Finding the area of an overlap between curves (python)

Question

Is it possible to calculate the area of the overlap of two curves? I found two answers here but they are written in R which I am not familiar with. Or struggling to convert them to python.

Area between the two curves and Find area of overlap between two curves

For example, for a given dataset with defined x, y points. (x1,y1,x2,y2)

I am able to get the area of each curve using :

np.trapz

However, to get the overlap only is challenging and I haven't found a solution to show. Any guidance or maths formulas will be appreciated.

Answer 1

So this can be done using the shapely module within Python.

Firstly, Join the two curves together to create one self-intersecting polygon (shown in code below).

Then using the unary_union() function from shapely, you will:

1. Split the complex polygon into seperate simple polygons.
2. Find the area of each simple polygon.
3. Sum it to find the overall area of the two curves.

Full code shown below:

    import numpy as np
    from shapely.geometry import LineString
    from shapely.ops import unary_union, polygonize
    
    avg_coords = [(0.0, 0.0), (4.872117, 2.29658), (5.268545, 2.4639225), (5.664686, 2.6485724), (6.059776, 2.8966842), (6.695151, 3.0986626), (7.728006, 3.4045217), (8.522297, 3.652668), (9.157002, 3.895031), (10.191483, 4.1028132), (10.827622, 4.258638), (11.38593, 4.2933016), (11.86478, 4.3048816), (12.344586, 4.258769), (12.984073, 4.2126703), (13.942729, 4.1781383), (14.58212, 4.137809), (15.542498, 3.99943), (16.502588, 3.878359), (17.182951, 3.7745714), (18.262657, 3.6621647), (19.102558, 3.567045), (20.061789, 3.497897), (21.139917, 3.4806826), (22.097425, 3.5153809), (23.65388, 3.5414772), (24.851482, 3.541581), (26.04966, 3.507069), (27.72702, 3.463945), (28.925198, 3.429433), (29.883854, 3.3949006), (31.08246, 3.3344274), (31.92107, 3.317192), (33.716183, 3.3952322), (35.63192, 3.4213595), (37.427895, 3.4474766), (39.343628, 3.473604), (41.49874, 3.508406), (43.773468, 3.5518723), (46.287716, 3.595359), (49.28115, 3.6302335), (52.633293, 3.6997545), (54.30922, 3.7431688), (55.8651, 3.8038807), (58.738773, 3.8387446), (60.893887, 3.8735466), (63.647655, 3.9170544), (66.760704, 3.960593), (68.79663, 3.9607692), (70.23332, 3.986855), (72.867905, 3.995737), (75.38245, 4.0219164), (77.778656, 3.9615464), (79.337975, 3.8145657), (80.41826, 3.6675436), (80.899734, 3.5204697), (81.62059, 3.38207), (82.34045, 3.3042476), (83.30039, 3.1918304), (84.38039, 3.062116), (84.50359, 2.854434), (83.906364, 2.7591898), (83.669716, 2.586092), (83.43435, 2.3351095), (83.19727, 2.1879735), (82.84229, 1.9283267), (82.48516, 1.7984879), (81.65014, 1.5993768), (80.454544, 1.4781193), (79.13962, 1.3308897), (77.944595, 1.1750168), (76.39001, 1.0364205), (74.59633, 0.87184185), (71.60447, 0.741775), (70.04903, 0.6551017), (58.3, 0.0)]
