Last week I was really surprised to find Functional JavaScript: Introducing Functional Programming with Underscore.js by Michael Fogus in the local library and I just finished it and wanted to leave a short review here.

This book really delivers what the title promises: an introduction to functional programming in JavaScript using the library Underscore.js. It doesn't teach you JavaScript nor Underscore.js but teaches what the different functional programming concepts are and how they can be implemented in JavaScript. This is done in less then 250 pages of densely, in a good way, written text and example code. Starting from the basics like first-class functions, applicative programming, variable scoping and closures, the book moves on to higher-order functions, currying and partial function application. Then some side steps are made with a great chapter on recursion which ends with the trampoline and a chapter on other important functional aspects like purity and immutability. Next is a chapter about flow-based programming, what it is, why it matters and different ways to define flows in your programs. The last chapter makes the connection with object-oriented programming and introduces mixins.

I really enjoyed reading this book because it is written very fluently without heavy (unnecessary) jargon and probably at a sweet spot on my learning curve. I've already read Real World Functional Programming: With Examples in F# and C# by Thomas Petricek and Jon Skeet and the first chapters of SICP but I haven't used it a lot in the wild. I've written my share of JavaScript programs but nothing very advanced, except maybe a Google Maps like library from scratch. If you're new to JavaScript AND functional programming then would advice against this book but otherwise, if you're motivated and don't let you get scared away by the first chapters then everything will be fine. But some playing around with the examples (like I did in this fiddle) and learning the basics of how to call passed in functions and how the often used Underscore.js functions (map, reduce, ...) work might be needed to get the most of this book. Overall this book is a very complete introduction to functional programming, the only thing I missed was a part on functional pattern matching. Note that this book is more about what introducing different functional programming techniques then about when and how to apply this techniques in your day-to-day programming.

Other books you might be interested in:

JavaScript: The Good Parts by Douglas Crockford (he popularized JSON and wrote JSLint and JSMin)

JavaScript: The Definitive Guide by David Flanagan

JavaScript Allongé by Reginald Braithwaite

Real World Functional Programming: With Examples in F# and C# by Thomas Petricek and Jon Skeet

Learn you a Haskell for Great Good! by Miran Lipovača

Learn you some Erlang for Great Good! by Fred Hébert

### Working in lat long: great circle distance, bearing, midpoint and centroid calculations

For my work in species distribution modeling I'm mostly working with lat long coordinates so I needed some great circle functions to calculate the point-point distance, point-to-line distance and the centroid of a group of points. Starting from 2 excellent resource http://www.movable-type.co.uk/scripts/latlong.html

and http://williams.best.vwh.net/ftp/avsig/avform.txt I created the following functions in Python, with some tests. As usual the full source code can be found online: GreatCircle.py.

I started with importing some functions from the math module and defining a small Point class:

Then came some very simple functions for calculating the distance (with the Haversine formula), the bearing and the midpoint. These are almost verbatim translations from the previously mentioned resources.

Then came the more puzzling part where I implemented the cross track error to calculate the great circle distance from a point to a line .

But the results didn't always match the distances that I calculated with PostGIS so I wrote a recursive binary search function consisting of 3 parts: calculating the midpoint, calculating the distance to two end and the midpoint and continuing again with the halve that is closest to the point until the difference in distance between the point and the two ends of the line is within a certain tolerance. The disadvantage of this function is that the rounding errors in calculating the midpoint do add up and the function is not very fast. So if anyone knows what wrong with my cross track implementation or knows a better method for calculating the great circle distance between a point and a line, feel free to comment below or send me an e-mail at mail@<thisdomainhere>.com.

The last function I needed was finding the centroid of a set of points in lat long coordinates. Below is an implementation from a response on gis.stackexchange.com. First all coordinates are converted to cartesian coordinates. Then the average of the x, y and z coordinates are calculated and finally these average coordinates are converted back to longitude and latitude.

For the implementation and running of my tests I used the built-in unittest module in the following way.

That's it for this post, the full code is on GitHub. If you'd like more posts like this on this blog then feel free to make some topic suggestions in the comments and make sure to subscribe to the updates of this site via RSS or via e-mail.

and http://williams.best.vwh.net/ftp/avsig/avform.txt I created the following functions in Python, with some tests. As usual the full source code can be found online: GreatCircle.py.

