Wikipedia:Reference desk/Archives/Mathematics/2019 May 31

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May 31

Rectangle on sphere?

What do you call a rectangle on a spherical surface? (E.g. your flight path if you fly 10 degrees of latitude due north, 10 degrees of longitude due east, 10 degrees of latitude due south, and finally 10 degrees of longitude due west again?) Is there a special name for such a geometric shape? 2601:646:8A00:A0B3:D572:F62A:ECB8:F316 (talk) 06:48, 31 May 2019 (UTC)

Those would be different figures, depending on a lattitude you start from... A corner case: if you begin at 80°N, then after initial fly 10° due north you are at the North Pole and there is no East from there; all directions are due South! --CiaPan (talk) 08:05, 31 May 2019 (UTC)

I'm not talking about the limiting case here (that would be a triangle, not a rectangle) -- my question is, what do you call the general class of these figures (excluding the limiting case)? 2601:646:8A00:A0B3:D572:F62A:ECB8:F316 (talk) 10:32, 31 May 2019 (UTC)

I suspect it's not a general class (weird degenerate edge cases make it rather a very specific class, IMHO) and as such it probably has no proper name. --CiaPan (talk) 10:59, 31 May 2019 (UTC)

Our article spherical trigonometry uses the term spherical polygon for the generic version of your figure. And based on spherical triangle, spherical rectangle seems legitimate. 91.141.0.175 (talk) 14:54, 31 May 2019 (UTC)

A triangle is a three-sided polygon; a four-sided polygon is a quadrilateral. Only certain special quadrilaterals are rectangles. I'm not sure how you decide which "spherical quadrilaterals" should be "spherical rectangles". All four interior angles equal? Opposite sides equal? Are those two criteria equivalent? (It seems like they should be, but there's something to prove.)

I'm assuming that all four sides should be geodesics, in this case great circle paths. The OP's description actually sounds like the east–west paths may be following latitude lines, in which case they're not "straight lines" at all in spherical geometry. --Trovatore (talk) 16:53, 31 May 2019 (UTC)

The edges are segments of the graticule on the sphere but as they're not geodesic, calling the figure a spherical rectangle doesn't fit the definition of a spherical polygon. 173.228.123.207 (talk) 18:29, 31 May 2019 (UTC)