[英]How to make Peirce projection algorithm allow for 360 degrees rotation?
I'm using this algorithm for Peirce world map projection in R. I'm able to do some fine maps, for instance using 28 as the value for the lambda_0
parameter in function toPeirceQuincuncial
, since this angle creates less land distortion and breaks no important islands (besides Antarctica, obviously). 我正在将此算法用于R中的Peirce世界地图投影。我能够做一些精细的地图,例如使用28作为函数toPeirceQuincuncial
lambda_0
参数的值,因为这个角度产生的土地失真较小并且没有重要岛屿(显然除了南极洲)。 The algorithm is used like this: 该算法使用如下:
# constants
pi<-acos(-1.0)
twopi<-2.0*pi
halfpi<-0.5*pi
degree<-pi / 180
halfSqrt2<-sqrt(2) / 2
quarterpi<-0.25 * pi
mquarterpi<--0.25 * pi
threequarterpi<-0.75 * pi
mthreequarterpi<--0.75 * pi
radian<-180/pi
sqrt2<-sqrt(2)
sqrt8<-2. * sqrt2
halfSqrt3<-sqrt(3) / 2
PeirceQuincuncialScale<-3.7081493546027438 ;# 2*K(1/2)
PeirceQuincuncialLimit<-1.8540746773013719 ;# K(1/2)
ellFaux<-function(cos_phi,sin_phi,k){
x<-cos_phi * cos_phi
y<-1.0 - k * k * sin_phi * sin_phi
z<-1.0
rf<-ellRF(x,y,z)
return(sin_phi * rf)
}
ellRF<-function(x,y,z){
if (x < 0.0 || y < 0.0 || z < 0.0) {
print("Negative argument to Carlson's ellRF")
print("ellRF negArgument")
}
delx<-1.0;
dely<-1.0;
delz<-1.0
while(abs(delx) > 0.0025 || abs(dely) > 0.0025 || abs(delz) > 0.0025) {
sx<-sqrt(x)
sy<-sqrt(y)
sz<-sqrt(z)
len<-sx * (sy + sz) + sy * sz
x<-0.25 * (x + len)
y<-0.25 * (y + len)
z<-0.25 * (z + len)
mean<-(x + y + z) / 3.0
delx<-(mean - x) / mean
dely<-(mean - y) / mean
delz<-(mean - z) / mean
}
e2<-delx * dely - delz * delz
e3<-delx * dely * delz
return((1.0 + (e2 / 24.0 - 0.1 - 3.0 * e3 / 44.0) * e2+ e3 / 14) / sqrt(mean))
}
toPeirceQuincuncial<-function(lambda,phi,lambda_0=20.0){
# Convert latitude and longitude to radians relative to the
# central meridian
lambda<-lambda - lambda_0 + 180
if (lambda < 0.0 || lambda > 360.0) {
lambda<-lambda - 360 * floor(lambda / 360)
}
lambda<-(lambda - 180) * degree
phi<-phi * degree
# Compute the auxiliary quantities 'm' and 'n'. Set 'm' to match
# the sign of 'lambda' and 'n' to be positive if |lambda| > pi/2
cos_phiosqrt2<-halfSqrt2 * cos(phi)
cos_lambda<-cos(lambda)
sin_lambda<-sin(lambda)
cos_a<-cos_phiosqrt2 * (sin_lambda + cos_lambda)
cos_b<-cos_phiosqrt2 * (sin_lambda - cos_lambda)
sin_a<-sqrt(1.0 - cos_a * cos_a)
sin_b<-sqrt(1.0 - cos_b * cos_b)
cos_a_cos_b<-cos_a * cos_b
sin_a_sin_b<-sin_a * sin_b
sin2_m<-1.0 + cos_a_cos_b - sin_a_sin_b
sin2_n<-1.0 - cos_a_cos_b - sin_a_sin_b
if (sin2_m < 0.0) {
sin2_m<-0.0
}
sin_m<-sqrt(sin2_m)
if (sin2_m > 1.0) {
sin2_m<-1.0
}
cos_m<-sqrt(1.0 - sin2_m)
if (sin_lambda < 0.0) {
sin_m<--sin_m
}
if (sin2_n < 0.0) {
sin2_n<-0.0
}
sin_n<-sqrt(sin2_n)
if (sin2_n > 1.0) {
sin2_n<-1.0
}
cos_n<-sqrt(1.0 - sin2_n)
if (cos_lambda > 0.0) {
sin_n<--sin_n
}
# Compute elliptic integrals to map the disc to the square
x<-ellFaux(cos_m,sin_m,halfSqrt2)
y<-ellFaux(cos_n,sin_n,halfSqrt2)
# Reflect the Southern Hemisphere outward
if(phi < 0) {
if (lambda < mthreequarterpi) {
y<-PeirceQuincuncialScale - y
} else if (lambda < mquarterpi) {
x<--PeirceQuincuncialScale - x
} else if (lambda < quarterpi) {
y<--PeirceQuincuncialScale - y
} else if (lambda < threequarterpi) {