    model_coords = [(0.0, 0.0), (0.6699889, 0.18807), (1.339894, 0.37499), (2.009583, 0.55966), (2.67915, 0.74106), (3.348189, 0.91826), (4.016881, 1.0904), (4.685107, 1.2567), (5.359344, 1.418), (6.026172, 1.5706), (6.685472, 1.714), (7.350604, 1.8508), (8.021434, 1.9803), (8.684451, 2.0996), (9.346408, 2.2099), (10.0066, 2.311), (10.66665, 2.4028), (11.32436, 2.4853), (11.98068, 2.5585), (12.6356, 2.6225), (13.29005, 2.6775), (13.93507, 2.7232), (14.58554, 2.7609), (15.23346, 2.7903), (15.87982, 2.8116), (16.52556, 2.8254), (17.16867, 2.832), (17.80914, 2.8317), (18.44891, 2.825), (19.08598, 2.8124), (19.72132, 2.7944), (20.35491, 2.7713), (20.98673, 2.7438), (21.61675, 2.7121), (22.24398, 2.677), (22.86939, 2.6387), (23.49297, 2.5978), (24.1147, 2.5548), (24.73458, 2.51), (25.3526, 2.464), (25.96874, 2.4171), (26.58301, 2.3697), (27.1954, 2.3223), (27.80491, 2.2751), (28.41354, 2.2285), (29.02028, 2.1829), (29.62512, 2.1384), (30.22809, 2.0954), (30.82917, 2.0541), (31.42837, 2.0147), (32.02669, 1.9775), (32.62215, 1.9425), (33.21674, 1.9099), (33.80945, 1.8799), (34.40032, 1.8525), (34.98933, 1.8277), (35.5765, 1.8058), (36.16283, 1.7865), (36.74733, 1.7701), (37.33002, 1.7564), (37.91187, 1.7455), (38.49092, 1.7372), (39.06917, 1.7316), (39.64661, 1.7285), (40.22127, 1.7279), (40.79514, 1.7297), (41.36723, 1.7337), (41.93759, 1.7399), (42.50707, 1.748), (43.07386, 1.7581), (43.63995, 1.7699), (44.20512, 1.7832), (44.76772, 1.7981), (45.3295, 1.8143), (45.88948, 1.8318), (46.44767, 1.8504), (47.00525, 1.8703), (47.55994, 1.8911), (48.11392, 1.9129), (48.6661, 1.9356), (49.21658, 1.959), (49.76518, 1.9832), (50.31305, 2.0079), (50.85824, 2.033), (51.40252, 2.0586), (51.94501, 2.0845), (52.48579, 2.1107), (53.02467, 2.1369), (53.56185, 2.1632), (54.09715, 2.1895), (54.63171, 2.2156), (55.1634, 2.2416), (55.69329, 2.2674), (56.22236, 2.2928), (56.74855, 2.3179), (57.27392, 2.3426), (57.7964, 2.3668), (58.31709, 2.3905), (58.83687, 2.4136), (59.35905, 2.4365), (59.87414, 2.4585), (60.38831, 2.4798), (60.8996, 2.5006), (61.40888, 2.5207), (61.91636, 2.5401), (62.42194, 2.5589), (62.92551, 2.577), (63.42729, 2.5945), (63.92607, 2.6113), (64.42384, 2.6275), (64.91873, 2.643), (65.4127, 2.658), (65.90369, 2.6724), (66.39266, 2.6862), (66.87964, 2.6995), (67.36373, 2.7123), (67.84679, 2.7246), (68.32689, 2.7364), (68.80595, 2.7478), (69.28194, 2.7588), (69.756, 2.7695), (70.22709, 2.7798), (70.69707, 2.7898), (71.16405, 2.7995), (71.62902, 2.809), (72.0919, 2.8183), (72.55277, 2.8273), (73.01067, 2.8362), (73.46734, 2.845), (73.92112, 2.8536), (74.37269, 2.8622), (74.82127, 2.8706), (75.26884, 2.8791), (75.71322, 2.8875), (76.15559, 2.8958), (76.59488, 2.9042), (77.03304, 2.9126), (77.46812, 2.921), (77.90111, 2.9294), (78.33199, 2.9379), (78.75986, 2.9464), (79.18652, 2.955), (79.60912, 2.9637), (80.03049, 2.9724), (80.44985, 2.9811), (80.86613, 2.99), (81.2802, 