I started with importing some functions from the math module and defining a small Point class:

from math import radians, degrees, cos, sin, sqrt, atan2, asin, fabs, pi class Point(object): def __init__(self,x,y): self.x = x self.y = y

Then came some very simple functions for calculating the distance (with the Haversine formula), the bearing and the midpoint. These are almost verbatim translations from the previously mentioned resources.

def distance_haversine(A, B, radius=6371000): """ note that the default distance is in meters """ dLat = radians(B.y-A.y) dLon = radians(B.x-A.x) lat1 = radians(A.y) lat2 = radians(B.y) a = sin(dLat/2) * sin(dLat/2) + sin(dLon/2) * sin(dLon/2) * cos(lat1) * cos(lat2) c = 2 * atan2(sqrt(a), sqrt(1-a)) return c * radius def bearing(A, B): dLon = radians(B.x-A.x) lat1 = radians(A.y) lat2 = radians(B.y) y = sin(dLon) * cos(lat2) x = cos(lat1)* sin(lat2) - sin(lat1) * cos(lat2)* cos(dLon) return atan2(y, x) def bearing_degrees(A,B): return degrees(bearing(A,B)) def midpoint(A,B): ## little shortcut if A.x == B.x: return Point(A.x, (A.y+B.y)/2) if A.y == B.y: return Point((A.x+B.x)/2, A.y) lon1, lat1 = radians(A.x), radians(A.y) lon2, lat2 = radians(B.x), radians(B.y) dLon = lon2-lon1 Bx = cos(lat2) * cos(dLon) By = cos(lat2) * sin(dLon) lat3 = atan2(sin(lat1)+sin(lat2), sqrt((cos(lat1)+Bx)*(cos(lat1)+Bx) + By*By)) lon3 = lon1 + atan2(By, cos(lat1) + Bx) return Point(degrees(lon3),degrees(lat3))

Then came the more puzzling part where I implemented the cross track error to calculate the great circle distance from a point to a line .

def crosstrack_error(p,A,B, radius=6371000): """ distance (in meters) from a point to the closest point along a track http://williams.best.vwh.net/avform.htm#XTE """ dAp = distance_haversine(A, p, radius=1) brngAp = bearing(A,p) brngAB = bearing(A,B) dXt = asin(sin(dAp)*sin(brngAp-brngAB)) return fabs(dXt) * radius

But the results didn't always match the distances that I calculated with PostGIS so I wrote a recursive binary search function consisting of 3 parts: calculating the midpoint, calculating the distance to two end and the midpoint and continuing again with the halve that is closest to the point until the difference in distance between the point and the two ends of the line is within a certain tolerance. The disadvantage of this function is that the rounding errors in calculating the midpoint do add up and the function is not very fast. So if anyone knows what wrong with my cross track implementation or knows a better method for calculating the great circle distance between a point and a line, feel free to comment below or send me an e-mail at mail@<thisdomainhere>

`def point_line_distance(p, A,B, radius=6371000, tolerance=0.1): `

# tolerance and radius in meters """ recursive function that halves the search space until result is within tolerance disadvantages: 1) the rounding errors in midpoint do add up 2) not very fast """ def rec_point_line_distance(p,A,B,dA,dB): C = midpoint(A,B) dC = distance_haversine(p,C) if fabs(dC-dA) < tolerance or fabs(dC-dB) < tolerance: return dC elif dA < dB: return point_line_distance(p,A,C,dA, dC) else: return point_line_distance(p,C,B,dC, dB) dA = distance_haversine(p,A) dB = distance_haversine(p,B) return rec_point_line_distance(p,A,B,dA,dB)

The last function I needed was finding the centroid of a set of points in lat long coordinates. Below is an implementation from a response on gis.stackexchange.com. First all coordinates are converted to cartesian coordinates. Then the average of the x, y and z coordinates are calculated and finally these average coordinates are converted back to longitude and latitude.

