sorting a complicated collection of 2d euclidian points in in clockwise/counterclockwise fashion to form a closed ring

Question

This looks like a repeated question but I tried the solution that already exists and none seems to work so far for me. .

this solution gives a hint but it works only for a regular geometry. I have a rather complicated geometry from which I extract boundary points which are unsorted.

Below is a picture of the geometry and of the boundary vertices that I extract from the geometry.

The (x,y) coordinates of points as in the image are :

import numpy as np

pts = np.array([[  30.        ,   -6.25      ],
                [  30.        ,   -8.10127917],
                [   0.        ,   -6.25      ],
                [  34.14082772,   -6.75584268],
                [  36.49784598,  -10.        ],
                [  44.43561524,  -10.        ],
                [ 100.        ,  -10.        ],
                [ 100.        ,   10.        ],
                [  84.1244615 ,  -10.        ],
                [  84.1244615 ,   10.        ],
                [  36.49784598,   10.        ],
                [  34.14082772,    6.75584268],
                [  44.43561524,   10.        ],
                [  30.        ,    8.10127917],
                [  30.        ,    6.25      ],
                [   0.        ,    6.25      ],
                [ -30.        ,    6.25      ],
                [ -30.        ,    8.10127917],
                [ -32.92183092,    9.05063958],
                [ -35.84366185,   10.        ],
                [ -51.88274638,   10.        ],
                [-100.        ,   10.        ],
                [-100.        ,  -10.        ],
                [ -83.96091546,   10.        ],
                [ -83.96091546,  -10.        ],
                [ -35.84366185,  -10.        ],
                [ -51.88274638,  -10.        ],
                [ -32.92183092,   -9.05063958],
                [ -30.        ,   -8.10127917],
                [ -30.        ,   -6.25      ],
                [ -67.92183092,   10.        ],
                [ -67.92183092,  -10.        ],
                [  68.24892299,   10.        ],
                [  52.37338449,   10.        ],
                [  68.24892299,  -10.        ],
                [  52.37338449,  -10.        ]])

In the boundary vertex data, we can see that the points are unordered. Is there a way the points could be ordered clockwise/counter clockwise so that these points form a closed ring when connected successively?

My goal is to create a polygon or a linear ring as described here and later find if an arbitrary euclidean point lies inside the polyogn/ring

Update: the approach of computing angle between centroid of the pts and the individual euclidean point in pts also doesn't work. Here is a sample code of what I tried :

def sort_circular(pts):
    cent = coords.mean(axis=0)
    idx = list(np.arange(0, len(pts)+1, dtype=int))
    angle = []
    for i, cc in enumerate(coords):
        dx,dy = cc[0] - center[0], cc[1]-center[1]
        angle.append(math.degrees(math.atan2(float(dy), float(dx))))
    #simultaneously sort angle and indices
    _, idx_sorted = (list(t) for t in zip(*sorted(zip(angle, idx))))
    pts_sorted = pts[idx_sorted]
    return pts_sorted

The result from this is still not the way i expect it to be (image below):

Answer 1

Method 1:

Define a center point, compute the angle between every coordinate and the center point, then order them by angle:

import pandas as pd

# Define function to compute angle between vectors
import math

def clockwiseangle_and_distance(point, origin = [0,0], refvec = [1,0]):
    # Vector between point and the origin: v = p - o
    vector = [point[0]-origin[0], point[1]-origin[1]]
    # Length of vector: ||v||
    lenvector = math.hypot(vector[0], vector[1])
    # If length is zero there is no angle
    if lenvector == 0:
        return -math.pi, 0
    # Normalize vector: v/||v||
    normalized = [vector[0]/lenvector, vector[1]/lenvector]
    dotprod  = normalized[0]*refvec[0] + normalized[1]*refvec[1]     # x1*x2 + y1*y2
    diffprod = refvec[1]*normalized[0] - refvec[0]*normalized[1]     # x1*y2 - y1*x2
    angle = math.atan2(diffprod, dotprod)
    # Negative angles represent counter-clockwise angles so we need to subtract them 
    # from 2*pi (360 degrees)
    if angle < 0:
        return 2*math.pi+angle, lenvector
    # I return first the angle because that's the primary sorting criterium
    # but if two vectors have the same angle then the shorter distance should come first.
    return angle, lenvector

