100 numpy exercises

100 numpy exercises

A joint effort of the numpy community

The goal is both to offer a quick reference for new and old users and to provide also a set of exercices for those who teach. If you remember having asked or answered a (short) problem, you can send a pull request. The format is:

#. Find indices of non-zero elements from [1,2,0,0,4,0]

   .. code:: python

      # Author: Somebody

      print(np.nonzero([1,2,0,0,4,0]))

Here is what the page looks like so far: http://www.labri.fr/perso/nrougier/teaching/numpy.100/index.html

Repository is at: https://github.com/rougier/numpy-100

Thanks to Michiaki Ariga, there is now a Julia version.

1. Import the numpy package under the name np (★☆☆)

   import numpy as np
   

2. Print the numpy version and the configuration (★☆☆)

   print(np.__version__)
   np.__config__.show()
   

3. Create a null vector of size 10 (★☆☆)

   Z = np.zeros(10)
   print(Z)
   

4. How to get the documentation of the numpy add function from the command line ? (★☆☆)

   python -c "import numpy; numpy.info(numpy.add)"
   

5. Create a null vector of size 10 but the fifth value which is 1 (★☆☆)

   Z = np.zeros(10)
   Z[4] = 1
   print(Z)
   

6. Create a vector with values ranging from 10 to 49 (★☆☆)

   Z = np.arange(10,50)
   print(Z)
   

7. Reverse a vector (first element becomes last) (★☆☆)

   Z = np.arange(50)
   Z = Z[::-1]
   

8. Create a 3x3 matrix with values ranging from 0 to 8 (★☆☆)

   Z = np.arange(9).reshape(3,3)
   print(Z)
   

9. Find indices of non-zero elements from [1,2,0,0,4,0] (★☆☆)

   nz = np.nonzero([1,2,0,0,4,0])
   print(nz)
   

10. Create a 3x3 identity matrix (★☆☆)

    Z = np.eye(3)
    print(Z)
    

11. Create a 3x3x3 array with random values (★☆☆)

    Z = np.random.random((3,3,3))
    print(Z)
    

12. Create a 10x10 array with random values and find the minimum and maximum values (★☆☆)

    Z = np.random.random((10,10))
    Zmin, Zmax = Z.min(), Z.max()
    print(Zmin, Zmax)
    

13. Create a random vector of size 30 and find the mean value (★☆☆)

    Z = np.random.random(30)
    m = Z.mean()
    print(m)
    

14. Create a 2d array with 1 on the border and 0 inside (★☆☆)

    Z = np.ones((10,10))
    Z[1:-1,1:-1] = 0
    

15. What is the result of the following expression ? (★☆☆)

    0 * np.nan
    np.nan == np.nan
    np.inf > np.nan
    np.nan - np.nan
    0.3 == 3 * 0.1
    

16. Create a 5x5 matrix with values 1,2,3,4 just below the diagonal (★☆☆)

    Z = np.diag(1+np.arange(4),k=-1)
    print(Z)
    

17. Create a 8x8 matrix and fill it with a checkerboard pattern (★☆☆)

    Z = np.zeros((8,8),dtype=int)
    Z[1::2,::2] = 1
    Z[::2,1::2] = 1
    print(Z)
    

18. Consider a (6,7,8) shape array, what is the index (x,y,z) of the 100th element ?

    print(np.unravel_index(100,(6,7,8)))
    

19. Create a checkerboard 8x8 matrix using the tile function (★☆☆)

    Z = np.tile( np.array([[0,1],[1,0]]), (4,4))
    print(Z)
    

20. Normalize a 5x5 random matrix (★☆☆)

    Z = np.random.random((5,5))
    Zmax, Zmin = Z.max(), Z.min()
    Z = (Z - Zmin)/(Zmax - Zmin)
    print(Z)
    

21. Multiply a 5x3 matrix by a 3x2 matrix (real matrix product) (★☆☆)

    Z = np.dot(np.ones((5,3)), np.ones((3,2)))
    print(Z)
    

22. Create a 5x5 matrix with row values ranging from 0 to 4 (★★☆)

    Z = np.zeros((5,5))
    Z += np.arange(5)
    print(Z)
    

23. Consider a generator function that generates 10 integers and use it to build an array (★☆☆)

    def generate():
        for x in xrange(10):
            yield x
    Z = np.fromiter(generate(),dtype=float,count=-1)
    print(Z)
    

24. Create a vector of size 10 with values ranging from 0 to 1, both excluded (★★☆)

    Z = np.linspace(0,1,12,endpoint=True)[1:-1]
    print(Z)
    

25. Create a random vector of size 10 and sort it (★★☆)

    Z = np.random.random(10)
    Z.sort()
    print(Z)
    

