Finding centre of a polygon using limited data

Question

I'm implementing Voronoi tesselation followed by smoothing. For the smoothing I was going to do Lloyd relaxation, but I've encountered a problem.

I'm using following module for calculation of Voronoi sides:

https://bitbucket.org/mozman/geoalg/src/5bbd46fa2270/geoalg/voronoi.py

For the smoothing I need to know the edges of each polygon so I can calculate the centre, which unfortunately this code doesn't provide.

Information I have access to consists of:

• A list of all nodes,
• A list of all edges (but just where they are, not what nodes they're associated with).

Can anyone see a relatively simple way to calculate this?

Answer 1

For finding a centroid, you can use the formula described on wikipedia:

import math

def area_for_polygon(polygon):
    result = 0
    imax = len(polygon) - 1
    for i in range(0,imax):
        result += (polygon[i]['x'] * polygon[i+1]['y']) - (polygon[i+1]['x'] * polygon[i]['y'])
    result += (polygon[imax]['x'] * polygon[0]['y']) - (polygon[0]['x'] * polygon[imax]['y'])
    return result / 2.

def centroid_for_polygon(polygon):
    area = area_for_polygon(polygon)
    imax = len(polygon) - 1

    result_x = 0
    result_y = 0
    for i in range(0,imax):
        result_x += (polygon[i]['x'] + polygon[i+1]['x']) * ((polygon[i]['x'] * polygon[i+1]['y']) - (polygon[i+1]['x'] * polygon[i]['y']))
        result_y += (polygon[i]['y'] + polygon[i+1]['y']) * ((polygon[i]['x'] * polygon[i+1]['y']) - (polygon[i+1]['x'] * polygon[i]['y']))
    result_x += (polygon[imax]['x'] + polygon[0]['x']) * ((polygon[imax]['x'] * polygon[0]['y']) - (polygon[0]['x'] * polygon[imax]['y']))
    result_y += (polygon[imax]['y'] + polygon[0]['y']) * ((polygon[imax]['x'] * polygon[0]['y']) - (polygon[0]['x'] * polygon[imax]['y']))
    result_x /= (area * 6.0)
    result_y /= (area * 6.0)

    return {'x': result_x, 'y': result_y}

def bottommost_index_for_polygon(polygon):
    bottommost_index = 0
    for index, point in enumerate(polygon):
        if (point['y'] < polygon[bottommost_index]['y']):
            bottommost_index = index
    return bottommost_index

def angle_for_vector(start_point, end_point):
    y = end_point['y'] - start_point['y']
    x = end_point['x'] - start_point['x']
    angle = 0

    if (x == 0):
        if (y > 0):
            angle = 90.0
        else:
            angle = 270.0
    elif (y == 0):
        if (x > 0):
            angle = 0.0
        else:
            angle = 180.0
    else:
        angle = math.degrees(math.atan((y+0.0)/x))
        if (x < 0):
            angle += 180
        elif (y < 0):
            angle += 360

    return angle

def convex_hull_for_polygon(polygon):
    starting_point_index = bottommost_index_for_polygon(polygon)
    convex_hull = [polygon[starting_point_index]]
    polygon_length = len(polygon)

    hull_index_candidate = 0 #arbitrary
    previous_hull_index_candidate = starting_point_index
    previous_angle = 0
    while True:
        smallest_angle = 360

        for j in range(0,polygon_length):
            if (previous_hull_index_candidate == j):
                continue
            current_angle = angle_for_vector(polygon[previous_hull_index_candidate], polygon[j])
            if (current_angle < smallest_angle and current_angle > previous_angle):
                hull_index_candidate = j
                smallest_angle = current_angle

        if (hull_index_candidate == starting_point_index): # we've wrapped all the way around
            break
        else:
            convex_hull.append(polygon[hull_index_candidate])
            previous_angle = smallest_angle
            previous_hull_index_candidate = hull_index_candidate

    return convex_hull

I used a gift-wrapping algorithm to find the outside points (a.k.a. convex hull). There are a bunch of ways to do this, but gift-wrapping is nice because of its conceptual and practical simplicity. Here's an animated gif explaining this particular implementation:

