In geography or spatial analyses, it is common that lengths are expressed in degree (minutes, seconds) instead of meter or km as we are more used to. Because of the spheroidal shape of the earth, the length in km of one degree of latitude or longitude depends on the latitude of the location. Formulas to do the conversion are available here that we use below.
The length of 1 degree of latitude in km, in function of the latitude x in
decimal degree is given by the following function:
latitude_length <- function(x) {
x <- pi * x / 180
(111132.92 - 559.82 * cos(2 * x) + 1.175 * cos(4 * x) - .0023 * cos(6 * x)) / 1000
}let’s generate a vector of latitude values between the equator and the north pole:
latitudes <- seq(0, 90, le = 512)We can verify that the length of 1 degree of latitude varies very little between the equator and the north pole:
range(latitude_length(latitudes))## [1] 110.5743 111.6939this is because the shape of the earth is very close to be spherical. Now, let’s
look at the length in km of 1 degree of longitude as a function of the latitude
x in decimal degree, which is given by the following function:
longitude_length <- function(x) {
x <- pi * x / 180
(111412.84 * cos(x) - 93.5 * cos(3 * x) + .118 * cos(5 * x)) / 1000
}let’s use this function to plot the length of 1 degree of longitude as a function of the latitude:
plot(longitude_length(latitudes), latitudes, type = "l",
xlab = "length of 1° of longitude (km)",
ylab = "latitude (°)",
xlim = c(0, 120),
ylim = c(0, 100))
We can thus verify that a cell close to the equator is almost of square shape and becomes more and more triangular as we get closer to the north pole. To finish, let’s have a look at the variation of the length of 1 degree of longitude in the intertropical zone:
tropics <- seq(0, 23.43682, le = 512)
range(longitude_length(tropics))## [1] 102.1896 111.3195And in Vietnam:
vietnam <- seq(8.563512, 23.392651, le = 512)
range(longitude_length(vietnam))## [1] 102.2236 110.0866