x<-PeirceQuincuncialScale - x
} else {
y<-PeirceQuincuncialScale - y
}
}
# Rotate the square by 45 degrees to fit the screen better
X<-(x - y) * halfSqrt2
Y<-(x + y) * halfSqrt2
res<-list(X,Y)
return(res)
}
library(rgdal)
ang <- 28
p <- readOGR('~/R/shp','TM_WORLD_BORDERS-0.3') # read world shapefile
for (p1 in 1:length(p@polygons)) {
for (p2 in 1:length(p@polygons[[p1]]@Polygons)) {
for (p3 in 1:nrow(p@polygons[[p1]]@Polygons[[p2]]@coords)) {
pos <- toPeirceQuincuncial(p@polygons[[p1]]@Polygons[[p2]]@coords[p3,1],
p@polygons[[p1]]@Polygons[[p2]]@coords[p3,2],ang)
p@polygons[[p1]]@Polygons[[p2]]@coords[p3,1] <- pos[[1]][1]
p@polygons[[p1]]@Polygons[[p2]]@coords[p3,2] <- pos[[2]][1]
}
}
}
z <- toPeirceQuincuncial(0,-90,ang)[[1]][1]
p@bbox[,1] <- -z
p@bbox[,2] <- z
# plotting the map
par(mar=c(0,0,0,0),bg='#a7cdf2',xaxs='i',yaxs='i')
plot(p,col='gray',lwd=.5)
for (lon in 15*1:24) { # meridians
pos <- 0
posAnt <- 0
for (lat in -90:90) {
if (length(pos) == 2) {
posAnt <- pos
}
pos <- toPeirceQuincuncial(lon,lat,ang)
if (length(posAnt) == 2) {
segments(pos[[1]][1],pos[[2]][1],posAnt[[1]][1],posAnt[[2]][1],col='white',lwd=.5)
}
}
}
lats <- 15*1:5
posS <- matrix(0,length(lats),2)
posST <- 0
pos0 <- 0
posN <- matrix(0,length(lats),2)
posNT <- 0
for (lon in 0:360) {
posAntS <- posS
posAntST <- posST
posAnt0 <- pos0
posAntN <- posN
posAntNT <- posNT
pos0 <- unlist(toPeirceQuincuncial(lon,0,ang))
posST <- unlist(toPeirceQuincuncial(lon,-23.4368,ang))
posNT <- unlist(toPeirceQuincuncial(lon,23.4368,ang))
for (i in 1:length(lats)) {
posS[i,] <- unlist(toPeirceQuincuncial(lon,-lats[i],ang))
posN[i,] <- unlist(toPeirceQuincuncial(lon,lats[i],ang))
}
if (lon > 0) {
segments(pos0[1],pos0[2],posAnt0[1],posAnt0[2],col='red',lwd=1)
segments(posNT[1],posNT[2],posAntNT[1],posAntNT[2],col='yellow')
for (i in 1:length(lats)) {
segments(posN[i,1],posN[i,2],posAntN[i,1],posAntN[i,2],col='white',lwd=.5)
}
if (!(lon %in% round(90*(0:3+.5)+ang))) {
for (i in 1:length(lats)) {
segments(posS[i,1],posS[i,2],posAntS[i,1],posAntS[i,2],col='white',lwd=.5)
}
segments(posST[1],posST[2],posAntST[1],posAntST[2],col='yellow')
} else {
for (i in 1:length(lats)) {
posS[i,] <- unlist(toPeirceQuincuncial(lon-0.001,-lats[i],ang))
segments(posS[i,1],posS[i,2],posAntS[i,1],posAntS[i,2],col='white',lwd=.5)
posS[i,] <- unlist(toPeirceQuincuncial(lon,-lats[i],ang))
}
posST <- unlist(toPeirceQuincuncial(lon-0.001,-23.4368,ang))
segments(posST[1],posST[2],posAntST[1],posAntST[2],col='yellow')
posST <- unlist(toPeirceQuincuncial(lon,-23.4368,ang))
}
}
}
Playing with different values for lambda_0
, I've found out that I apparently cannot choose any value I want. 使用lambda_0
不同值,我发现我显然无法选择任何我想要的值。 It seems that the function will only work with half the possibilities I thought it did. 似乎该功能只能用我认为它的一半可能性。
Numbers indicate values of lambda_0
. 数字表示lambda_0
值。 As you can see, North America moves clockwise from right to left between 80° and 200°, and starts the same movement again between 240° and 40°. 如您所见,北美在80°和200°之间从右向左顺时针移动,并在240°和40°之间再次开始相同的移动。
How can I change the algorithm to allow for any angle I want (for instance, North America pointing up)? 如何更改算法以允许我想要的任何角度(例如,北美指向上方)?
