Back in June I wrote a story about the 45th parallel, which runs through New Hampshire. The state has two (two!) historical markers noting that you are halfway between the equator and the North Pole at that point. Except you’re not really because of geometry, as you can learn by reading my sterling prose.
I got an email today pointing me to a 2000 article in the Journal of the Royal Astronomical Society of Canada (http://adsabs.harvard.edu/full/2000JRASC..94…48B) about this very topic. The email is from a guy named Thomas Murray in Scotland who runs a website about this topic called latitudes information.
Here is the article:
“Several years ago there was a small controversy in Nova Scotia concerning the correct location of a highway monument near Stewiacke, Nova Scotia. In the 1930s, a roadside marker along Route 2 was erected to indicate that it was one place in Nova Scotia that is midway between the North Pole and the Equator. That marker was 16 kilometres north of the 45th parallel, where some people felt it should be located. The old marker is correct, and if you are driving south on Highway 102 toward Halifax, you will see another sign just north of exit 11 to Stewiacke announcing that you are at that unique line of latitude: 45° 8 ́.65 N, midway from the pole to the equator. The controversy occurs because a midway point can be defined in a couple of ways.
“The most obvious definition is to use a point on the 45th parallel (45°N),which is the latitude numerically midway between the Equator at 0°N latitude and the North Pole at 90°N latitude. The correct method, however, is to halve the distance along the surface of the Earth between the Equator and North Pole. The latter procedure involves more difficult calculations than is the case with the first definition. Only if the Earth were a perfect sphere would the midpoint specified by both methods occur at 45°N latitude.”
“Note that the halfway point is not at 45°N, but falls slightly north of that parallel. Halfway is determined to be at 45° 8 ́ 39” of latitude.”
45° of latitude is obviously halfway in latitude between the Equator at 0° and Poles at 90°. However it is not halfway in distance. In the WGS84 (1984) reference ellipsoid, 45° North due north to the North Pole results in a distance of 5017.021351334979 kilometers or 3117.432538559176 international miles, similarly from 45° South to the South Pole. 45° N due south to the Equator results in 4984.944377977744 km or 3097.500831380826 miles, similarly from 45° S due north to the Equator. Therefore in the WGS84 ellipsoid, 45° North / 45° South is 32.076973357235 km or 19.931707178350 miles closer to the Equator, than to the North / South Pole.
This is because the Earth is not a sphere, but an approximate ellipsoid of revolution (oblate spheroid), since it bulges at the Equator. Calculated WGS84 reference ellipsoid, the surface at the Poles is 1 part in 298.257223563 (21.3846857548 km or 13.2878276831 miles) closer to the Earth’s center, than at the surface at the Equator. In WGS84, the Equatorial circumference is 40075.016685578486 km or 24901.460896848956 miles, and the Polar or meridional circumference is 40007.862917250892 km or 24859.733479760009 miles. Therefore the Equatorial circumference in WGS84 is 67.153768327594 km or 41.727417088947 miles longer than the Polar or meridional circumference. From a Pole to a Pole the distance in WGS84 is 20003.931458625446 km or 12429.866739880005 miles, and from the Equator to a Pole is 10001.965729312723 km or 6214.933369940002 miles.
Therefore different degrees of latitude are not equal in distance. They are increasingly longer the farther you are from the Equator.
For example in WGS84:
Latitude 0° to 1° distance 110.574388557799 km or 68.707739649074 miles.
Latitude 89° to 90° distance 111.693864914200 km or 69.403350007332 miles.
Equator to a Pole in WGS84 = 10001965.729312723 meters.
10001965.729312723 meters ÷ 2 = 5000982.8646563615
0° of latitude due north or south for 5000982.8646563615
meters = WGS84 45° 08’ 39.5437411966” (45 degrees 8 minutes 39.5437411966 seconds) N or S / 45° 08.659062353275666’ (45 degrees 8.659062353278 minutes) N or S / 45.14431770588793° (45.14431770588793 degrees) N or S. Obviously also, from the North Pole the same distance due south, and from the South Pole the same distance due north. Therefore in the WGS84 ellipsoid, precisely midway between the Equator and the North / South Pole it is a distance of 5000.9828646563615 km or 3107.4666849700011 miles to both the Equator and the North / South Pole.
