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    Re: Medieval Navigation Problem
    From: David Pike
    Date: 2019 Nov 26, 11:38 -0800

    Frank you wrote:

    Want to know the actual distance and angles and all that? Then you do as I said earlier and seek out one of the early derivations of the law of reflection based on least time. The simplest trick is to look at the target "in the mirror". So flip the barn over to the other side of the river to its exact mirror image. That image is 200 yards east of the river. The house remains 400 yards west of the river and the north-south distance (parallel to the river/mirror) is still 800 yards. So the distance N/S is 800, the distance E/W (to the mirror image) is 600. Divide both of those numbers by 200, and you get 4 and 3... Therefore, you have a 3-4-5 triangle so the straight line distance from the house to the mirror image of the barn is 5 units or 1000 yards. 

    Yes, that’s the slick answer.  I took a somewhat longer path initially.  I decided to solve it graphically.  First, I got rid of the hundreds, then I started a table of distance travelled for every unit along the river bank.  It soon became apparent that the ideal distance was between 5 and 6 units up the river, so I started taking progressively smaller intervals until I found that the shortest distance was 10 units when you aimed 5.33 units up the river bank.  I didn’t actually need to draw the graph.

    However, I couldn’t help noticing that the position 2/3 of the way up the river towards the barn was also the ratio of the two buildings from the river, so I started substituting other ratios of distance from the river and distance between buildings (see attached spreadsheet).  Sure enough, the only thing that seemed to matter was the ratio of the distances of the two buildings from the river, so  I started to look for slick solution.

    The best I could come up with was to draw a line from the house to a point on the river opposite the barn and a line from the barn to a point on the river opposite the house.  Then I drew a line from where these two lines crossed to the river.  This was the point to aim at.  I tried looking for something like a 3,4,5 triangle, but being restricted to one side of the river I never spotted it.  I did however notice the similar triangles and got the total distance of 1000yds that way

    I was sure there must be a neater answer, so I rang the author up.  As soon as he said, “imagine the barn on the opposite side of the river”, it all became clear, as did the 3,4,5 triangle for distance.  DaveP

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