Plotting Pole of Rotation Grids
To visualize the theoretical trajectories of the direction of \(\sigma_{Hmax}\) (great circles, small
circles, and loxodomes), we need to transform the locations from the
geographical coordinate system into the PoR coordinate system.
The transformations are done through the function functions
geographical_to_PoR() and
PoR_to_geographical(). They are the base of the functions
eulerpole_smallcircles(),
eulerpole_greatcircles(), and
eulerpole_loxodromes() that allow to draw the theoretical
trajectories in geographical coordinates.
Small Circles
Function eulerpole_smallcircles(x, gridsize) returns
small circles as as simple feature(sf) by giving a
data.frame of the PoR coordinates in lat and lon
(x) and the number of small circles (n).
For example the small circles around the pole of the relative motion of the Indian plate relative to the Eurasian plate (transformed from the from the NUVEL1 model).
The returnclass option in
eulerpole_smallcircles() provides the output types
"sf" (for a simple feature) and "sp"
(Spatial* object) for the small circles.
To eventually plot the small circles with ggplot, I
recommend to extract a sf feature and plot the it with
geom_sf():
por.sm <- eulerpole_smallcircles(por)
data("plates") # load plate boundary data set
# world <- rnaturalearth::ne_countries(scale = "small", returnclass = "sf")
ggplot() +
# geom_sf(data = world, alpha = .5) +
geom_sf(
data = plates,
color = "red",
alpha = .5
) +
labs(title = "India relative to Eurasia", subtitle = "source: NUVEL1") +
geom_sf(
data = por.sm,
aes(lty = "small circles"),
color = "darkblue", fill = NA,
alpha = .5
) +
geom_point(
data = por,
aes(lon, lat),
shape = 21,
colour = "lightblue",
size = 2,
fill = "darkblue",
stroke = 1
) +
geom_point(
data = euler,
aes(lon + 180, -lat),
shape = 21,
colour = "lightblue",
size = 2,
fill = "darkblue",
stroke = 1
) +
coord_sf(default_crs = "WGS84", crs = sf::st_crs("ESRI:54030"))
Great Circles
Great circles are lines that cut the small circles at 90\(^{\circ}\) and the PoR. Function
eulerpole_greatcircles(x, n) returns great circles as
sf object by giving a data.frame of the Pole
of Rotation (PoR) coordinates in lat and lon (x) and the
number of great circles n, or the great circle angles
(360/d).
por.gm <- eulerpole_greatcircles(por)
ggplot() +
# geom_sf(data = world, alpha = .5) +
geom_sf(
data = plates,
color = "red",
alpha = .5
) +
labs(title = "India relative to Eurasia", subtitle = "source: NUVEL1") +
geom_sf(
data = por.sm,
aes(lty = "small circles"),
color = "darkblue",
alpha = .5
) +
geom_sf(
data = por.gm,
aes(lty = "great circles"),
color = "darkblue"
) +
geom_point(
data = por,
aes(lon, lat),
shape = 21,
colour = "lightblue",
size = 2,
fill = "darkblue",
stroke = 1
) +
geom_point(
data = por,
aes(lon + 180, -lat),
shape = 21,
colour = "lightblue",
size = 2,
fill = "darkblue",
stroke = 1
) +
coord_sf(default_crs = "WGS84", crs = sf::st_crs("ESRI:54030"))
Loxodromes
Loxodrome (also called Rhumb Line) is a curve cutting the small circles at a constant angle. Thus, small and great circles are 0\(^{\circ}\) and 90\(^{\circ}\) loxodromes, respectively.
Function eulerpole_loxodromes(x, n) returns loxodromes
as sf object by giving a data.frame of the PoR
coordinates in lat and lon (x) and the angle between the
loxodromes, the direction, and the sense.
por.ld <- eulerpole_loxodromes(x = por, angle = 45, n = 10, cw = TRUE)
ggplot() +
labs(title = "India relative to Eurasia", subtitle = "source: NUVEL1") +
# geom_sf(data = world, alpha = .5) +
geom_sf(
data = plates,
color = "red",
alpha = .5
) +
geom_sf(
data = por.sm,
aes(lty = "small circles"),
color = "darkblue",
alpha = .5
) +
geom_sf(
data = por.ld,
aes(lty = "clockwise loxodromes"),
color = "darkblue"
) +
geom_point(
data = por,
aes(lon, lat),
shape = 21,
colour = "lightblue",
size = 2,
fill = "darkblue",
stroke = 1
) +
geom_point(
data = por,
aes(lon + 180, -lat),
shape = 21,
colour = "lightblue",
size = 2,
fill = "darkblue",
stroke = 1
) +
coord_sf(default_crs = "WGS84", crs = sf::st_crs("ESRI:54030"))