# Find position B from position A and delta

Source:`R/n_EA_E_and_p_AB_E2n_EB_E.R`

`n_EA_E_and_p_AB_E2n_EB_E.Rd`

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`

).

## Arguments

- n_EA_E
n-vector of position A, decomposed in E (3x1 vector) (no unit)

- p_AB_E
Position vector from A to B, decomposed in E (3x1 vector) (m)

- z_EA
Depth of system A, relative to the ellipsoid (z_EA = -height) (m, default 0)

- a
Semi-major axis of the Earth ellipsoid (m, default [WGS-84] 6378137)

- f
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.

## References

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

## 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
#> [1] 0.2667916 0.5343565 0.8020507
#>
#> $z_EB
#> [1] -406.0072
#>
```