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)



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)


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)


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


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 #>