2.9989), (81.69118, 3.0078), (82.10006, 3.0168), (82.50674, 3.0259), (82.91132, 3.035), (83.31379, 3.0441), (83.71307, 3.0533), (84.10925, 3.0625), (84.50421, 3.0717), (84.8961, 3.0809), (85.28577, 3.0901), (85.67334, 3.0993), (86.05771, 3.1085), (86.43989, 3.1176), (86.81896, 3.1267), (87.19585, 3.1358), (87.57063, 3.1448), (87.94319, 3.1537), (88.31257, 3.1626), (88.67973, 3.1713), (89.04372, 3.18), (89.40659, 3.1886), (89.7652, 3.197), (90.12457, 3.2053), (90.47256, 3.2135), (90.82946, 3.2216), (91.17545, 3.2295), (91.52045, 3.2373), (91.86441, 3.2449), (92.20641, 3.2524), (92.54739, 3.2597), (92.88728, 3.2669), (93.21538, 3.2739), (93.55325, 3.2807), (93.87924, 3.2874), (94.20424, 3.2939), (94.52822, 3.3002), (94.85012, 3.3064), (95.16219, 3.3123), (95.48208, 3.3182), (95.79107, 3.3238), (96.09807, 3.3293), (96.40505, 3.3346), (96.71003, 3.3397), (97.01401, 3.3447), (97.31592, 3.3496), (97.60799, 3.3542), (97.90789, 3.3587), (98.19686, 3.3631), (98.48386, 3.3673), (98.77085, 3.3714), (99.05574, 3.3753), (99.32983, 3.3791), (99.6127, 3.3828), (99.8837, 3.3863), (100.1538, 3.3897), (100.4326, 3.393), (100.6897, 3.3961), (100.9566, 3.3991), (101.2215, 3.402), (101.4756, 3.4048), (101.7375, 3.4075), (101.9885, 3.4101), (102.2385, 3.4126), (102.4875, 3.4149), (102.7354, 3.4172), (102.9714, 3.4194), (103.2163, 3.4214), (103.4493, 3.4234), (103.6823, 3.4253), (103.9133, 3.4271), (104.1433, 3.4288), (104.3712, 3.4304), (104.5882, 3.4319), (104.8141, 3.4333), (105.0291, 3.4346), (105.2421, 3.4358), (105.4541, 3.437), (105.6651, 3.438), (105.8751, 3.439), (106.083, 3.4399), (106.28, 3.4407), (106.4759, 3.4414), (106.6699, 3.442), (106.8629, 3.4425), (107.0549, 3.443), (107.2458, 3.4433), (107.4249, 3.4435), (107.6128, 3.4437), (107.7897, 3.4438), (107.9647, 3.4437), (108.1387, 3.4436), (108.3116, 3.4433), (108.4737, 3.443), (108.6436, 3.4426), (108.8027, 3.4421), (108.9706, 3.4414), (109.1265, 3.4407), (109.2814, 3.4399), (109.4255, 3.439), (109.5784, 3.4379), (109.7195, 3.4368), (109.8694, 3.4356), (110.0084, 3.4342), (110.1454, 3.4328), (110.2813, 3.4313), (110.4162, 3.4296), (110.5403, 3.4279), (110.6722, 3.426), (110.7932, 3.424), (110.9132, 3.422), (111.0322, 3.4198), (111.1492, 3.4175), (111.2651, 3.4151), (111.3701, 3.4127), (111.483, 3.4101), (111.585, 3.4074), (111.686, 3.4046), (111.786, 3.4017), (111.884, 3.3987), (111.9809, 3.3956), (112.0669, 3.3924), (112.1608, 3.3891), (112.2448, 3.3857), (112.3268, 3.3822), (112.4078, 3.3786), (112.4867, 3.3749), (112.5548, 3.3711), (112.6317, 3.3672), (112.6978, 3.3632), (112.7726, 3.3591), (112.8356, 3.3549), (112.8975, 3.3506), (112.9476, 3.3462), (113.0076, 3.3417), (113.0655, 3.3372), (113.1125, 3.3325), (113.1584, 3.3278), (113.2024, 3.3229), (113.2464, 3.318), (113.2884, 3.313), (113.3283, 3.3079), (113.3584, 3.3027), (113.3963, 3.2974), (113.4233, 3.292), (113.4492, 3.2865), (113.4742, 3.281), (113.4972, 