Great circles path not curved (in R)? Conventions and convertions problems?

Question

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).

Answer 1

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)