Rotating caliper algorithm is used to find a rectangle that fits a convex hull. Below is a python implementation that finds all rotated rectangles for a given convex hull points. It is up to the user to select which rectangle to use since it returns all possible rotating caliper rectangles.

import numpy as np def get_rotating_caliper_bbox_list(hull_points_2d): """ hull_points_2d: array of hull points. each element should have [x,y] format """ # Compute edges (x2-x1,y2-y1) edges = np.zeros( (len(hull_points_2d)-1,2) ) # empty 2 column array for i in range( len(edges) ): edge_x = hull_points_2d[i+1,0] - hull_points_2d[i,0] edge_y = hull_points_2d[i+1,1] - hull_points_2d[i,1] edges[i] = [edge_x,edge_y] # Calculate edge angles atan2(y/x) edge_angles = np.zeros( (len(edges)) ) # empty 1 column array for i in range( len(edge_angles) ): edge_angles[i] = np.arctan2( edges[i,1], edges[i,0] ) # Check for angles in 1st quadrant for i in range( len(edge_angles) ): edge_angles[i] = np.abs( edge_angles[i] % (np.pi/2) ) # want strictly positive answers #print "Edge angles in 1st Quadrant: \n", edge_angles # Remove duplicate angles edge_angles = np.unique(edge_angles) #print "Unique edge angles: \n", edge_angles bbox_list=[] for i in range( len(edge_angles) ): # Create rotation matrix to shift points to baseline # R = [ cos(theta) , cos(theta-PI/2) # cos(theta+PI/2) , cos(theta) ] R = np.array([ [ np.cos(edge_angles[i]), np.cos(edge_angles[i]-(np.pi/2)) ], [ np.cos(edge_angles[i]+(np.pi/2)), np.cos(edge_angles[i]) ] ]) # Apply this rotation to convex hull points rot_points = np.dot(R, np.transpose(hull_points_2d) ) # 2x2 * 2xn # Find min/max x,y points min_x = np.nanmin(rot_points[0], axis=0) max_x = np.nanmax(rot_points[0], axis=0) min_y = np.nanmin(rot_points[1], axis=0) max_y = np.nanmax(rot_points[1], axis=0) # Calculate height/width/area of this bounding rectangle width = max_x - min_x height = max_y - min_y area = width*height # Calculate center point and restore to original coordinate system center_x = (min_x + max_x)/2 center_y = (min_y + max_y)/2 center_point = np.dot( [ center_x, center_y ], R ) # Calculate corner points and restore to original coordinate system corner_points = np.zeros( (4,2) ) # empty 2 column array corner_points[0] = np.dot( [ max_x, min_y ], R ) corner_points[1] = np.dot( [ min_x, min_y ], R ) corner_points[2] = np.dot( [ min_x, max_y ], R ) corner_points[3] = np.dot( [ max_x, max_y ], R ) bbox_info = [edge_angles[i], area, width, height, min_x, max_x, min_y, max_y, corner_points, center_point] bbox_list.append(bbox_info) return bbox_list

this code was based on code from here. I only updated the code to return all possible rectangles instead of the one with minimum area since the user may be interested in other rectangles depending on their needs. Also I’ve updated it to use explicit functions from numpy.

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