3.2753), (113.5201, 3.2696), (113.5312, 3.2638), (113.5501, 3.2579), (113.5591, 3.2519), (113.5661, 3.2459), (113.5721, 3.2397), (113.577, 3.2335), (113.5809, 3.2272), (113.573, 3.2208), (113.5749, 3.2143), (113.5649, 3.2077), (113.5539, 3.2011), (113.5409, 3.1944), (113.5278, 3.1876), (113.5128, 3.1807), (113.4967, 3.1737), (113.4697, 3.1667), (113.4418, 3.1596), (113.4227, 3.1524), (113.3917, 3.145), (113.3597, 3.1375), (113.3266, 3.1298), (113.2827, 3.1218), (113.2475, 3.1136), (113.2016, 3.1051), (113.1635, 3.0964), (113.1155, 3.0873), (113.0655, 3.0779), (113.0144, 3.0683), (112.9525, 3.0583), (112.8994, 3.048), (112.8345, 3.0373), (112.7793, 3.0264), (112.7123, 3.0152), (112.6453, 3.0037), (112.5763, 2.9919), (112.5063, 2.9798), (112.4352, 2.9674), (112.3533, 2.9548), (112.2801, 2.9419), (112.1952, 2.9287), (112.1102, 2.9153), (112.034, 2.9017), (111.9361, 2.8879), (111.8481, 2.8739), (111.7581, 2.8597), (111.667, 2.8453), (111.5661, 2.8307), (111.473, 2.816), (111.3689, 2.801), (111.2639, 2.786), (111.1579, 2.7708), (111.0509, 2.7555), (110.9428, 2.74), (110.8239, 2.7245), (110.7138, 2.7088), (110.5928, 2.6931), (110.4709, 2.6772), (110.3578, 2.6613), (110.2338, 2.6453), (110.1087, 2.6292), (109.9826, 2.613), (109.8457, 2.5968), (109.7176, 2.5805), (109.5787, 2.5642), (109.4496, 2.5478), (109.3086, 2.5314), (109.1666, 2.5149), (109.0236, 2.4984), (108.8806, 2.4819), (108.7355, 2.4653), (108.5905, 2.4488), (108.4434, 2.4322), (108.2865, 2.4155), (108.1384, 2.3989), (107.9794, 2.3822), (107.8195, 2.3655), (107.6684, 2.3488), (107.5063, 2.3321), (107.3374, 2.3156), (107.1744, 2.2989), (107.0104, 2.2822), (106.8442, 2.2654), (106.6683, 2.2487), (106.5012, 2.232), (106.3242, 2.2152), (106.1452, 2.1985), (105.9662, 2.1818), (105.7862, 2.165), (105.6052, 2.1483), (105.4232, 2.1316), (105.2402, 2.1149), (105.0572, 2.0981), (104.8721, 2.0814), (104.6772, 2.0647), (104.492, 2.048), (104.295, 2.0313), (104.098, 2.0147), (103.9, 1.998), (103.701, 1.9813), (103.502, 1.9647), (103.301, 1.948), (103.1, 1.9314), (102.899, 1.9148), (102.6959, 1.8982), (102.483, 1.8816), (102.2789, 1.865), (102.0649, 1.8484), (101.8588, 1.8318), (101.6428, 1.8153), (101.4268, 1.7988), (101.2098, 1.7822), (100.9918, 1.7657), (100.7728, 1.7492), (100.5538, 1.7328), (100.3338, 1.7163), (100.1128, 1.6999), (99.89169, 1.6834), (99.65978, 1.667), (99.43769, 1.6506), (99.20477, 1.6343), (98.98066, 1.6179), (98.74665, 1.6016), (98.51164, 1.5852), (98.27574, 1.5689), (98.04964, 1.5527), (97.81264, 1.5364), (97.57562, 1.5202), (97.33752, 1.5039), (97.08962, 1.4877), (96.8506, 1.4716), (96.61061, 1.4554), (96.37051, 1.4393), (96.12058, 1.4232), (95.87949, 1.4071), (95.62759, 1.391), (95.38547, 1.375), (95.13258, 1.359), (94.88946, 1.343), (94.63548, 1.3271), (94.38145, 1.3111), (94.12645, 1.2952), (93.87144, 1.2793), (93.61545, 1.2635), (93.35946, 1.2477), (93.10343, 1.2319), (92.84642, 