WGS84 latitude 45° 08’ 39.5437411966” North / 45° 08.659062353278’ N / 45.14431770588793° N is 16.038486678618 km or 9.965853589176 International miles farther north than latitude 45° North. Similarly, WGS84 latitude 45° 08’ 39.5437411966” South / 45° 08.659062353278’ S / 45.14431770588793° S is 16.038486678618 km or 9.965853589176 miles farther south than latitude 45° South.
Personally computed using Charles Karney’s “Online geodesic calculations using the GeodSolve utility”: https://geographiclib.sourceforge.io/cgi-bin/GeodSolve
“GeodSolve is accurate to about 15 nanometers (for the WGS84 ellipsoid)” or 0.000015 of a millimeter or 0.00000059 of an inch. In an email to me from Charles Karney: “The accuracy of 15 nanometers that I quote is for paths up to half-way round the earth.”
WGS84 45° 08’ 39.5437411966” (45 degrees 8 minutes 39.5437411966 seconds North or South / 45° 08.659062353278’ (45 degrees 8.659062353278 minutes) N or S / 45.14431770588793° (45.14431770588793 degrees) N or S, was calculated by me in the WGS84 ellipsoid, and confirmed to me by Mario Bérubé, Team Leader, Geodetic Survey Division, Natural Resources Canada in September 2013.
Currently the best fit reference ellipsoids for the Earth / geoid, that have latitude-longitude based on them, are the GRS80 (1980) ellipsoid, and the WGS84 (1984) ellipsoid. Therefore it makes sense to use latitude-longitude based on the GRS80 ellipsoid or the WGS84 ellipsoid, for the halfway latitude. An example if one uses a local ellipsoid: Computed in the Airy 1830 ellipsoid (Great Britain), the Earth’s equatorial circumference would be 3.604 km or 2.239 international miles less, than computed in the WGS84 / GRS80 ellipsoid, and the Earth’s polar or meridional circumference 3.359 km or 2.087 miles less. Therefore using a local datum such as NAD27 (Clarke 1866 ellipsoid) for halfway, does not make sense. The current datum for the USA, and Canada, is NAD83, which is based on the GRS80 ellipsoid. Though NAD83 will be replaced in North America by NATRF2022 in 2022, but NATRF2022 will retain the GRS80 ellipsoid. GRS80 is civilian and has replaced local ellipsoids in many countries, whereas WGS84 is run by the United States Department of Defense. However WGS84 is used by handheld navigation grade, commercial grade, consumer grade GPSr / GNSSr; automotive navigation systems; smartphones; navigation systems for aircraft, ships; Google Earth; Wikipedia etc. Whereas GRS80 is used by Survey-Grade GNSS receivers for accurate and precise surveying, i.e. GRS80 is used by surveyors, surveying & mapping organizations.
Using Charles Karney’s GeodSolve, in the GRS80 ellipsoid, halfway computes as: 45° 08’ 39.5437437477” North or South / 45° 08.6590623957954’ North or South / 45.14431770659659° North or South.
The difference between halfway computed in both WGS84, and GRS80 ellipsoidal distance, in latitude, i.e due north-south, is 0.078757 of a millimeter or 0.003101 of an inch. The WGS84 latitude is 0.078757 mm or 0.003101 inches farther north, than the GRS80 latitude. Incidentally the Earth’s polar or meridional circumference computed in the GRS80 ellipsoid, is only 0.32906 of a millimeter or 0.01296 of an inch shorter, than that computed in the WGS84 ellipsoid.
The National Geodetic Survey (USA) INVERSE Computation, and its FORWARD Computation: https://www.ngs.noaa.gov/cgi-bin/Inv_Fwd/inverse2.prl
, and https://www.ngs.noaa.gov/TOOLS/Inv_Fwd/Inv_Fwd.html
assume that distances computed in the GRS80 (1980) ellipsoid, and the WGS84 (1984) ellipsoid are identical “GRS80 / WGS84”, however they actually compute in the GRS80 ellipsoid. Using
National Geodetic Survey (USA) INVERSE Computation, and its FORWARD Computation: “Ellipsoid : GRS80 / WGS84 (NAD83)”, halfway computes as “LAT = 45 8 39.54374 North” (Latitude 45 degrees 8 minutes 39.54374 seconds North), which to the same precision is identical to that computed in the WGS84
ellipsoid. “LAT = 45 8 39.54374 North” (45° 08’ 39.54374” N) is in latitude, i.e. due north-south, to a precision of 0.308707 of a millimeter or 0.012154 of an inch.