How it works
The earth is close to an ellipsoid of revolution: an ellipse turned about its short
axis, the polar axis. Its equatorial radius is a and its polar radius b, and its
flattening f = (a - b) / a is about 1/298 (WGS84, GRS80). That is 21 km in 6378, too little
to see: at 1x the drawing is the true shape, and it sits on the sphere of radius a. The slider
multiplies f, keeping a, so the drawing can show what the numbers say. Every number is given for
the true shape and for the drawn one.
On an ellipsoid one point has three latitudes, and they differ because the surface is not
round:
- Geodetic latitude (yellow) is the angle the normal to the surface makes with
the equator. It is the latitude of a map, a GPS and PROJ's longlat, and the one a plumb line
and the stars give (up to the geoid). The normal misses the centre: it crosses the polar axis on
the other side of the equator, and its length from the surface to the axis is N, the
radius of curvature across the meridian (in the prime vertical). A parallel is a circle of
radius N cos of the latitude.
- Geocentric latitude (cyan) is the angle at the centre. It is the latitude of
Earth-centred x, y, z (PROJ's
+proj=cart, ECEF) and of a sphere. tan of it is
(1 - e^2) tan of the geodetic latitude, so the two agree at the equator and the poles and part
most near 45 degrees, by 11.5 minutes of arc on WGS84 (an angle that spans about 21 km of meridian).
- Parametric or reduced latitude (green) is the angle at the centre of the point on the
circle of radius a straight out from the point, parallel to the equator: the ellipse is that
circle squashed by b/a. tan of it is (1 - f) tan of the geodetic latitude, and it lies between the
other two. Geodesics on the ellipsoid are worked in it (Bessel, Vincenty, Karney).
M (pink) is the radius of the circle that best fits the meridian at the point: it is
smallest at the equator, where the ellipse is most curved, and largest at the poles, where it is
flattest. That is why a degree of latitude is about 1.1 km longer at the pole than at the equator,
the measurement (in Lapland and Peru, in the 1730s) that settled that the earth is flattened at the
poles rather than drawn out.
Formulas as in Snyder (1987), in src/core/ellipsoid.js, checked against PROJ in
test/check-ellipsoid.mjs: each ellipsoid's a and b from PROJ's ellps table, x, y, z
against +proj=cart, and meridian lengths against PROJ's geodesics, on the true shapes
and on WGS84 with its flattening times 100.
Hash keys: ell (a PROJ ellps name), x (the exaggeration),
lat, lon, sph, par, osc,
coast, cam.
- Snyder, J. P. (1987) Map Projections: A Working Manual, USGS Professional Paper 1395
(the auxiliary latitudes, radii of curvature). Ellipsoid parameters from
PROJ (
proj -le). Karney, C. F. F. (2013) Algorithms
for geodesics, J. Geodesy 87, 43, as PROJ has them.
- Coastlines from Natural Earth.