1.2161), (92.58843, 1.2004), (92.33042, 1.1846), (92.07232, 1.169), (91.8034, 1.1533), (91.54331, 1.1377), (91.2744, 1.1221), (91.0133, 1.1065), (90.7434, 1.091), (90.48229, 1.0755), (90.21139, 1.0601), (89.9493, 1.0446), (89.67728, 1.0292), (89.40428, 1.0139), (89.13137, 0.99855), (88.86826, 0.98325), (88.59427, 0.96799), (88.32026, 0.95277), (88.04527, 0.93758), (87.77126, 0.92242), (87.4972, 0.90731), (87.21732, 0.89222), (86.94719, 0.87718), (86.66711, 0.86217), (86.3773, 0.8472), (86.10719, 0.83227), (85.82721, 0.81738), (85.5472, 0.80252), (85.26721, 0.7877), (84.9872, 0.77292), (84.7071, 0.75819), (84.41721, 0.74349), (84.1371, 0.72883), (83.84721, 0.71421), (83.5671, 0.69963), (83.27721, 0.68509), (82.99711, 0.6706), (82.70711, 0.65615), (82.41721, 0.64173), (82.1371, 0.62736), (81.8471, 0.61304), (81.55722, 0.59875), (81.27709, 0.58451), (80.98712, 0.57031), (80.697, 0.55616), (80.39711, 0.54205), (80.10722, 0.52798), (79.8271, 0.51396), (79.53701, 0.49999), (79.23711, 0.48605), (78.9471, 0.47217), (78.65701, 0.45833), (78.3571, 0.44453), (78.06712, 0.43078), (77.77701, 0.41708), (77.4771, 0.40343), (77.18701, 0.38982), (76.8871, 0.37626), (76.59711, 0.36274), (76.30701, 0.34928), (76.0071, 0.33586), (75.7169, 0.32249), (75.4071, 0.30917), (75.11701, 0.29589), (74.8171, 0.28267), (74.52701, 0.26949), (74.22711, 0.25636), (73.937, 0.24329), (73.63691, 0.23026), (73.3271, 0.21728), (73.03699, 0.20436), (72.73712, 0.19148), (72.4469, 0.17865), (72.13712, 0.16588), (71.84701, 0.15315), (71.547, 0.14048), (71.24701, 0.12786), (70.947, 0.11528), (70.64701, 0.10277), (70.3471, 0.090298), (70.05691, 0.077883), (69.74712, 0.06552), (69.457, 0.05321), (69.1569, 0.040952), (68.84709, 0.028747), (68.557, 0.016595), (68.25701, 0.0)]
    
    polygon_points = [] #creates a empty list where we will append the points to create the polygon
    
    for xyvalue in avg_coords:
        polygon_points.append([xyvalue[0],xyvalue[1]]) #append all xy points for curve 1
    
    for xyvalue in model_coords[::-1]:
        polygon_points.append([xyvalue[0],xyvalue[1]]) #append all xy points for curve 2 in the reverse order (from last point to first point)
    
    for xyvalue in avg_coords[0:1]:
        polygon_points.append([xyvalue[0],xyvalue[1]]) #append the first point in curve 1 again, to it "closes" the polygon
    
    avg_poly = [] 
    model_poly = []
    
    for xyvalue in avg_coords:
        avg_poly.append([xyvalue[0],xyvalue[1]]) 
    
    for xyvalue in model_coords:
        model_poly.append([xyvalue[0],xyvalue[1]]) 
    
    
    line_non_simple = LineString(polygon_points)
    mls = unary_union(line_non_simple)
    
    Area_cal =[]
    
    for polygon in polygonize(mls):
        Area_cal.append(polygon.area)
        print(polygon.area)# print area of each section 
        Area_poly = (np.asarray(Area_cal).sum())
        
    print(Area_poly)#print combined area