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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).

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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

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  • 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? Commented Mar 9, 2016 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. Commented Mar 9, 2016 at 14:38
  • The great circle line cuts the sphere in half - nicely illustrated by Wikipedia. Commented Mar 9, 2016 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)? Commented Mar 9, 2016 at 14:46
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    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. Commented Mar 9, 2016 at 15:00

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