Wikipedia:Reference desk/Archives/Mathematics/2024 April 22

= April 22 =

How are arbitrary hemispheres defined on theWGS84 ellipsoid?
With spheres there's only 1 right answer, there's also formulas that can be used to find points 0.5π radians from another point, is there a formula for the line with half the ellipsoid surface area on each side? Is there a formula for the line where an infinitely far Star of Bethlehem and an anti-Star of Bethlehem at the other end of the ellipsoid normal line would have equal zenith distances ignoring refraction and geoids? Are these lines the same? How far apart can they be? Sagittarian Milky Way (talk) 18:30, 22 April 2024 (UTC)


 * Unless an ellipsoid is a sphere (which the WGS 84 reference ellipsoid is not), no portion of it is a mathematical hemisphere. Any plane through its centre divides it though into two equal (congruent) parts. Usually the plane will be a meridional or the equatorial plane. In more general geodetic systems the equator and meridians, although not ellipses, also lie in a plane and can be used for a fairly fair cutting into two parts, which however will normally not be congruent. Calling the two parts "hemispheres", although not correct in a strictly mathematical sense, is nevertheless conventional. --Lambiam 19:11, 22 April 2024 (UTC)
 * Yes but that's the easy way out, hemispheres centered on the equator or pole are exactly zero percent of all possible centers. Sagittarian Milky Way (talk) 21:56, 22 April 2024 (UTC)
 * Is division by any plane through the centre not general enough?
 * The sight lines to a point on the celestial sphere and to its celestial opposite are parallel. So are the directions to the respective zeniths from a given place on the ellipsoid and its antipodal place. Therefore the angular distances are the same. --Lambiam 11:03, 23 April 2024 (UTC)
 * I don't know how to do the center plane. Either finding points of surface tangency from the point of surface perpendicularity or the point of surface perpendicularity from 2 surface points on the plane. There seems to be a u and a v involved I keep seeing u and v but don't know what that is, or if that's needed when the plane is not arbitrary but has 1 of 3 defining points fixed to the ellipsoid center. I stupidly dropped out before learning u, v and pseudo-delta swirl. Sagittarian Milky Way (talk) 22:31, 23 April 2024 (UTC)
 * If they are boldfaced and, these variables probably stand for some 3D-vector ($x, y, z$).  --Lambiam 16:51, 24 April 2024 (UTC)
 * Likely two such vectors orthogonal to each other. —Tamfang (talk) 02:57, 1 May 2024 (UTC)


 * I must admit to being intrigued by the idea that the Star of Bethlehem might have been arranged by angels using WGS 84! However as far as I can see there's just ocean at the antipodal point for the birth of Damien Thorn. Pehaps we're safe for a while yet ;) NadVolum (talk) 19:42, 22 April 2024 (UTC)
 * If one wanted to start a rival franchise, one could look for holy sites (holy to some cult) in the dark patches of this map. —Tamfang (talk) 02:52, 1 May 2024 (UTC)
 * I just want to make orthographic projections, not start a religion. Or equivalently, find star of Bethlehem umbras for anywhere like a lone star over Texas. I'm not sure I didn't make a mistake interpreting the orthographic projection formulae I found. Sagittarian Milky Way (talk) 04:58, 3 May 2024 (UTC)
 * I meant a movie series based on some other existing religion's holy site. —Tamfang (talk) 05:15, 4 May 2024 (UTC)
 * Ah I see. The Left Behind Antichrist was not born in the southernmost part of the 145°E area of South Pacific archipelago either for some reason (he was born in what is now the EU), and he wasn't the most incompetent sentient on Earth at carpentry, so there should be room for at least a 3rd franchise where he is. There's many Dracula films after all. For some religions an Antichrist-type demon would be extracanonical, I suppose someone could make a film where a boy never follows the middle path, always sleeping and staying awake twice the normal hours for instance, and celibacy then BDSM orgies. Sagittarian Milky Way (talk) 14:54, 4 May 2024 (UTC)
 * Or praying to anti-Makkah belly-up in tongues but someone gets the idea to tape-record and play backwards and it's the correct prayers in perfect Arabic pronunciation when he only knows language(s) French Polynesia teaches in school. Someone might try to murder the filmmaker though. Sagittarian Milky Way (talk) 15:11, 4 May 2024 (UTC)
 * The anti-Messiah would presumably try to eat as little Kashrut as possible (consuming dairy as close in time as possible if fed beef or poultry), walk at least 1 mile per Shabbat and do all weekend homework then (when it's also Shabbat in Jerusalem), and try to cause Holocaust II. Sagittarian Milky Way (talk) 16:19, 4 May 2024 (UTC)


 * The normal on the ellipsoidal through Bethlehem won't go through the center of the Earth and so won't go though the antipodal point. NadVolum (talk) 20:04, 22 April 2024 (UTC)
 * It doesn't really matter though cause the idealized celestial sphere/astronomical coordinate system is infinitely far, the lines to the star from anywhere on Earth would be parallel. It would matter for the "ranking all points by distance and picking the nearer half" way as an extremely flattened ellipse could have the (geographic, not geocentric) latitude minus 90 be only a few miles away (plus 90 in the southern hemisphere) Sagittarian Milky Way (talk) 22:15, 22 April 2024 (UTC)