## ----setup, include = FALSE--------------------------------------------------- knitr::opts_chunk$set( collapse = TRUE, comment = "#>", out.width = "220px", fig.align = 'center' ) library(png) library(nvctr) ## ----example-01-fig,fig.cap='A and B to delta.',fig.pos='hbt',fig.width=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex1img.png") ## ----example-01-init---------------------------------------------------------- # Position A: lat_EA <- rad(1) lon_EA <- rad(2) z_EA <- 3 # Position B: lat_EB <- rad(4) lon_EB <- rad(5) z_EB <- 6 ## ----example-01-step01-------------------------------------------------------- (n_EA_E <- lat_lon2n_E(lat_EA, lon_EA)) (n_EB_E <- lat_lon2n_E(lat_EB, lon_EB)) ## ----example-01-step02-------------------------------------------------------- (p_AB_E <- n_EA_E_and_n_EB_E2p_AB_E(n_EA_E, n_EB_E, z_EA, z_EB)) ## ----example-01-step03-------------------------------------------------------- (R_EN <- n_E2R_EN(n_EA_E)) ## ----example-01-step04-------------------------------------------------------- # (Note the transpose of R_EN: The "closest-rule" says that when # decomposing, the frame in the subscript of the rotation matrix that is # closest to the vector, should equal the frame where the vector is # decomposed. Thus the calculation R_NE*p_AB_E is correct, since the vector # is decomposed in E, and E is closest to the vector. In the above example # we only had R_EN, and thus we must transpose it: base::t(R_EN) = R_NE) (p_AB_N <- base::t(R_EN) %*% p_AB_E %>% as.vector()) ## ----line-of-sight_distance--------------------------------------------------- (los_distance <- norm(p_AB_N, type = "2")) ## ----elevation---------------------------------------------------------------- (elevation <- atan2(-p_AB_N[3], p_AB_N[2]) %>% deg()) ## ----example-01-step05-------------------------------------------------------- (azimuth <- atan2(p_AB_N[2], p_AB_N[1]) %>% # positive angle about down-axis deg()) ## ----example-02-fig,fig.cap='B and delta to C.',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex2img.png") ## ----example-02-init---------------------------------------------------------- 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))) # the three angles are yaw, pitch, and roll # A custom reference ellipsoid is given (replacing WGS-84): # (WGS-72) a <- 6378135 f <- 1 / 298.26 ## ----example-02-step01-------------------------------------------------------- (R_EN <- n_E2R_EN(n_EB_E)) ## ----example-02-step02-------------------------------------------------------- (R_EB <- R_EN %*% R_NB) # Note: closest frames cancel ## ----example-02-step03-------------------------------------------------------- (p_BC_E <- R_EB %*% p_BC_B) # no transpose of R_EB, since the vector is in B) ## ----example-02-step04-------------------------------------------------------- l <- n_EA_E_and_p_AB_E2n_EB_E(n_EB_E, p_BC_E, z_EB, a, f) (n_EB_E <- l[['n_EB_E']]) (z_EB <- l[['z_EB']]) ## ----example-02-final--------------------------------------------------------- lat_lon_EB <- n_E2lat_lon(n_EB_E) (latitude <- lat_lon_EB[1]) (longitude <- lat_lon_EB[2]) # height (= - depth) (height <- -z_EB) ## ----example-03-fig,fig.cap='ECEF-vector to geodetic latitude.',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex3img.png") ## ----example-03-init---------------------------------------------------------- (p_EB_E <- 6371e3 * c(0.9, -1, 1.1)) # m ## ----example-03-step01-------------------------------------------------------- l <- p_EB_E2n_EB_E(p_EB_E) (n_EB_E <- l[['n_EB_E']]) (z_EB <- l[['z_EB']]) ## ----example-03-final--------------------------------------------------------- lat_lon_EB <- n_E2lat_lon(n_EB_E) (latEB <- lat_lon_EB[1]) (lonEB <- lat_lon_EB[2]) # height (= - depth) (hEB <- -z_EB) ## ----example-04-fig,fig.cap='Geodetic latitude to ECEF-vector.',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex4img.png") ## ----example-04-init---------------------------------------------------------- lat_EB <- rad(1) lon_EB <- rad(2) h_EB <- 3 ## ----example-04-step01-------------------------------------------------------- (n_EB_E <- lat_lon2n_E(lat_EB, lon_EB)) ## ----example-04-step02-------------------------------------------------------- (p_EB_E <- n_EB_E2p_EB_E(n_EB_E, -h_EB)) ## ----example-05-fig,fig.cap='Surface distance.',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex5img.png") ## ----example-05-init---------------------------------------------------------- n_EA_E <- lat_lon2n_E(rad(88), rad(0)); n_EB_E <- lat_lon2n_E(rad(89), rad(-170)) r_Earth <- 6371e3 ## ----example-05-spherical----------------------------------------------------- (s_AB <- (atan2(base::norm(pracma::cross(n_EA_E, n_EB_E), type = "2"), pracma::dot(n_EA_E, n_EB_E)) * r_Earth)) ## ----example-05-step02-------------------------------------------------------- (d_AB <- base::norm(n_EB_E - n_EA_E, type = "2") * r_Earth) ## ----example-05-geodesic------------------------------------------------------ geosphere::distGeo(c(0, 88), c(-170, 89)) ## ----example-06-fig,fig.cap='Interpolated position.',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex6img.png") ## ----example-06-init---------------------------------------------------------- n_EB_E_t0 <- lat_lon2n_E(rad(89.9), rad(-150)) n_EB_E_t1 <- lat_lon2n_E(rad(89.9), rad(150)) # The times are given as: t0 <- 10 t1 <- 20 ti <- 16 # time of interpolation ## ----exemple-06-interpolation------------------------------------------------- t_frac <- (ti - t0) / (t1 - t0) (n_EB_E_ti <- unit(n_EB_E_t0 + t_frac * (n_EB_E_t1 - n_EB_E_t0) )) ## ----example-06-conversion---------------------------------------------------- (l <- n_E2lat_lon(n_EB_E_ti) %>% deg()) (latitude <- l[1]) (longitude <- l[2]) ## ----example-07-fig,fig.cap='Mean position (center/midpoint).',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex7img.png") ## ----example-07-init---------------------------------------------------------- n_EA_E <- lat_lon2n_E(rad(90), rad(0)) n_EB_E <- lat_lon2n_E(rad(60), rad(10)) n_EC_E <- lat_lon2n_E(rad(50), rad(-20)) ## ----example-07-geographical-mean--------------------------------------------- (n_EM_E <- unit(n_EA_E + n_EB_E + n_EC_E)) ## ----example-07-conversion---------------------------------------------------- (l <- n_E2lat_lon(n_EM_E) %>% deg()) (latitude <- l[1]) (longitude <- l[2]) ## ----example-08-fig,fig.cap='A and azimuth/distance to B.',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex8img.png") ## ----example-08-init---------------------------------------------------------- n_EA_E <- lat_lon2n_E(rad(80),rad(-90)) azimuth <- rad(200) s_AB <- 1000 # distance (m) r_Earth <- 6371e3 # mean Earth radius (m) ## ----------------------------------------------------------------------------- k_east_E <- unit(pracma::cross(base::t(R_Ee()) %*% c(1, 0, 0) %>% as.vector(), n_EA_E)) k_north_E <- pracma::cross(n_EA_E, k_east_E) ## ----------------------------------------------------------------------------- d_E <- k_north_E * cos(azimuth) + k_east_E * sin(azimuth) ## ----------------------------------------------------------------------------- n_EB_E <- n_EA_E * cos(s_AB / r_Earth) + d_E * sin(s_AB / r_Earth) ## ----example-08-conversion---------------------------------------------------- (l <- n_E2lat_lon(n_EB_E) %>% deg()) (latitude <- l[1]) (longitude <- l[2]) ## ----example-09-fig,fig.cap='Intersection of two paths.',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex9img.png") ## ----example-09-init---------------------------------------------------------- n_EA1_E <- lat_lon2n_E(rad(50), rad(180)) n_EA2_E <- lat_lon2n_E(rad(90), rad(180)) n_EB1_E <- lat_lon2n_E(rad(60), rad(160)) n_EB2_E <- lat_lon2n_E(rad(80), rad(-140)) # These are from the python version (results are the same ;-) # n_EA1_E <- lat_lon2n_E(rad(10), rad(20)) # n_EA2_E <- lat_lon2n_E(rad(30), rad(40)) # n_EB1_E <- lat_lon2n_E(rad(50), rad(60)) # n_EB2_E <- lat_lon2n_E(rad(70), rad(80)) ## ----------------------------------------------------------------------------- n_EC_E_tmp <- unit(pracma::cross( pracma::cross(n_EA1_E, n_EA2_E), pracma::cross(n_EB1_E, n_EB2_E))) ## ----------------------------------------------------------------------------- n_EC_E <- sign(pracma::dot(n_EC_E_tmp, n_EA1_E)) * n_EC_E_tmp ## ----example-09-conversion---------------------------------------------------- (l <- n_E2lat_lon(n_EC_E) %>% deg()) (latitude <- l[1]) (longitude <- l[2]) ## ----example-10-fig,fig.cap='Cross track distance (cross track error).',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex10img.png") ## ----------------------------------------------------------------------------- n_EA1_E <- lat_lon2n_E(rad(0), rad(0)) n_EA2_E <- lat_lon2n_E(rad(10),rad(0)) n_EB_E <- lat_lon2n_E(rad(1), rad(0.1)) r_Earth <- 6371e3 # mean Earth radius (m) ## ----------------------------------------------------------------------------- c_E <- unit(pracma::cross(n_EA1_E, n_EA2_E)) ## ----solution-10-fig, fig.cap='Vectors for cross track distance calculation.', fig.pos='hbt', fig.height=4, echo=FALSE, warning=FALSE, message=FALSE---- knitr::include_graphics("solution10img.png") ## ----------------------------------------------------------------------------- (s_xt <- (acos(pracma::dot(c_E, n_EB_E)) - pi / 2) * r_Earth) ## ----------------------------------------------------------------------------- (d_xt <- -pracma::dot(c_E, n_EB_E) * r_Earth) ## ----example-11-fig,fig.cap='Cross track intersection.',fig.pos='hbt',fig.height=4,echo=FALSE,warning=FALSE,message=FALSE---- knitr::include_graphics("ex11img.png") ## ----------------------------------------------------------------------------- n_EA1_E <- lat_lon2n_E(rad(0), rad(3)) n_EA2_E <- lat_lon2n_E(rad(0),rad(10)) n_EB_E <- lat_lon2n_E(rad(-1), rad(-1)) ## ----------------------------------------------------------------------------- n_EN_E <- unit(pracma::cross(n_EA1_E, n_EA2_E)) ## ----------------------------------------------------------------------------- n_EC_E_tmp <- unit( pracma::cross( n_EN_E, pracma::cross(n_EN_E, n_EB_E) ) ) ## ----------------------------------------------------------------------------- n_EC_E <- sign(pracma::dot(n_EC_E_tmp, n_EB_E)) * n_EC_E_tmp ## ----example-11-conversion---------------------------------------------------- (l <- n_E2lat_lon(n_EC_E) %>% deg()) (latitude <- l[1]) (longitude <- l[2])