import math import numpy as np """ Minimal Enclosing Parallelogram area, v1, v2, v3, v4 = mep(convex_polygon) convex_polygon - array of points. Each point is a array [x, y] (1d array of 2 elements) points should be presented in clockwise order. the algorithm used is described in the following paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.53.9659&rep=rep1&type=pdf """ def distance(p1, p2, p): return abs(((p2[1] - p1[1]) * p[0] - (p2[0] - p1[0]) * p[1] + p2[0] * p1[1] - p2[1] * p1[0]) / math.sqrt((p2[1] - p1[1]) ** 2 + (p2[0] - p1[0]) ** 2)) def antipodal_pairs(convex_polygon): l = [] n = len(convex_polygon) p1, p2 = convex_polygon[0], convex_polygon[1] t, d_max = None, 0 for p in range(1, n): d = distance(p1, p2, convex_polygon[p]) if d > d_max: t, d_max = p, d l.append(t) for p in range(1, n): p1, p2 = convex_polygon[p % n], convex_polygon[(p + 1) % n] _p, _pp = convex_polygon[t % n], convex_polygon[(t + 1) % n] while distance(p1, p2, _pp) > distance(p1, p2, _p): t = (t + 1) % n _p, _pp = convex_polygon[t % n], convex_polygon[(t + 1) % n] l.append(t) return l # returns score, area, points from top-left, clockwise , favouring low area def mep(convex_polygon): def compute_parallelogram(convex_polygon, l, z1, z2): def parallel_vector(a, b, c): v0 = [c[0] - a[0], c[1] - a[1]] v1 = [b[0] - c[0], b[1] - c[1]] return [c[0] - v0[0] - v1[0], c[1] - v0[1] - v1[1]] # finds intersection between lines, given 2 points on each line. # (x1, y1), (x2, y2) on 1st line, (x3, y3), (x4, y4) on 2nd line. def line_intersection(x1, y1, x2, y2, x3, y3, x4, y4): px = ((x1 * y2 - y1 * x2) * (x3 - x4) - (x1 - x2) * (x3 * y4 - y3 * x4)) / ( (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4)) py = ((x1 * y2 - y1 * x2) * (y3 - y4) - (y1 - y2) * (x3 * y4 - y3 * x4)) / ( (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4)) return px, py # from each antipodal point, draw a parallel vector, # so ap1->ap2 is parallel to p1->p2 # aq1->aq2 is parallel to q1->q2 p1, p2 = convex_polygon[z1 % n], convex_polygon[(z1 + 1) % n] q1, q2 = convex_polygon[z2 % n], convex_polygon[(z2 + 1) % n] ap1, aq1 = convex_polygon[l[z1 % n]], convex_polygon[l[z2 % n]] ap2, aq2 = parallel_vector(p1, p2, ap1), parallel_vector(q1, q2, aq1) a = line_intersection(p1[0], p1[1], p2[0], p2[1], q1[0], q1[1], q2[0], q2[1]) b = line_intersection(p1[0], p1[1], p2[0], p2[1], aq1[0], aq1[1], aq2[0], aq2[1]) d = line_intersection(ap1[0], ap1[1], ap2[0], ap2[1], q1[0], q1[1], q2[0], q2[1]) c = line_intersection(ap1[0], ap1[1], ap2[0], ap2[1], aq1[0], aq1[1], aq2[0], aq2[1]) s = distance(a, b, c) * math.sqrt((b[0] - a[0]) ** 2 + (b[1] - a[1]) ** 2) return s, a, b, c, d z1, z2 = 0, 0 n = len(convex_polygon) # for each edge, find antipodal vertice for it (step 1 in paper). l = antipodal_pairs(convex_polygon) so, ao, bo, co, do, z1o, z2o = 100000000000, None, None, None, None, None, None # step 2 in paper. for z1 in range(0, n): if z1 >= z2: z2 = z1 + 1 p1, p2 = convex_polygon[z1 % n], convex_polygon[(z1 + 1) % n] a, b, c = convex_polygon[z2 % n], convex_polygon[(z2 + 1) % n], convex_polygon[l[z2 % n]] if distance(p1, p2, a) >= distance(p1, p2, b): continue while distance(p1, p2, c) > distance(p1, p2, b): z2 += 1 a, b, c = convex_polygon[z2 % n], convex_polygon[(z2 + 1) % n], convex_polygon[l[z2 % n]] st, at, bt, ct, dt = compute_parallelogram(convex_polygon, l, z1, z2) if st < so: so, ao, bo, co, do, z1o, z2o = st, at, bt, ct, dt, z1, z2 return so, ao, bo, co, do, z1o, z2o def sort_rectangle(poly): # sort the four coordinates of the polygon, points in poly should be sorted clockwise # First find the lowest point p_lowest = np.argmax(poly[:, 1]) if np.count_nonzero(poly[:, 1] == poly[p_lowest, 1]) == 2: # - if the bottom line is parallel to x-axis, then p0 must be the upper-left corner p0_index = np.argmin(np.sum(poly, axis=1)) p1_index = (p0_index + 1) % 4 p2_index = (p0_index + 2) % 4 p3_index = (p0_index + 3) % 4 return poly[[p0_index, p1_index, p2_index, p3_index]], 0. else: #- find the point that sits right to the lowest point p_lowest_right = (p_lowest - 1) % 4 p_lowest_left = (p_lowest + 1) % 4 angle = np.arctan( -(poly[p_lowest][1] - poly[p_lowest_right][1]) / (poly[p_lowest][0] - poly[p_lowest_right][0])) # assert angle > 0 # if angle <= 0: # print(angle, poly[p_lowest], poly[p_lowest_right]) if angle / np.pi * 180 > 45: #p2 - this point is p2 p2_index = p_lowest p1_index = (p2_index - 1) % 4 p0_index = (p2_index - 2) % 4 p3_index = (p2_index + 1) % 4 return poly[[p0_index, p1_index, p2_index, p3_index]], -(np.pi / 2 - angle) else: #p3 - this point is p3 p3_index = p_lowest p0_index = (p3_index + 1) % 4 p1_index = (p3_index + 2) % 4 p2_index = (p3_index + 3) % 4 return poly[[p0_index, p1_index, p2_index, p3_index]], angle