I made a simple hack that "solved" the problem. 我做了一个简单的黑客“解决”了这个问题。 First I've found out which angles were a repetition of the others: between 46 and 225 degrees. 首先,我发现哪些角度是其他角度的重复:在46到225度之间。 Then, for these angles, I just had to flip the bounding box of the shapefile: 然后,对于这些角度,我只需要翻转shapefile的边界框:
z <- toPeirceQuincuncial(0,-90,ang)[[1]][1]
if (ang > 45 & ang < 226) {
p@bbox[,1] <- z
p@bbox[,2] <- -z
} else {
p@bbox[,1] <- -z
p@bbox[,2] <- z
}
Other thing: I improved the R code of toPeirceQuincuncial
, since it was converting coordinate by coordinate. 其他的事情:我改进了toPeirceQuincuncial
的R代码,因为它是通过坐标转换坐标。 Since R is a vector language, I adapted it to convert a group of coordinates at once, which made the code extremely fast. 由于R是一种矢量语言,我将其改编为一次转换一组坐标,这使得代码非常快。 Now lambda
and phi
may both be vectors (same size, of course), while lambda_0
is still a single number. 现在lambda
和phi
都可以是向量(当然大小相同),而lambda_0
仍然是单个数字。
toPeirceQuincuncial2<-function(lambda,phi,lambda_0=20.0){
# Convert latitude and longitude to radians relative to the
# central meridian
lambda<-lambda - lambda_0 + 180
lambda[which(lambda<0.0 | lambda>360.0)] <-
lambda[which(lambda<0.0 | lambda>360.0)] - 360*floor(lambda[which(lambda<0.0 | lambda>360.0)]/360)
lambda<-(lambda - 180) * degree
phi<-phi * degree
# Compute the auxiliary quantities 'm' and 'n'. Set 'm' to match
# the sign of 'lambda' and 'n' to be positive if |lambda| > pi/2
cos_phiosqrt2<-halfSqrt2 * cos(phi)
cos_lambda<-cos(lambda)
sin_lambda<-sin(lambda)
cos_a<-cos_phiosqrt2 * (sin_lambda + cos_lambda)
cos_b<-cos_phiosqrt2 * (sin_lambda - cos_lambda)
sin_a<-sqrt(1.0 - cos_a * cos_a)
sin_b<-sqrt(1.0 - cos_b * cos_b)
cos_a_cos_b<-cos_a * cos_b
sin_a_sin_b<-sin_a * sin_b
sin2_m<-1.0 + cos_a_cos_b - sin_a_sin_b
sin2_n<-1.0 - cos_a_cos_b - sin_a_sin_b
sin2_m[which(sin2_m < 0.0)] <- 0.0
sin_m<-sqrt(sin2_m)
sin2_m[which(sin2_m > 1.0)] <- 1.0
cos_m<-sqrt(1.0 - sin2_m)
sin_m[which(sin_lambda < 0.0)] <- -sin_m[which(sin_lambda < 0.0)]
sin2_n[which(sin2_n < 0.0)] <- 0.0
sin_n<-sqrt(sin2_n)
sin2_n[which(sin2_n > 1.0)] <- 1.0
cos_n<-sqrt(1.0 - sin2_n)
sin_n[which(cos_lambda > 0.0)] <- -sin_n[which(cos_lambda > 0.0)]
# Compute elliptic integrals to map the disc to the square
x<-ellFaux(cos_m,sin_m,halfSqrt2)
y<-ellFaux(cos_n,sin_n,halfSqrt2)
# Reflect the Southern Hemisphere outward
y[which(phi<0 & lambda<mthreequarterpi)] <- PeirceQuincuncialScale - y[which(phi<0 & lambda<mthreequarterpi)]
x[which(phi<0 & lambda>=mthreequarterpi & lambda<mquarterpi)] <- -PeirceQuincuncialScale - x[which(phi<0 & lambda>=mthreequarterpi & lambda<mquarterpi)]
y[which(phi<0 & lambda>=mquarterpi & lambda<quarterpi)] <- -PeirceQuincuncialScale - y[which(phi<0 & lambda>=mquarterpi & lambda<quarterpi)]
x[which(phi<0 & lambda>=quarterpi & lambda<threequarterpi)] <- PeirceQuincuncialScale - x[which(phi<0 & lambda>=quarterpi & lambda<threequarterpi)]
y[which(phi<0 & lambda>=threequarterpi)] <- PeirceQuincuncialScale - y[which(phi<0 & lambda>=threequarterpi)]
# Rotate the square by 45 degrees to fit the screen better
X<-(x - y) * halfSqrt2
Y<-(x + y) * halfSqrt2
res<-list(X,Y)
return(res)
}
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