be used to find the intersection point.

Supplemental (May 3): The choice of angles for arc along the great circle and the angular distance of a point away from it may not have been the best ones since the analogy with latitude and longitude could be a little confusing. A better model would replace (θ,φ) with the angles (α,δ) which are used to represent the Right Ascension and Declination on the Celestial Sphere. Like the great circles of constant Right Ascension those of constant arc length along the great circle will converge to a point that is analogous to the Celestial Pole, the pole of rotation. The analogy identifies the great circle with the Celestial Equator.

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