Given the n-vector for position A (n_EA_E) and the position-vector from position A to position B (p_AB_E), the output is the n-vector of position B (n_EB_E) and depth of B (z_EB).

n_EA_E_and_p_AB_E2n_EB_E(
n_EA_E,
p_AB_E,
z_EA = 0,
a = 6378137,
f = 1/298.257223563
)

## Arguments

n_EA_E n-vector of position A, decomposed in E (3x1 vector) (no unit) Position vector from A to B, decomposed in E (3x1 vector) (m) Depth of system A, relative to the ellipsoid (z_EA = -height) (m, default 0) Semi-major axis of the Earth ellipsoid (m, default [WGS-84] 6378137) Flattening of the Earth ellipsoid (no unit, default [WGS-84] 1/298.257223563)

## Value

a list with n-vector of position B, decomposed in E (3x1 vector) (no unit) and the depth of system B, relative to the ellipsoid (z_EB = -height)

## Details

The calculation is exact, taking the ellipticity of the Earth into account.

It is also nonsingular as both n-vector and p-vector are nonsingular (except for the center of the Earth). The default ellipsoid model used is WGS-84, but other ellipsoids (or spheres) might be specified.

Kenneth Gade A Nonsingular Horizontal Position Representation. The Journal of Navigation, Volume 63, Issue 03, pp 395-417, July 2010.

## See also

n_EA_E_and_n_EB_E2p_AB_E, p_EB_E2n_EB_E and n_EB_E2p_EB_E

## Examples

p_BC_B <- c(3000, 2000, 100)

# Position and orientation of B is given:
n_EB_E <- unit(c(1,2,3))  # unit to get unit length of vector
z_EB <- -400
R_NB  <- zyx2R(rad(10), rad(20), rad(30)) # yaw, pitch, and roll
R_EN <- n_E2R_EN(n_EB_E)
R_EB <- R_EN %*% R_NB

# Decompose the delta vector in E:
p_BC_E <- (R_EB %*% p_BC_B) %>% as.vector() # no transpose of R_EB, since the vector is in B

# Find the position of C, using the functions that goes from one
# position and a delta, to a new position:
(n_EB_E <- n_EA_E_and_p_AB_E2n_EB_E(n_EB_E, p_BC_E, z_EB))
#> $n_EB_E #>  0.2667916 0.5343565 0.8020507 #> #>$z_EB
#>  -406.0072
#>