If we leave out the constraint about the distance to the closest tower, then there is a continuum of solutions. Each solution requires a different aspect ratio for the rectangular power station. If you fix one edge of the station (letting the other telescope in and out as needed), what does the locus of viewpoints look like? Is this a classic curve, or is it something new ("Splictrix of Hess")? On Tue, Jul 17, 2018 at 11:50 PM, Bill Gosper <billgosper@gmail.com> wrote:
gosper.org/battersea_run.txt gosper.org/battersea.pdf should be up now. --rwg
On Tue, Jul 17, 2018 at 7:29 PM Bill Gosper <billgosper@gmail.com> wrote:
Fred: Pls email me these pix and I'll host them. --rwg I cannot refrain from adding: Screw DropBox.
<< Scale diagrams of these solutions might be appreciated here ...? >>
Quite so. In a pleasant inversion of the scholastic dictum that a diagram does not constitute a proof, a clutch of accurate diagrams immediately discloses that:
The obvious interpretation of a signed sightline distance --- that the correct angle is made only with a line segment directed _away_ from the tower --- was after all perfectly correct.
Hence --- assuming an observer lacking eyes in the back of the head --- my second and fourth solutions were duds, and only the other two remain standing (outside the rectangle).
Corrected summary & diagrams, and program & results, are posted at https://www.dropbox.com/s/gva6w9ni9y6svp1/battersea.pdf https://www.dropbox.com/s/175plix3vkvxqoj/battersea_run.txt
This has turned out to be an unexpectedly involved exercise: in particular, a neat elementary application of Gröbner bases, as well as an excellent topic for an undergraduate project; perhaps in a situation calling for an especially --- ahem --- challenging one ...
Fred Lunnon
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