The Tissot Indicatrix

The Tissot Indicatrix is used to characterize local distortions within map projections.

I have derived the code in this package (with permission) from Bill Huber’s wonderful online answer here:

http://gis.stackexchange.com/questions/31651/an-example-tissot-ellipse-for-an-equirectangular-projection

Also see

https://gis.stackexchange.com/questions/5068/how-to-create-an-accurate-tissot-indicatrix

Installation

Can be installed with

remotes::install_github("hypertidy/tissot")

Minimal example

library(tissot)
# NAD 27 in
# World Robinson projection out
r <- tissot(130, 54,
           proj.in= "EPSG:4267",  
           proj.out= "ESRI:54030")
i0 <- indicatrix(r, scale=1e2, n=71)
plot(i0)

plot(tissot:::.prj(earthcircle::earthcircle(130, 54, scale = 5e2), tissot_get_proj()))
#> assuming WGS84 for unprojected angular coordinates
tissot_abline(130, 54)
#> assuming WGS84 for unprojected angular coordinates

tissot_map(add = FALSE, xlim = c(8.5e6, 1.3e7), ylim = c(4e6, 7e6))
i1 <- indicatrix(r, scale=1e6, n=71)
plot(i1, add = T)
tissot_abline(130, 54)
#> assuming WGS84 for unprojected angular coordinates
lines(tissot:::.prj(earthcircle::earthcircle(130, 54, scale = 2e6), tissot_get_proj()) %*% (diag(2) * 2))
#> assuming WGS84 for unprojected angular coordinates

Since an original port of whuber’s code we have now made it much easier to create many indicatrixes and plot them in one step. Or we can still just grab one and plot it on its own. Note that the scale is quite different in these plots.

x <- seq(-172.5, 172.5, by = 15)
y <- seq(-82.5, 82.5, by = 15)
xy <- expand.grid(x, y)
r <- tissot(xy,
            proj.in= "OGC:CRS84",
            proj.out= "+proj=robin")

j <- which.min(abs(135 - r$lon) + abs(54 - r$lat))
i <- indicatrix0(r[j, ], scale= 1e4, n=71)
plot(i, add = FALSE)

ii <- indicatrix(r, scale=4e5, n=71)
tissot_map(add = FALSE)
plot(ii, add = TRUE)
tissot_abline(xy[j, 1], xy[j, 2])
#> assuming WGS84 for unprojected angular coordinates

Mollweide.

m <- tissot(xy,
            proj.in= "OGC:CRS84",
            proj.out= "+proj=moll")


plot(indicatrix(m, scale=4e5, n=71), add = FALSE)
tissot_map()

Eckhert III

e <- tissot(xy,
            proj.in= "OGC:CRS84",
            proj.out= "+proj=eck3")


plot(indicatrix(e, scale=4e5, n=71), add = FALSE)

Equidistant

aeqd <- tissot(xy,
            proj.in= "OGC:CRS84",
            proj.out= "+proj=aeqd")


plot(indicatrix(aeqd, scale=4e5, n=71), add = FALSE)

Cassini-Soldner (spherical because ellipsoidal seems broken)

xx <- tissot(xy,
            proj.in= "OGC:CRS84",
            proj.out= "+proj=cass +R=6378137")


plot(indicatrix(xx, scale=4e5, n=71), add = FALSE)
points(tissot_map(col = "transparent"), pch = ".")

Sinusoidal

s <- tissot(xy,
            proj.in= "OGC:CRS84",
            proj.out= "+proj=sinu")


plot(indicatrix(s, scale=3e5, n=71), add = FALSE)
tissot_map()

Polar example

p <- tissot(xy[xy[,2] < -30, ],
            proj.in= "OGC:CRS84",
            proj.out= "+proj=stere +lon_0=147 +lat_ts-71 +lat_0=-90 +datum=WGS84")

plot(indicatrix(p, scale = 3e5))
tissot_map()
tissot_abline(147, -42)
#> assuming WGS84 for unprojected angular coordinates

laea <- tissot(xy[xy[,2] < 20, ],
            proj.in= "OGC:CRS84",
            proj.out= "+proj=laea +lon_0=147 +lat_0=-90 +datum=WGS84")

plot(indicatrix(laea, scale = 3e5))

Oblique Mercator

You get the idea … many projections need extra attention for real data.

mp0 <- do.call(cbind, maps::map(plot = FALSE)[1:2])
omerc <- "+proj=omerc +lonc=147 +gamma=9 +alpha=9 +lat_0=-80 +ellps=WGS84"
mp <- tissot:::.prj(mp0, omerc, proj.in = "OGC:CRS84")
o <- tissot(xy,
            proj.in= "OGC:CRS84",
            proj.out= omerc)

plot(indicatrix(o, scale = 3e5))
lines(mp)

VicGrid

vgrid <- "+proj=lcc +lat_1=-36 +lat_2=-38 +lat_0=-37 +lon_0=145 +x_0=2500000 +y_0=2500000 +ellps=GRS80 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs"
mp <- tissot:::.prj(mp0, vgrid, proj.in = "OGC:CRS84")
v <- tissot(as.matrix(expand.grid(seq(120, 165, by =5 ), 
                                          seq(-45, -35, by = 5))),
            proj.in= "OGC:CRS84",
            proj.out= vgrid)

plot(indicatrix(v, scale = 2e5))
lines(mp)

UTM Zone 54 (Hobart)

utm <- "+proj=utm +zone=54 +south"
mp <- tissot:::.prj(mp0, utm, proj.in = "OGC:CRS84")
u <- tissot(as.matrix(expand.grid(seq(108, 162, by =6 ), 
                                          seq(-65, 55, by = 15))),
            proj.in= "OGC:CRS84",
            proj.out= utm)

plot(indicatrix(u, scale = 2e5))
lines(mp)

Code of Conduct

Please note that the tissot project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.