Download or view MollweideProjection.frink in plain text format
/** This program demonstrates the use of the country polygons in
Country.frink to draw a map of the world. It can be readily altered to
draw the map in your favorite projection. This demonstrates the
Mollweide projection:
https://mathworld.wolfram.com/MollweideProjection.html
https://en.wikipedia.org/wiki/Mollweide_projection
*/
use Country.frink
g = new graphics
g.stroke[0.001]
g.font["SansSerif", "bold", 1 degree]
// Iterate through all countries.
for [code, country] = Country.getCountryList[]
{
cc = new color[randomFloat[0,1], randomFloat[0,1], randomFloat[0,1], .8]
first = true
for poly = country.borders // Iterate through polygons in a country.
{
p = new filledPolygon // This polygon is the filled country
po = new polygon // This is the outline of the country
for [long, lat] = poly // Iterate through points in polygon
{
[x,y] = latLongToXYMollweide[lat degree, long degree]
p.addPoint[x, -y]
po.addPoint[x, -y]
}
// Draw filled countries
g.color[cc]
g.add[p]
// The polygons in Country.frink are sorted so the first polygon is the
// largest. Just label the largest.
if first
{
[cx, cy] = p.getCentroid[]
g.color[0,0,0]
g.text[code, cx, cy]
}
// Draw country outlines
g.color[0.2, 0.2, 0.2, 0.8]
g.add[po]
first = false
}
}
// Optional: Draw the ellipse that makes the boundary
// You could do this first but with fillEllipse... to draw water
//g.color[0,0,0,.3]
//g.drawEllipseSides[-2 sqrt[2], -sqrt[2], 2 sqrt[2], sqrt[2]]
g.show[]
g.write["Mollweide.svg", 1000, undef]
g.write["Mollweide.png", 1000, undef]
g.write["Mollweide.html", 1000, 500]
/** Convert a latitude, longitude, and optional center longitude (long0)
into x,y coordinates.
The x coordinate ranges from -2 sqrt[2] to 2 sqrt[2]
The y cooridnate ranges from -sqrt[2] to sqrt[2]]
returns
[x ,y]
*/
latLongToXYMollweide[lat, long, long0 = 0 degrees West] :=
{
// The Mollweide projection uses an "auxiliary angle" theta where theta is
// given by
// 2 theta + sin(2 theta) = pi sin[lat]
//
// Where theta is found iteratively by iterating:
// theta = theta-(2 theta + sin[2 theta] - pi sin[lat])/(2 + 2 cos[2 theta])
//
// or, by introducing a different supplementary angle theta1 where
// theta = 1/2 theta1
// The equations can be iterated as
// theta1 = theta1 - (theta1 + sin[theta1] - pi sin[lat]) / (1 + cos[theta1])
// with an initial guess
// theta1 = 2 arcsin[2 lat / pi]
theta1 = 2 arcsin[2 lat / pi]
slat = sin[lat]
do
{
ct = cos[theta1]
if ct == -1
break
delta = - (theta1 + sin[theta1] - pi slat) / (1 + ct)
theta1 = theta1 + delta
} while abs[delta] > 1e-7 // error of 0.63 m on the earth
theta = 1/2 theta1
x = 2 sqrt[2]/pi (long - long0) cos[theta]
y = sqrt[2] sin[theta]
return [x,y]
}
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This is a program written in the programming language Frink.
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Alan Eliasen was born 19966 days, 9 hours, 57 minutes ago.