Azimuthal equidistant projection: An implementation in F#, Python and Julia

Recently I needed to calculate distance from one point to another set of points in order to find the nearest point and its distance to the origin point. I opted to project all points to the azimuthal equidistant projection. The most important property of this projection is that all distances from the center of the projection to all other points represents the true distance (assuming a spheric earth) to all other projected points. To calculate the distance you just have to calculate the euclidean distance and multiply it with the average earth radius (6371000.0 meters).

The formulas for the azimuthal equidistant projection can be found at mathworld.wolfram.com. This formula can then be directly translated into F# like this:

open System
module AzimuthalEquidistantProjection =

    let inline degToRad d = 0.0174532925199433 * d; // (1.0/180.0 * Math.PI) * d

    let project centerlon centerlat lon lat =
        // http://mathworld.wolfram.com/AzimuthalEquidistantProjection.html
        // http://www.radicalcartography.net/?projectionref
        let t:float = degToRad lat
        let l:float = degToRad lon
        let t1 = degToRad centerlat // latitude center of projection
        let l0 = degToRad centerlon // longitude center of projection
        let c = acos ((sin t1) * (sin t) + (cos t1) * (cos t) * (cos (l-l0)))
        let k = c / (sin c)
        let x = k * (cos t) * (sin (l-l0))
        let y = k * ((cos t1) * (sin t) - (sin t1) * (cos t) * (cos (l-l0)))
        (x, y)

I also implemented the azimuthal equidistant projection by using units of measure. I create one set of measures to distinguish between degrees and radians and another one to distinguish between x and y.


open System
module AzimuthalEquidistantProjectionWithMeasures = 
    [<Measure>] type deg // degrees
    [<Measure>] type rad // radians
    [<Measure>] type x
    [<Measure>] type y
    type lon = float<x deg>
    type lat = float<y deg>
    
    let inline degToRad (d:float<'u deg>) = d*0.0174532925199433<rad/deg>
    let cos (d:float<'u rad>) = Math.Cos(float d)
    let sin (d:float<'u rad>) = Math.Sin(float d)
    
    let project (centerlon:lon) (centerlat:lat) (lon:lon) (lat:lat) =
        // http://mathworld.wolfram.com/AzimuthalEquidistantProjection.html
        // http://www.radicalcartography.net/?projectionref
        let t = degToRad lat
        let l = degToRad lon
        let t1 = degToRad centerlat // latitude center of projection
        let l0 = degToRad centerlon // longitude center of projection
        let c = acos ((sin t1) * (sin t) + (cos t1) * (cos t) * (cos (l-l0)))
        let k = c / (sin c)
        let x:float<x> = k * (cos t) * (sin (l-l0)) 
                         |> LanguagePrimitives.FloatWithMeasure
        let y:float<y> = k * ((cos t1) * (sin t) - (sin t1) * (cos t) * (cos (l-l0))) 
                         |> LanguagePrimitives.FloatWithMeasure
        (x, y)

In the past I've used the python OGR bindings and the ESRI Projection Engine for my map projection needs but this time I needed a pure python implementation so I translated the above code and optimized it a bit by precalculating some values that we'll need when we project the points during the initialization of my projection class. A similarly optimized version in F# is on fssnip.net.

from math import cos, sin, acos, radians

class Point(object):
    def __init__(self,x,y):
        self.x = x
        self.y = y       

class AzimuthalEquidistantProjection(object):
    """ 
        http://mathworld.wolfram.com/AzimuthalEquidistantProjection.html
        http://www.radicalcartography.net/?projectionref
    """
    def __init__(self, center):
        self.center = center
        self.t1 = radians(center.y) ## latitude center of projection
        self.l0 = radians(center.x) ## longitude center of projection
        self.cost1 = cos(self.t1)
        self.sint1 = sin(self.t1)
        
    def project(self, point):
        t = radians(point.y)
        l = radians(point.x)
        costcosll0 = cos(t) * cos(l-self.l0)
        sint = sin(t)
        
        c = acos ((self.sint1) * (sint) + (self.cost1) * costcosll0)
        k = c / sin(c)
        x = k * cos(t) * sin(l-self.l0)
        y = k * (self.cost1 * sint - self.sint1 * costcosll0)
        return Point(x, y)

import unittest
class Test_AzimuthalEquidistantProjection(unittest.TestCase):
    def test_project(self):
        p = AzimuthalEquidistantProjection(Point(0.0,0.0))
        r = p.project(Point(1.0,1.0))
        self.assertAlmostEqual(0.01745152022, r.x)
        self.assertAlmostEqual(0.01745417858, r.y)

        p = AzimuthalEquidistantProjection(Point(1.0,2.0))
        r = p.project(Point(3.0,4.0))
        self.assertAlmostEqual(0.03482860733, r.x)
        self.assertAlmostEqual(0.03494898734, r.y)

        p = AzimuthalEquidistantProjection(Point(-10.0001, 80.0001))
        r = p.project(Point(7.935, 63.302))
        self.assertAlmostEqual(0.1405127567, r.x)
        self.assertAlmostEqual(-0.263406547, r.y)

if __name__ == '__main__':
    unittest.main()

I've also implemented the straightforward projection code in Julia. Julia is a promising language built for technical computing. It's still only on version 0.2 but a lot of packages have already been created for the language.


function degToRad(d)
    d*0.0174532925199433
end

function project(centerlon, centerlat, lon, lat)
  t = degToRad(lat)
  l = degToRad(lon)
  t1 = degToRad(centerlat)
  l0 = degToRad(centerlon)
  c = acos(sin(t1) * sin(t) + cos(t1) * cos(t) * cos(l-l0))
  k = c / sin(c)
  x = k * cos(t) * sin(l-l0)
  y = k * (cos(t1) * sin(t) - sin(t1) * cos(t) * cos(l-l0))
  x, y
end

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2 comments:

Kerry Zhang said...

Comparing with equation 2 in the Wolfram link, aren't you forgetting the parenthesis in calculating y?

Should be:
y = k * (self.cost1 * sint - self.sint1 * costcosll0)

Samuel Bosch said...

You're right, I updated the code.