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################################
#### Lambert's equal area projection
#### Tsagris Michail 3/2014
#### mtsagris@yahoo.gr
#### References: Kent John T. (1982, JRSS Series C)
#### The Fisher-Bingham distribution on the sphere.
################################
lambert <- function(y) {
## y contains the data in degrees, latitude and longitude
u <- Directional::euclid(y) ## transform the data into euclidean coordinates
m <- Rfast::colmeans(u)
m <- m / sqrt(sum( m^2) ) ## the mean direction
b <- c(0, 0, 1) ## the north pole
H <- rotation(m, b) ## the rotation matrix
u1 <- tcrossprod(u, H) ## rotating the data so that their mean
## direction is the north pole
u2 <- Directional::euclid.inv(u1) ## bring the data into degrees again
u2 <- pi * u2 / 180 ## from degrees to radians
theta <- u2[, 1]
phi <- u2[, 2]
rho <- 2 * sin(theta / 2) ## radius of the disk is sqrt(2)
z1 <- rho * cos(phi) ## x coordinate
z2 <- rho * sin(phi) ## y coordinate
cbind(z1, z2) ## the Lambert equal area projected data on the disk
}
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