Here's some naive code to find centroids of the individual voronoi cells based on a collection of nodes and edges for a voronoi diagram. It introduces a method to find edges belonging to a node and relies on the previous centroid and convex-hull code:

def midpoint(edge):
    x1 = edge[0][0]
    y1 = edge[0][9]
    x2 = edge[1][0]
    y2 = edge[1][10]

    mid_x = x1+((x2-x1)/2.0)
    mid_y = y1+((y2-y1)/2.0)

    return (mid_x, mid_y)

def ccw(A,B,C): # from http://www.bryceboe.com/2006/10/23/line-segment-intersection-algorithm/
    return (C[1]-A[1])*(B[0]-A[0]) > (B[1]-A[1])*(C[0]-A[0])

def intersect(segment1, segment2): # from http://www.bryceboe.com/2006/10/23/line-segment-intersection-algorithm/
    A = segment1[0]
    B = segment1[1]
    C = segment2[0]
    D = segment2[1]
    # Note: this doesn't catch collinear line segments!
    return ccw(A,C,D) != ccw(B,C,D) and ccw(A,B,C) != ccw(A,B,D)

def points_from_edges(edges):
    point_set = set()
    for i in range(0,len(edges)):
          point_set.add(edges[i][0])
          point_set.add(edges[i][11])

    points = []
    for point in point_set:
          points.append({'x':point[0], 'y':point[1]})

    return list(points)

def centroids_for_points_and_edges(points, edges):

    centroids = []

    # for each voronoi_node,
    for i in range(0,len(points)):
        cell_edges = []

        # for each edge
        for j in range(0,len(edges)):
            is_cell_edge = True

            # let vector be the line from voronoi_node to the midpoint of edge
            vector = (points[i],midpoint(edges[j]))

            # for each other_edge
            for k in range(0,len(edges)):

                # if vector crosses other_edge
                if (k != j and intersect(edges[k], vector)):
                    # edge is not in voronoi_node's polygon
                    is_cell_edge = False
                    break

            # if the vector didn't cross any other edges, it's an edge for the current node
            if (is_cell_edge):
                cell_edges.append(edges[j])

        # find the hull for the cell
        convex_hull = convex_hull_for_polygon(points_from_edges(cell_edges))

        # calculate the centroid of the hull
        centroids.append(centroid_for_polygon(convex_hull))

    return centroids

edges = [
  ((10,  200),(30,  50 )),
  ((10,  200),(100, 140)),
  ((10,  200),(200, 180)),
  ((30,  50 ),(100, 140)),
  ((30,  50 ),(150, 75 )),
  ((30,  50 ),(200, 10 )),
  ((100, 140),(150, 75 )),
  ((100, 140),(200, 180)),
  ((150, 75 ),(200, 10 )),
  ((150, 75 ),(200, 180)),
  ((150, 75 ),(220, 80 )),
  ((200, 10 ),(220, 80 )),
  ((200, 10 ),(350, 100)),
  ((200, 180),(220, 80 )),
  ((200, 180),(350, 100)),
  ((220, 80 ),(350, 100))
]

points = [
  (50,130),
  (100,95),
  (100,170),
  (130,45),
  (150,130),
  (190,55),
  (190,110),
  (240,60),
  (245,120)
]

centroids = centroids_for_points_and_edges(points, edges)
print "centroids:"
for centroid in centroids:
    print "  (%s, %s)" % (centroid['x'], centroid['y'])

Below is an image of the script results. The blue lines are edges. The black squares are nodes. The red squares are vertices that the blue lines are derived from. The vertices and nodes were chosen arbitrarily. The red crosses are centroids. While not an actual voronoi tesselation, the method used to procure the centroids should hold for tessalations composed of convex cells:

Here's the html to render the image:

<html>
<head>
  <script>
    window.onload = draw;
    function draw() {
      var canvas = document.getElementById('canvas').getContext('2d');

      // draw polygon points
      var polygon = [ 
        {'x':220, 'y':80},
        {'x':200, 'y':180},
        {'x':350, 'y':100},
        {'x':30, 'y':50}, 
        {'x':100, 'y':140},
        {'x':200, 'y':10},
        {'x':10, 'y':200},
        {'x':150, 'y':75}
      ];  
      plen=polygon.length;
      for(i=0; i<plen; i++) {
        canvas.fillStyle = 'red';
        canvas.fillRect(polygon[i].x-4,polygon[i].y-4,8,8);
        canvas.fillStyle = 'yellow';
        canvas.fillRect(polygon[i].x-2,polygon[i].y-2,4,4);
      }   

      // draw edges
      var edges = [ 
        [[10,  200],[30,  50 ]], 
        [[10,  200],[100, 140]],
        [[10,  200],[200, 180]],
        [[30,  50 ],[100, 140]], 
        [[30,  50 ],[150, 75 ]], 
        [[30,  50 ],[200, 10 ]], 
        [[100, 140],[150, 75 ]], 
        [[100, 140],[200, 180]],
        [[150, 75 ],[200, 10 ]], 
        [[150, 75 ],[200, 180]],
        [[150, 75 ],[220, 80 ]], 
        [[200, 10 ],[220, 80 ]], 
        [[200, 10 ],[350, 100]],
        [[200, 180],[220, 80 ]], 
        [[200, 180],[350, 100]],
        [[220, 80 ],[350, 100]]
      ];  
      elen=edges.length;
      canvas.beginPath();
      for(i=0; i<elen; i++) {
        canvas.moveTo(edges[i][0][0], edges[i][0][1]);
        canvas.lineTo(edges[i][13][0], edges[i][14][1]);
      }   
      canvas.closePath();
      canvas.strokeStyle = 'blue';
      canvas.stroke();

      // draw center points
      var points = [ 
        [50,130],
        [100,95],
        [100,170],
        [130,45],
        [150,130],
        [190,55],
        [190,110],
        [240,60],
        [245,120]
      ]   
      plen=points.length;
      for(i=0; i<plen; i++) {
        canvas.fillStyle = 'black';
        canvas.fillRect(points[i][0]-3,points[i][15]-3,6,6);
        canvas.fillStyle = 'white';
        canvas.fillRect(points[i][0]-1,points[i][16]-1,2,2);
      }   

      // draw centroids
      var centroids = [ 
        [46.6666666667, 130.0],
        [93.3333333333, 88.3333333333],
        [103.333333333, 173.333333333],
        [126.666666667, 45.0],
        [150.0, 131.666666667],
        [190.0, 55.0],
        [190.0, 111.666666667],
        [256.666666667, 63.3333333333],
        [256.666666667, 120.0]
      ]
      clen=centroids.length;
      canvas.beginPath();
      for(i=0; i<clen; i++) {
        canvas.moveTo(centroids[i][0], centroids[i][17]-5);
        canvas.lineTo(centroids[i][0], centroids[i][18]+5);
        canvas.moveTo(centroids[i][0]-5, centroids[i][19]);
        canvas.lineTo(centroids[i][0]+5, centroids[i][20]);
      }
      canvas.closePath();
      canvas.strokeStyle = 'red';
      canvas.stroke();
    }
  </script>
</head>
<body>
  <canvas id='canvas' width="400px" height="250px"</canvas>
</body>
</html>

This will likely get the job done. A more robust algo for finding which edges belong to a cell would be to use an inverse gift-wrapping method where edges are linked end-to-end and path choice at a split would be determined by angle. That method would not have a susceptibility to concave polygons and it would have the added benefit of not relying on the nodes.