3 ms·

Mercator maps are conformal, which means that over small distances, shapes are preserved, though you do have to correctly adjust the distance scale. So how much

Mercator maps are conformal, which means that over small distances, shapes are preserved, though you do have to correctly adjust the distance scale. So how much does that matter over 76 km, would the bend in the line even be visible?

ryandrake 3y ago
Yea, admittedly at these distances (<100km) the difference will be very small unless at extreme latitudes. Looks like the points are at approximately N57°18.280', W3°55.948' -> N56°54.152', W4°56.664' if anyone wants to do the math. My guess is the rhumb line and GC route differ by no more than a few meters. I was hoping the article would go more into how they searched the data and found the points. An exhaustive search of all points in the U.K. would obviously be computationally prohibitive. Maybe create millions of polygons where the edges are roads, and sort them by area to reduce the search space? From the article's text it looks like he "eyeballed" the map to find empty areas to focus on, which is unsatisfying :)
- bluenose69 3y ago
  I get a bit over 200 metres from the following R code. (Edit: I forgot how to format code for HM, sorry.) library(oce) lat <- c(57+18.280/60, 56+54.152/60) lon <- -c(3+55.948/60, 4+56.664/60) m <- lm(lat ~ lon) # Find points along a great-circle route gc <- oce::geodGc(lon, lat, 0.001) # 'scale' is distance associated with 1 degree latitude shift scale <- geodDist(mean(lon), mean(lat), mean(lon), mean(lat)+1) # 'e' is latitude shift from great-circle route to 'straight' line route e <- scale\*sapply(seq_along(gc$longitude), function(i) gc$latitude[i] - predict(m, data.frame(lon=gc$longitude[i]))) plot(gc$longitude, e, type="l", xlab="Longitude [degE]", ylab="Distance error [km]", col=2) max(e)
  - codeflo 3y ago
    I’m not familiar with R, but the lm(…) line looks like it’s linearly interpolating the latitude and longitude. However, that’s not what straight lines in a Mercator projection are. You might be confusing it with the cylindrical projection. The Mercator projection stretches things away from the equator vertically to compensate the horizontal stretching, in order to preserve local shapes. This makes it a bit better for these kinds of local measurements than one might think at first glance.
    - bluenose69 3y ago
      Yes, `lm()` linearly interpolates. Thanks for pointing out my error.
      - codeflo 3y ago
        To be fair, I’m not sure this will actually make a significant difference. :) Even 200m wouldn’t be a huge error.
        bluenose69 3y ago
        I tried a computation that I think may be correct. It gives 319 metres of northing separation between rhumb and creat-circle paths. In case it's of any interest, the R code is below. # Northing distance between rhumb and great-circle paths. library(oce) lat <- c(57+18.280/60, 56+54.152/60) lon <- -c(3+55.948/60, 4+56.664/60) # Great-circle path, first in longitude-latitude space, and then # (for Mercator projection) in easting-northing space. gc <- oce::geodGc(lon, lat, 0.01) # same results for 3rd arg any value < 0.01 gcXY <- lonlat2map(gc$longitude, gc$latitude, projection="+proj=merc") # Create function, f(easting), that computes northing value on # a rhumb line. We need this to make the two lines share easting # values. p <- oce::lonlat2map(lon, lat, "+proj=merc") m <- lm(y ~ x, data=p) X <- seq(p$x[1], p$x[2], length.out=length(gc$longitude)) Y <- predict(m, data.frame(x=X)) f <- approxfun(X, Y) # interpolating function # Compute the northing error. northingError <- f(gcXY$x) - gcXY$y maxNorthingError <- round(max(abs(northingError)), 1) message("maxNorthingError=", maxNorthingError, " [m]") plot(gcXY$x, northingError, type="l", xlab="Easting [m]", ylab="Northing error [m]") mtext(paste("max northing error ", maxNorthingError, " [m]"))

ryandrake 3y ago

Yea, admittedly at these distances (<100km) the difference will be very small unless at extreme latitudes. Looks like the points are at approximately N57°18.280', W3°55.948' -> N56°54.152', W4°56.664' if anyone wants to do the math. My guess is the rhumb line and GC route differ by no more than a few meters. I was hoping the article would go more into how they searched the data and found the points. An exhaustive search of all points in the U.K. would obviously be computationally prohibitive. Maybe create millions of polygons where the edges are roads, and sort them by area to reduce the search space? From the article's text it looks like he "eyeballed" the map to find empty areas to focus on, which is unsatisfying :)

bluenose69 3y ago
I get a bit over 200 metres from the following R code. (Edit: I forgot how to format code for HM, sorry.) library(oce) lat <- c(57+18.280/60, 56+54.152/60) lon <- -c(3+55.948/60, 4+56.664/60) m <- lm(lat ~ lon) # Find points along a great-circle route gc <- oce::geodGc(lon, lat, 0.001) # 'scale' is distance associated with 1 degree latitude shift scale <- geodDist(mean(lon), mean(lat), mean(lon), mean(lat)+1) # 'e' is latitude shift from great-circle route to 'straight' line route e <- scale\*sapply(seq_along(gc$longitude), function(i) gc$latitude[i] - predict(m, data.frame(lon=gc$longitude[i]))) plot(gc$longitude, e, type="l", xlab="Longitude [degE]", ylab="Distance error [km]", col=2) max(e)
- codeflo 3y ago
  I’m not familiar with R, but the lm(…) line looks like it’s linearly interpolating the latitude and longitude. However, that’s not what straight lines in a Mercator projection are. You might be confusing it with the cylindrical projection. The Mercator projection stretches things away from the equator vertically to compensate the horizontal stretching, in order to preserve local shapes. This makes it a bit better for these kinds of local measurements than one might think at first glance.
  - bluenose69 3y ago
    Yes, `lm()` linearly interpolates. Thanks for pointing out my error.
    - codeflo 3y ago
      To be fair, I’m not sure this will actually make a significant difference. :) Even 200m wouldn’t be a huge error.
      - bluenose69 3y ago
        I tried a computation that I think may be correct. It gives 319 metres of northing separation between rhumb and creat-circle paths. In case it's of any interest, the R code is below. # Northing distance between rhumb and great-circle paths. library(oce) lat <- c(57+18.280/60, 56+54.152/60) lon <- -c(3+55.948/60, 4+56.664/60) # Great-circle path, first in longitude-latitude space, and then # (for Mercator projection) in easting-northing space. gc <- oce::geodGc(lon, lat, 0.01) # same results for 3rd arg any value < 0.01 gcXY <- lonlat2map(gc$longitude, gc$latitude, projection="+proj=merc") # Create function, f(easting), that computes northing value on # a rhumb line. We need this to make the two lines share easting # values. p <- oce::lonlat2map(lon, lat, "+proj=merc") m <- lm(y ~ x, data=p) X <- seq(p$x[1], p$x[2], length.out=length(gc$longitude)) Y <- predict(m, data.frame(x=X)) f <- approxfun(X, Y) # interpolating function # Compute the northing error. northingError <- f(gcXY$x) - gcXY$y maxNorthingError <- round(max(abs(northingError)), 1) message("maxNorthingError=", maxNorthingError, " [m]") plot(gcXY$x, northingError, type="l", xlab="Easting [m]", ylab="Northing error [m]") mtext(paste("max northing error ", maxNorthingError, " [m]"))