javascript geometry coordinates grahams-scan

Sort latitude and longitude coordinates into clockwise ordered quadrilateral

Problem

Users can provide up to four latitude and longitude coordinates, in any order. They do so with Google Maps. Using Google's Polygon API (v3), the coordinates they select should highlight the selected area between the four coordinates.

Question

How do you sort an array of latitude and longitude coordinates in (counter-)clockwise order?

Solutions and Searches

StackOverflow Questions

Related Sites

Known Algorithms

Graham's scan (too complicated)
Jarvis March algorithm (handles N points)
Recursive Convex Hull (removes a point)

Code

Here is what I have so far:

// Ensures the markers are sorted: NW, NE, SE, SW
function sortMarkers() {
  var ns = markers.slice( 0 );
  var ew = markers.slice( 0 );

  ew.sort( function( a, b ) {
    if( a.position.lat() < b.position.lat() ) {
      return -1;
    }
    else if( a.position.lat() > b.position.lat() ) {
      return 1;
    }

    return 0;
  });

  ns.sort( function( a, b ) {
    if( a.position.lng() < b.position.lng() ) {
      return -1;
    }
    else if( a.position.lng() > b.position.lng() ) {
      return 1;
    }

    return 0;
  });

  var nw;
  var ne;
  var se;
  var sw;

  if( ew.indexOf( ns[0] ) > 1 ) {
    nw = ns[0];
  }
  else {
    ne = ns[0];
  }

  if( ew.indexOf( ns[1] ) > 1 ) {
    nw = ns[1];
  }
  else {
    ne = ns[1];
  }

  if( ew.indexOf( ns[2] ) > 1 ) {
    sw = ns[2];
  }
  else {
    se = ns[2];
  }

  if( ew.indexOf( ns[3] ) > 1 ) {
    sw = ns[3];
  }
  else {
    se = ns[3];
  }

  markers[0] = nw;
  markers[1] = ne;
  markers[2] = se;
  markers[3] = sw;
}

Thank you.

Solution

Given the points:

   4  +        [d]            [g]                 
      |                             
   3 [a]            [e]                 
      |                             
   2  +                  [f]       [h]    
      |                             
   1  +   [b]                             
      |                             
   0  +----+---[c]---+----+----+----+
      0    1    2    3    4    5    6

you want to find the following bound walk:

   4  +     ___[d]------------[g]                 
      |  __/                     \    
   3 [a]/           [e]__         \       
      | \             \_ ```---    \  
   2  +  \              `[f]   \___[h]    
      |   \           __/            
   1  +   [b]      __/                   
      |      \    /                
   0  +----+--`[c]---+----+----+----+
      0    1    2    3    4    5    6

If this is correct, here's a way:

find the upper most point, P_top, in the set of points. In case of a tie, pick the point with the smallest x coordinate
sort all points by comparing the slopes m_i and m_j of the lines each pair of points (excluding P_top!) P_i and P_j make when passing through P_top
- if m_i and m_j are equal, let the point P_i or P_j closest to P_top come first
- if m_i is positive and m_j is negative (or zero), P_j comes first
- if both m_i and m_j are either positive or negative, let the point belonging to the line with the largest slope come first

Here's a quick demo for the map:

enter image description here

(I know little JavaScript, so I might, or probably have, violated some JavaScript code conventions...):

var points = [
    new Point("Stuttgard", 48.7771056, 9.1807688),
    new Point("Rotterdam", 51.9226899, 4.4707867),
    new Point("Paris", 48.8566667, 2.3509871),
    new Point("Hamburg", 53.5538148, 9.9915752),
    new Point("Praha", 50.0878114, 14.4204598),
    new Point("Amsterdam", 52.3738007, 4.8909347),
    new Point("Bremen", 53.074981, 8.807081),
    new Point("Calais", 50.9580293, 1.8524129),
];
var upper = upperLeft(points);

print("points :: " + points);
print("upper  :: " + upper);
points.sort(pointSort);
print("sorted :: " + points);

// A representation of a 2D Point.
function Point(label, lat, lon) {

    this.label = label;
    this.x = (lon + 180) * 360;
    this.y = (lat + 90) * 180;

    this.distance=function(that) {
        var dX = that.x - this.x;
        var dY = that.y - this.y;
        return Math.sqrt((dX*dX) + (dY*dY));
    }

    this.slope=function(that) {
        var dX = that.x - this.x;
        var dY = that.y - this.y;
        return dY / dX;
    }

    this.toString=function() {
        return this.label;
    }
}

// A custom sort function that sorts p1 and p2 based on their slope
// that is formed from the upper most point from the array of points.
function pointSort(p1, p2) {
    // Exclude the 'upper' point from the sort (which should come first).
    if(p1 == upper) return -1;
    if(p2 == upper) return 1;

    // Find the slopes of 'p1' and 'p2' when a line is 
    // drawn from those points through the 'upper' point.
    var m1 = upper.slope(p1);
    var m2 = upper.slope(p2);

    // 'p1' and 'p2' are on the same line towards 'upper'.
    if(m1 == m2) {
        // The point closest to 'upper' will come first.
        return p1.distance(upper) < p2.distance(upper) ? -1 : 1;
    }

    // If 'p1' is to the right of 'upper' and 'p2' is the the left.
    if(m1 <= 0 && m2 > 0) return -1;

    // If 'p1' is to the left of 'upper' and 'p2' is the the right.
    if(m1 > 0 && m2 <= 0) return 1;

    // It seems that both slopes are either positive, or negative.
    return m1 > m2 ? -1 : 1;
}

// Find the upper most point. In case of a tie, get the left most point.
function upperLeft(points) {
    var top = points[0];
    for(var i = 1; i < points.length; i++) {
        var temp = points[i];
        if(temp.y > top.y || (temp.y == top.y && temp.x < top.x)) {
            top = temp;
        }
    }
    return top;
}

Note: your should double, or triple check the conversions from lat,lon to x,y as I am a novice if it comes to GIS!!! But perhaps you don't even need to convert anything. If you don't, the upperLeft function might just return the lowest point instead of the highest, depending on the locations of the points in question. Again: triple check these assumptions!

When executing the snippet above, the following gets printed:

points :: Stuttgard,Rotterdam,Paris,Hamburg,Praha,Amsterdam,Bremen,Calais
upper  :: Hamburg
sorted :: Hamburg,Praha,Stuttgard,Paris,Bremen,Calais,Rotterdam,Amsterdam

Alternate Distance Function

function distance(lat1, lng1, lat2, lng2) {
  var R = 6371; // km
  var dLat = (lat2-lat1).toRad();
  var dLon = (lng2-lng1).toRad();
  var a = Math.sin(dLat/2) * Math.sin(dLat/2) +
          Math.cos(lat1.toRad()) * Math.cos(lat2.toRad()) *
          Math.sin(dLon/2) * Math.sin(dLon/2);
  var c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1-a));
  return R * c;
}