import pandas as pd

# Compute the center point
center = pts.mean(axis=0)

angle = []
for i in range(len(pts)):
    ang, dist = clockwiseangle_and_distance(pts[i,:] - center, origin=[0,0], refvec=[1,0])
    angle.append(ang)

df = pd.DataFrame(pts)
df['angle'] = np.degrees(angle)

df = df.sort_values(by='angle')
df['clockwise_order'] = np.arange(len(df))

import matplotlib.pyplot as plt

# Create plot to show the ordering of the points
plt.figure()
df.plot(kind='scatter', x=0, y=1, s=100, alpha=0.5)
plt.title('Points by clockwise order')

for idx, row in df.iterrows():
    plt.gca().annotate('{:.0f}'.format(row['clockwise_order']), (row[0], row[1]),
            ha='center', va='center_baseline', fontsize=6, color='k', fontweight='bold')

plt.gca().annotate('Center', center,
        ha='center', va='center')

If this clockwise ordering doesn't give you what you want, try Method 2.

Method 2:

To sort the points for the given geometry in clockwise order such that they form a closed ring, you can do the following:

1. Divide the data set in quadrants
2. Choose a center point such that the remaining points of the quadrant lie on about the arc of a circle centered at the center point
3. Order each quadrant by clockwise angle
4. Place each quadrant in clockwise order

# Compute the center point
center = pts.mean(axis=0)

df = pd.DataFrame(pts)

# Group points into quadrants
df['quadrant'] = 0
df.loc[(df[0] > center[0]) & (df[1] > center[1]), 'quadrant'] = 0
df.loc[(df[0] > center[0]) & (df[1] < center[1]), 'quadrant'] = 1
df.loc[(df[0] < center[0]) & (df[1] < center[1]), 'quadrant'] = 2
df.loc[(df[0] < center[0]) & (df[1] > center[1]), 'quadrant'] = 3

quadrant = {}
for i in range(4):
    quadrant[i] = df[df.quadrant == i]

# Intelligently choose the quadrant centers
x = 35
y = 5
subcenter = [[ x,  y],
             [ x, -y],
             [-x, -y],
             [-x,  y]]

# Compute the angle between each quadrant and respective center point
angle = {}
points = {}
df_sub = {}
for j in range(len(quadrant)):
    angle[j] = []
    points[j] = quadrant[j][[0,1]]
    for i in range(len(points[j])):
        ang, dist = clockwiseangle_and_distance(points[j].values[i,:] - subcenter[j], origin=[0,0], refvec=[1,0])
        angle[j].append(ang)

    df_sub[j] = quadrant[j]
    df_sub[j]['angle'] = np.degrees(angle[j])
    df_sub[j] = df_sub[j].sort_values(by='angle')

# Combine the data frames
df = pd.concat(df_sub)
df['clockwise_order'] = np.arange(len(df))

# Plot the points by clockwise order
import matplotlib.pyplot as plt

# Create plot to show the ordering of the points
fig, axis = plt.subplots()
df[[0,1]].plot(x=0, y=1, ax=axis, c='lightblue', legend=False, clip_on=False)
df.plot(kind='scatter', x=0, y=1, s=100, ax=axis, c='lightblue', clip_on=False)
plt.title('Points by quadrant in clockwise order')
plt.axis('off')

for idx, row in df.iterrows():
    plt.gca().annotate('{:.0f}'.format(row['clockwise_order']), (row[0], row[1]),
            ha='center', va='center_baseline', fontsize=6, color='k', fontweight='bold')

plt.gca().annotate('Center', center,
        ha='center', va='center')

for i in range(len(subcenter)):
    plt.scatter(subcenter[i][0], subcenter[i][1], alpha=0.5, s=80, marker='s')
    plt.gca().annotate('Quadrant \n'+str(i)+'\n', subcenter[i],
        ha='center', va='center_baseline', color='k', fontsize=8)

# Plot with axis equally-spaced
df2 = df[[0,1]].reset_index(drop=True)
df2.loc[len(df2),:] = df2.loc[0,:]
df2.plot(x=0, y=1, c='k', legend=False, clip_on=False)
plt.axis('equal')
plt.axis('off')

If this doesn't give you what you want, you may have to order the coordinates by hand.