Peirce quincuncial projection

ja: パース・クインカンシャル図法

en: Peirce quincuncial projection

https://upload.wikimedia.org/wikipedia/commons/6/6b/Peirce_quincuncial_projection_SW_20W.JPG

225° = 15° × 15

https://pbs.twimg.com/media/FYGWi2RacAAQzrb.jpg

デフォ設定だと オーストラリアとニュージーランドが分かれてしまう。

def.

$ \mathrm{sd}\left(\sqrt{2}\,w , \frac{1}{\sqrt{2}}\right) = \sqrt{2}\,r

$ w = \frac{1}{\sqrt{2}}\,\mathrm{arcsd}\left(\sqrt{2}r , \frac{1}{\sqrt{2}}\right)

$ = \frac{1}{\sqrt{2}} \int_0^{\sqrt{2}r} \frac{\mathrm{d}t}{ \sqrt{ 1 - \frac{1}{4}t^4 } }

$ = \int_0^{r} \frac{\mathrm{d}t}{ \sqrt{ 1 - z^4 } }

where z ≔ t/√2

t = z √2

dt = dz √2

sd を使う。

$ \mathrm{arcsd}(x \mid k^2) = \int_0^x \frac{\mathrm{d}t}{ \sqrt{ (1 - (1-k^2) t^2)(1 + k^2 t^2) } } = \int_0^x \frac{\mathrm{d}t}{ \sqrt{ 1 - (1 - 2k^2 - (1 - k^2) k^2 t^2) t^2 } }

sl を使う。

cf. sd版

$ \sqrt{2}\;\mathrm{sl}(w) = \sqrt{2}\;r

sl: lemniscate sine

$ w = \mathrm{arcsl}(r) = \int_0^r \frac{dz}{ \sqrt{1 - z^4} }

$ = k\,F\left(\arcsin\left(\frac{r}{k\sqrt{1+r^2}}\right); k\right)

where

$ m = k^2 = 1/2, α = τ/12

F: incomplete elliptic integral of the first kind

in Jacobi's algebraic form

類似

Spilhaus projection

偶数次 Chebyshev 級数で第一種楕円積分を近似計算できるらしい。

algo

l = long, p = lat

in radian

sign of m, n

sm = if sin(l) < 0 then -1 else 1

≈ sign(sin(l))

= で良い

sn = if cos(l) > 0 then -1 else 1

≈ -sign(cos(l))

= で良い

angle a, b

a = arccos(cos(p) * (sin(l) + cos(l)))

= arccos(cos(p) * sin(l+pi/4))

b = arccos(cos(p) * (sin(l) - cos(l)))

= arccos(cos(p) * -sin(pi/4-l))

= arccos(cos(p) * -cos(l+pi/4))

assert cos(a+b) ≤ 0, cos(a-b) ≥ 0

clamp sqrt to domain of arcsin = ±1

angle m, n as amplitude

m = arcsin(sqrt(1 + min(0,cos(a+b))))

= arcsin(sqrt(1 + cos(a+b)))

誤差対応

n = arcsin(sqrt(abs(1 - max(0,cos(a-b)))))

= arcsin(sqrt(1 - cos(a-b)))

誤差対応

calc F by amplitude and parameter=1/2

x = F(sm * m, 1/2)

y = F(sn * n, 1/2)

shift 南半球 by K(1/2)

scale = K(.5)

bins: l in (-4, -3, -1, 1, 3, 4)/4*π

1. scale - y

2. -scale - x

rotate 45deg from diamond

(x,y) := 1/√2 (x-y, x+y)

ref.

incomplete elliptic integral of the first kind

binary

(complete) elliptic integral of the first kind

unary