def get_centroid(points): """ http://www.geomidpoint.com/example.html http://gis.stackexchange.com/questions/6025/find-the-centroid-of-a-cluster-of-points """ sum_x,sum_y,sum_z = 0,0,0 for p in points: lat = radians(p.y) lon = radians(p.x) ## convert lat lon to cartesian coordinates sum_x = sum_x + cos(lat) * cos(lon) sum_y = sum_y + cos(lat) * sin(lon) sum_z = sum_z + sin(lat) avg_x = sum_x / float(len(points)) avg_y = sum_y / float(len(points)) avg_z = sum_z / float(len(points)) center_lon = atan2(avg_y,avg_x) hyp = sqrt(avg_x*avg_x + avg_y*avg_y) center_lat = atan2(avg_z, hyp) return Point(degrees(center_lon), degrees(center_lat))

For the implementation and running of my tests I used the built-in unittest module in the following way.

import unittest class Test_gc(unittest.TestCase): def test_distance_haversine(self): d = distance_haversine(Point(-10.0001, 80.0001), Point(7.935, 63.302)) self.assertAlmostEqual(d, 1939037.0, places=0) def test_bearing(self): b = bearing(Point(-10.0001, 80.0001), Point(7.935, 63.302)) self.assertAlmostEqual(b, 2.661709,places=6) def test_bearing_degrees(self): b = bearing_degrees(Point(-10.0001, 80.0001), Point(7.935, 63.302)) self.assertAlmostEqual(b, 152.504694,places=6) def test_midpoint(self): m = midpoint(Point(-20.0,5.0), Point(-10.0,5.0)) self.assertAlmostEqual(-15.0, m.x, places=6) m = midpoint(Point(7.0,5.0), Point(7.0,15.0)) self.assertAlmostEqual(10.0, m.y, places=6) def test_crosstrack_error(self): p,A,B = Point(0.0, 1.0), Point(1.0, 1.0), Point(2.0, 1.0) d = crosstrack_error(p,A,B) self.assertAlmostEqual(d, distance_haversine(p,A), places=0) ## Apparently crosstrack_error returns incorrect results in some very specific cases (when p is completly off track) def test_point_line_distance(self): p,A,B = Point(0.0, 1.0), Point(1.0, 1.0), Point(2.0, 1.0) d = point_line_distance(p,A,B) self.assertAlmostEqual(d, distance_haversine(p,A), places=0) p,A,B = Point(2.0, 1.2), Point(1.0, 1.0), Point(3.0, 1.0) d = point_line_distance(p,A,B) self.assertAlmostEqual(d, distance_haversine(p,Point(2.0,1.0)), places=0) p,A,B = Point(0.0, 90.0), Point(-179.0, -21.0), Point(179.5, -22.0) d = point_line_distance(p,A,B) self.assertAlmostEqual(d, distance_haversine(p,A), places=0) p,A,B = Point(0.0, 89.0), Point(-179.0, -22.0), Point(179.5, -22.0) d = point_line_distance(p,A,B) self.assertAlmostEqual(d, distance_haversine(p,Point(0, -22)), places=0) def test_get_centroid(self): ## check crosses the north pole center = get_centroid([Point(0.0, 80.0), Point(180.0,80.0)]) self.assertEqual(90.0, center.y) ## check multiple points center = get_centroid([Point(0.0, 0.0), Point(10.0,0.0), Point(5.0,10.0)]) self.assertAlmostEqual(5.0, center.x) ## even more points center = get_centroid([Point(0.0, 0.0), Point(10.0,0.0),Point(10.0,30.0), Point(10.0,10.0), Point(20.0, 0.0)]) self.assertAlmostEqual(10.0, center.x) ## not lat lon average center = get_centroid([Point(0.0, 30.0), Point(10.0,30.0),Point(10.0,60.0), Point(0.0,60.0)]) self.assertNotAlmostEqual(45.0, center.y) self.assertAlmostEqual(5.0, center.x) ## crosses date line center = get_centroid([Point(170.0, 30.0), Point(-160.0,30.0),Point(170.0,60.0), Point(-160.0,60.0)]) self.assertNotAlmostEqual(45.0, center.y) self.assertAlmostEqual(-175.0, center.x) if __name__ == '__main__': unittest.main()

That's it for this post, the full code is on GitHub. If you'd like more posts like this on this blog then feel free to make some topic suggestions in the comments and make sure to subscribe to the updates of this site via RSS or via e-mail.

Labels:
GIS,
GreatCircle,
Python

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