NavList:

Frank Reed wrote:
“We have two lines of position defined by two lat/lon pairs each:
LOP with Parachute Tower and E.S.B.: (40.57297°N, 73.9844°W), (40.74841°N, 73.98563°W),
LOP with Ferris Wheel and 550 Madison: (40.57403°N, 73.97914°W), (40.7615°N, 73.97308°W).
How do we determine where they cross?? That in itself is an interesting mathematical puzzle. What's the best way to proceed?”

After taking Frank Reed’s excellent Modern Celestial Navigation workshops last May, I became interested in finding a method to calculate the intersection point of two straight lines, given the (latitude, longitude) coordinates for two points in each line, as above. Naturally, this method would be useful only for the relatively short distances in which a straight line approximation of a circle of position is valid.  This approach would obviate the need to plot a graph to find a fix, though clearly a plot would  still be very useful as a sanity check.  I first saw an appealing old paper by one Clarence Woodman, merely a century after is was published in 1922 (attached).  He proposes a straightforward plane trigonometry approach.  I then found what I assume is a more recent algorithm in Wikipedia (https://en.wikipedia.org/wiki/Line%E2%80%93line_intersection), which uses determinants, and seems to work well. I’ve used the first formula listed in this Wikipedia article.

I’ve tried it for a handful of examples, and it gave accurate results.  I assume it would not work for “peculiar” coordinates (e.g., if longitude values for the two points on one LOP are 179.99 E and 179.99 W), without first adjusting these coordinates. The “Navigate by Skyline” example is worked out in the Excel spreadsheet I’m attaching. The formula is long-winded but reasonably straightforward.  I’ve typed it in the spreadsheet in text format to show the algorithm. The cells in the last two rows (C20 and C21) are the ones which implement this algorithm to compute the intersection coordinates.  As you may see, this results in a fix at 40° 26.4’N,  73°59.0’W.  These values agree vey well with those obtained by Bill Ritchie, Frank Reed, and Lars Bergman.

Although these results are very satisfactory, something that bothers me is that I don’t know how this algorithm was derived.  The Wikipedia article cites a MathWorld article (https://mathworld.wolfram.com/Line-LineIntersection.html), which contains essentially the same information, again without a derivation. Since I prefer to understand the rationale of the equations I use, instead of following a “cookbook recipe”,  I would be very grateful if any of you with more knowledge of  linear algebra than I have would point me to a source where I may find this derivation, or to an alternative equivalent algorithm.

Best regards,
Rafael C