Sorting list of two-dimensional coordinates by clockwise angle using Python?

Question

I have a set of points with x and y coordinates that can be seen in the figure below. The coordinates of the 9 points were stored in a list as follows:

L = [[5,2         


        

Answer 1

With a bit of trigonometry it's not that hard. Maybe you know but the angle between two (normalized) vectors is acos(vec1 * vec2). However this calculates only the projected angle but one could use atan2 to calculate the direction-aware angle.

To this means a function calculating it and then using it as key for sorting would be a good way:

import math

pts = [[2,3], [5,2],[4,1],[3.5,1],[1,2],[2,1],[3,1],[3,3],[4,3]]
origin = [2, 3]
refvec = [0, 1]

def clockwiseangle_and_distance(point):
    # 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

A sorted run:

>>> sorted(pts, key=clockwiseangle_and_distance)
[[2, 3], [3, 3], [4, 3], [5, 2], [4, 1], [3.5, 1], [3, 1], [2, 1], [1, 2]]

and with a rectangular grid around the origin this works as expected as well:

>>> origin = [2,3]
>>> refvec = [0, 1]
>>> pts = [[1,4],[2,4],[3,4],[1,3],[2,3],[3,3],[1,2],[2,2],[3,2]]
>>> sorted(pts, key=clockwiseangle_and_distance)
[[2, 3], [2, 4], [3, 4], [3, 3], [3, 2], [2, 2], [1, 2], [1, 3], [1, 4]]

even if you change the reference vector:

>>> origin = [2,3]
>>> refvec = [1,0]  # to the right instead of pointing up
>>> pts = [[1,4],[2,4],[3,4],[1,3],[2,3],[3,3],[1,2],[2,2],[3,2]]
>>> sorted(pts, key=clockwiseangle_and_distance)
[[2, 3], [3, 3], [3, 2], [2, 2], [1, 2], [1, 3], [1, 4], [2, 4], [3, 4]]

Thanks @Scott Mermelstein for the better function name and @f5r5e5d for the atan2 suggestion.