Question: I have a text file with xy-coordinates called xy.txt.  29.66150677 -98.39336541 29.66150677 -98.39337576 29.66150651 -98.39336541 29.66150328 -98.39337576 29.66150677 -98.39336475 29.66150677 -98.39338611 29.66150393 -98.39338611 29.66150677 -98.39339646 29.66150659 -98.39339646 29.66150677 -98.39339693 29.66151576 -98.39334472 29.66151576 -98.39335506 29.66151511 -98.39334472 29.66151058 -98.39335506 29.66151576 -98.39334322 29.66151576 -98.39336541 29.66151576 -98.39337576 29.66151576 -98.39338611 29.66151576 -98.39339646 29.66151576 -98.39340681 29.66151067 -98.39340681 29.66151576 -98.39341515 29.66152475 -98.39332402 29.66152475 -98.39333437 29.66152443 -98.39332402 29.66151973 -98.39333437 29.66152475 -98.39332332 29.66152475 -98.39334472 29.66152475 -98.39335506 29.66152475 -98.39336541 29.66152475 -98.39337576 29.66152475 -98.39338611 29.66152475 -98.39339646 29.66152475 -98.39340681 29.66152475 -98.39341716 29.66151699 -98.39341716 29.66152475 -98.39342722 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-98.39326192 29.66174059 -98.393242 29.66174059 -98.39327227 29.66174059 -98.39328262 29.66174059 -98.39329297 29.66174059 -98.39330332 29.66174059 -98.39331367 29.66174059 -98.39332402 29.66174059 -98.39333437 29.66174059 -98.39333807 29.66174561 -98.39333437 29.66174958 -98.39325157 29.66174958 -98.39326192 29.66174958 -98.39324293 29.66174958 -98.39327227 29.66174958 -98.39328262 29.66174958 -98.39329297 29.66174958 -98.39330332 29.66174958 -98.39331367 29.66174958 -98.39332402 29.66174958 -98.39333143 29.66175858 -98.39325157 29.66175858 -98.39326192 29.66176663 -98.39325157 29.66176757 -98.39326192 29.66175858 -98.39324585 29.66175858 -98.39327227 29.66175858 -98.39328262 29.66175858 -98.39329297 29.66175858 -98.39330332 29.66175858 -98.39331367 29.66175858 -98.39332402 29.66175858 -98.39332427 29.6617589 -98.39332402 29.66176757 -98.39327227 29.66177412 -98.39326192 29.66177656 -98.39327227 29.66176757 -98.3932525 29.66176757 -98.39328262 29.66176757 -98.39329297 29.66176757 -98.39330332 29.66176757 -98.39331367 29.66177543 -98.39330332 29.66176974 -98.39331367 29.66176757 -98.3933162 29.66177656 -98.39328262 29.66177775 -98.39327227 29.66177872 -98.39328262 29.66177656 -98.39326599 29.66177656 -98.39329297 29.66177855 -98.39329297 29.66177656 -98.39330028   I read the file using import numpy as np xy = np.loadtxt('xy.txt') x, y = xy[:, 0], xy[:, 1]  and I can plot the points with import matplotlib.pyplot as plt plt.plot(x, y, 'o', color='black', markersize=6) plt.show()  Visually the data looks as follows:  I want to retrieve the xy-coordinates of the points that form the boundary of the shape. Answers on similar questions suggest to use Concave Hull. With the help of this blog I write the following code: from scipy import spatial hull = spatial.ConvexHull(xy, incremental=False, qhull_options='Qt') hull_indices = hull.vertices boundary_x = [] boundary_y = [] for i in range(len(hull_indices)): index = hull_indices[i] boundary_x.append(xy[index, 0].astype('float32')) boundary_y.append(xy[index, 1].astype('float32')) plt.plot(boundary_x, boundary_y, 'o', color='red', markersize=6) plt.show()  The output looks like  Clearly the red points are not the boundary points. Some are enclosed by other points and they all are different from the original points. How do I define the boundary points in terms of the original points? Please advice

Answer: When I try to reproduce your code the convexHull function works perfectly. I changed your code so that the positions of black and red circles are rounded in the same way. And I reduced the radius of the red circles so you can better see if everything fits. import numpy as np import matplotlib.pyplot as plt from scipy import spatial xy = np.loadtxt('xy.txt') x, y = xy[:, 0].astype('float64'), xy[:, 1].astype('float64') hull = spatial.ConvexHull(xy, incremental=False, qhull_options='Qt') hull_indices = hull.vertices boundary_x = [] boundary_y = [] for i in range(len(hull_indices)): index = hull_indices[i] boundary_x.append(xy[index, 0].astype('float64')) boundary_y.append(xy[index, 1].astype('float64')) plt.plot(x, y, 'o', color='black', markersize=6) plt.plot(boundary_x, boundary_y, 'o', color='red', markersize=4) plt.show()  Result:  Update: If you need the concave parts of the boundary, too, you can use the python package alphashape instead and calculate the alpha shape. The points in your test data are very close, so I had to normalize the coordinates and adjust the alpha value to get reasonable results. import numpy as np import matplotlib.pyplot as plt import alphashape data = np.loadtxt('xy.txt') xy = (data + [-29.0, 98.0]) * 10 x, y = xy[:, 0], xy[:, 1] shape = alphashape.alphashape(xy, alpha=500.0) shape_x, shape_y = shape.exterior.coords.xy plt.plot(x, y, 'o', color='black', markersize=6) plt.plot(shape_x, shape_y, 'o', color='red', markersize=4) plt.show()  Result: