4

I am given the great circle equations of two paths. The first one is

Longitude = 168 degree.

and the second one is:

tan(Lattitude)=.11 cos(Longitude) + .64 sin(Longitude)

However, I am not sure I understand what these represent.

For example, when I then try to depict these two great circles on a map (For the code below, I use as model the example found here):

library(CircStats)
library(geosphere)
library(maps)

great_circle_1 <- function( lattitude ){
    return( 168 ) 
}
great_circle_2 <- function( longitude ){
    return( atan( ( 0.11 * cos(longitude * pi / 180) + 0.64 * sin( longitude * pi / 180 )) ) * 180 / pi) 
}
map("world", col="#f2f2f2", fill=TRUE, bg="white", lwd=0.05)
lat_c1_1 <- -50
lon_c1_1 <- great_circle_1( lat_c1_1 )
lat_c1_2 <- 90
lon_c1_2 <- great_circle_1( lat_c1_2 )
inter_c1 <- gcIntermediate(c(lon_c1_1, lat_c1_1), c(lon_c1_2, lat_c1_2), n=50, addStartEnd=TRUE)
lines(inter_c1)
lon_c2_1 <- 150
lat_c2_1 <- great_circle_2(lon_c2_1)
lon_c2_2 <- -50
lat_c2_2 <- great_circle_2(lon_c2_2)
inter_c2 <- gcIntermediate(c(lon_c2_1, lat_c2_1), c(lon_c2_2, lat_c2_2), n=50, addStartEnd=TRUE)
lines(inter_c2)

My problem is that contrary to the examples on the website (and what my intuition would dictate), the two grand circles appear as a flat lines (rather than curved ones). I wanted to make sure that this is not due to a comprehension error on my behalf over how to understand the great circle equation as formulated.

I am assuming spherical earth.

I gather that there are different conventions as to the coordinate systems. Since this is a question about conversions, for information, I use the longitude and latitudes as given on google map, for example the original Waterloo has coordinates (50.7167, 4.3833).

4

In that particular projection - Platte Carre - i.e. just plotting latitude and longitude as if they are cartesian coordinates, your great circle line is a straight line.

> gcIntermediate(c(168,-50),c(168,90),n=5,addStartEnd=TRUE)
     lon        lat
[1,] 168 -50.000000
[2,] 168 -26.666667
[3,] 168  -3.333333
[4,] 168  20.000000
[5,] 168  43.333333
[6,] 168  66.666667
[7,] 168  90.000000

Here you are following a particular meridian, it is just not terminating at the north and south pole.

If you go east-west then it is not a straight line (in that projection) - with the exception of following the equator exactly:

> gcIntermediate(c(5,52), c(20,52), n=6, addStartEnd=TRUE)
           lon      lat
[1,]  5.000000 52.00000
[2,]  7.133499 52.11666
[3,]  9.276333 52.19464
[4,] 11.424814 52.23369
[5,] 13.575186 52.23369
[6,] 15.723667 52.19464
[7,] 17.866501 52.11666
[8,] 20.000000 52.00000

So there is nothing wrong off-hand with your examples. Your second example inter_c2 looks like a straight line when zoomed so far out, but it is not:

Line2 <- gcIntermediate(c(150,0.2210643),c(-50,-0.3972554),n=50, addStartEnd=TRUE)
plot(Line2)

enter image description here

  • so, if I understand correctly, the result I see is just a combination of the projection I used and the fact that the first equation describes the path of an object (say a bird) that flies exactly north south and the second equation describes the path of an object (another bird) that flies exactly east-west? – user189035 Mar 9 '16 at 14:37
  • Yes, except the second bird does not have to fly exactly east-west for the line to be curved, and if the east-west bird follows the line of the equator it will be straight as well. – Andy W Mar 9 '16 at 14:38
  • The great circle line cuts the sphere in half - nicely illustrated by Wikipedia. – Andy W Mar 9 '16 at 14:39
  • Just to be sure: the second path in my example definitely appears straight, so that means that the second equation is describing a path parallel to the equator (on that projection)? – user189035 Mar 9 '16 at 14:46
  • 1
    It appears straight on the map because it is so zoomed out. If you shift it down one degree it becomes more clear that it is not a straight line. Line2 <- gcIntermediate(c(150,0.2210643-1),c(-50,-0.3972554-1),n=5, addStartEnd=TRUE);points(Line2,col='blue',pch=".",cex=10) If you simply do plot(inter_c2) you can clearly see it is not a straight line. – Andy W Mar 9 '16 at 15:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.