File talk:How far away is the horizon.png

Impossible Chart.

Regarding http://en.wikipedia.org/wiki/File:How_far_away_is_the_horizon.png, what this picture purports to show, a function that has the same feet-to-miles slope as meters-to-kilometers slope, is impossible.

Since one coordinate system has a built-in coordinate-axis scale relationship of 5280:1 while the other has a built-in coordinate-axis scale relationship of 1000:1, the subject function will have one slope in one coordinate system and a differing slope in the other coordinate system. A line (slope) common to both (as shown) requires either, 1, that the two coordinate systems be non-orthogonal (i.e., one rotated with respect to the other), or 2, that one of them have non-square decades (i.e., differing scales).

I haven't bothered to determine which slope is correct because I'm not really interested in the subject, but one is wrong (or possibly both are wrong).

A trivial check with the formulas given in the article shows that the graph is perfectly correct, and there is no problem with units. After all, this graph shows ratios, not absolute measurements -- when the viewer's height above the horizon is changed, the new distance to the horizon will be the same regardless of the units. (For example, near the earth's surface, doubling h will make d increase by sqrt(2), regardless of how they're measured.) The slope of the line is not 1:5280 or 1:1000, but rather 1:sqrt(2) Citynoise (talk) 16:29, 1 December 2014 (UTC)