Menemukan Luas Tumpang Tindih Antar Kurva (python)

Question

Apakah mungkin untuk menghitung luas tumpang tindih dua kurva? Saya menemukan dua jawaban di sini tetapi itu ditulis dalam R yang tidak saya kenal. Atau berjuang untuk mengubahnya menjadi python.

Area antara dua kurva dan Temukan area tumpang tindih antara dua kurva

Misalnya, untuk kumpulan data tertentu dengan titik x, y yang ditentukan. (x1, y1, x2, y2)

Saya bisa mendapatkan luas setiap kurva menggunakan:

np.trapz

Namun, untuk mendapatkan tumpang tindih saja itu menantang dan saya belum menemukan solusi untuk ditunjukkan. Panduan atau rumus matematika apa pun akan dihargai.

Answer 1

Jadi ini bisa dilakukan dengan menggunakan modul berbentuk dalam Python.

Pertama, Gabungkan kedua kurva bersama-sama untuk membuat satu poligon berpotongan sendiri (ditunjukkan pada kode di bawah).

Kemudian menggunakan unary_union()fungsi dari bentuk, Anda akan membagi poligon kompleks menjadi poligon sederhana dan mencari luas setiap poligon dan menjumlahkannya untuk mencari luas keseluruhan dari kedua kurva.

Kode yang ditunjukkan di bawah ini:

    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

Answer 2

Jika memungkinkan, tunjukkan wilayah tumpang tindih Anda sebagai poligon. Dari sana luas poligon dapat dihitung dengan rumus yang sangat ringkas seperti yang dijelaskan di situs Paul Bourke .

Misalkan (x [i], y [i]), i = 0, ..., N, adalah simpul poligon, dengan (x [0], y [0]) = (x [N], y [N ]) sehingga poligon ditutup, dan secara konsisten semua dalam urutan searah jarum jam atau semua dalam urutan berlawanan arah jarum jam. Kemudian areanya

area = |0.5 * sum_i (x[i] * y[i+1] - x[i+1] * y[i])|

dimana jumlahnya melebihi i = 0, ..., N-1. Ini berlaku bahkan untuk poligon nonkonveks. Rumus ini pada dasarnya adalah prinsip yang sama di balik bagaimana planimeter bekerja untuk mengukur luas suatu bentuk dua dimensi yang berubah-ubah, kasus khusus dari teorema Green .

Answer 3

Jika fungsi Anda sebenarnya adalah "fungsi" yang berarti tidak ada garis vertikal yang memotong fungsi lebih dari satu kali, maka mencari tumpang tindih adalah masalah mencari nol.

import numpy as np
import matplotlib.pyplot as plt

dx = 0.01
x = np.arange(-2, 2, dx)
f1 = np.sin(4*x)
f2 = np.cos(4*x)

plt.plot(x, f1)
plt.plot(x, f2)

eps = 1e-1; # threshold of intersection points.
df = f1 - f2
idx_zeros = np.where(abs(df) <= eps)[0]

area = 0
for i in range(len(idx_zeros) - 1):
    idx_left = idx_zeros[i]
    idx_rite = idx_zeros[i+1]
    area += abs(np.trapz(df[idx_left:idx_rite])) * dx

• Saya telah mengasumsikan bidang-bidang yang dianggap positif.
• Nilai analitis untuk contoh yang saya gunakan adalah

cukup dekat dengan nilai yang dihitung ( area=2.819). Tentu saja, Anda dapat meningkatkan ini jika kisi Anda lebih halus, dan ambang batas epslebih kecil.