26. Consider two random array A anb B, check if they are equal (★★☆)

    A = np.random.randint(0,2,5)
    B = np.random.randint(0,2,5)
    equal = np.allclose(A,B)
    print(equal)
    

27. Make an array immutable (read-only) (★★☆)

    Z = np.zeros(10)
    Z.flags.writeable = False
    Z[0] = 1
    

28. Consider a random 10x2 matrix representing cartesian coordinates, convert them to polar coordinates (★★☆)

    Z = np.random.random((10,2))
    X,Y = Z[:,0], Z[:,1]
    R = np.sqrt(X**2+Y**2)
    T = np.arctan2(Y,X)
    print(R)
    print(T)
    

29. Create random vector of size 10 and replace the maximum value by 0 (★★☆)

    Z = np.random.random(10)
    Z[Z.argmax()] = 0
    print(Z)
    

30. Create a structured array with x and y coordinates covering the [0,1]x[0,1] area (★★☆)

    Z = np.zeros((10,10), [('x',float),('y',float)])
    Z['x'], Z['y'] = np.meshgrid(np.linspace(0,1,10),
                                 np.linspace(0,1,10))
    print(Z)
    

31. Print the minimum and maximum representable value for each numpy scalar type (★★☆)

    for dtype in [np.int8, np.int32, np.int64]:
       print(np.iinfo(dtype).min)
       print(np.iinfo(dtype).max)
    for dtype in [np.float32, np.float64]:
       print(np.finfo(dtype).min)
       print(np.finfo(dtype).max)
       print(np.finfo(dtype).eps)
    

32. How to print all the values of an array ? (★★☆)

    np.set_printoptions(threshold=np.nan)
    Z = np.zeros((25,25))
    print(Z)
    

33. How to print all the values of an array ? (★★☆)

    np.set_printoptions(threshold=np.nan)
    Z = np.zeros((25,25))
    print(Z)
    

34. How to find the closest value (to a given scalar) in an array ? (★★☆)

    Z = np.arange(100)
    v = np.random.uniform(0,100)
    index = (np.abs(Z-v)).argmin()
    print(Z[index])
    

35. Create a structured array representing a position (x,y) and a color (r,g,b) (★★☆)

     Z = np.zeros(10, [ ('position', [ ('x', float, 1),
                                       ('y', float, 1)]),
                        ('color',    [ ('r', float, 1),
                                       ('g', float, 1),
                                       ('b', float, 1)])])
    print(Z)
    

36. Consider a random vector with shape (100,2) representing coordinates, find point by point distances (★★☆)

    Z = np.random.random((10,2))
    X,Y = np.atleast_2d(Z[:,0]), np.atleast_2d(Z[:,1])
    D = np.sqrt( (X-X.T)**2 + (Y-Y.T)**2)
    print(D)
    
    # Much faster with scipy
    import scipy
    # Thanks Gavin Heverly-Coulson (#issue 1)
    import scipy.spatial
    
    Z = np.random.random((10,2))
    D = scipy.spatial.distance.cdist(Z,Z)
    print(D)
    

37. How to convert a float (32 bits) array into an integer (32 bits) in place ?

    Z = np.arange(10, dtype=np.int32)
    Z = Z.astype(np.float32, copy=False)
    

38. Consider the following file:

    1,2,3,4,5
    6,,,7,8
    ,,9,10,11
    

    How to read it ? (★★☆)

    Z = np.genfromtxt("missing.dat", delimiter=",")
    

39. What is the equivalent of enumerate for numpy arrays ? (★★☆)

    Z = np.arange(9).reshape(3,3)
    for index, value in np.ndenumerate(Z):
        print(index, value)
    for index in np.ndindex(Z.shape):
        print(index, Z[index])
    

40. Generate a generic 2D Gaussian-like array (★★☆)

    X, Y = np.meshgrid(np.linspace(-1,1,10), np.linspace(-1,1,10))
    D = np.sqrt(X*X+Y*Y)
    sigma, mu = 1.0, 0.0
    G = np.exp(-( (D-mu)**2 / ( 2.0 * sigma**2 ) ) )
    print(G)
    

41. How to randomly place p elements in a 2D array ? (★★☆)

    # Author: Divakar
    
    n = 10
    p = 3
    Z = np.zeros((n,n))
    np.put(Z, np.random.choice(range(n*n), p, replace=False),1)
    

42. Subtract the mean of each row of a matrix (★★☆)

    # Author: Warren Weckesser
    
    X = np.random.rand(5, 10)
    
    # Recent versions of numpy
    Y = X - X.mean(axis=1, keepdims=True)
    
    # Older versions of numpy
    Y = X - X.mean(axis=1).reshape(-1, 1)
    

43. How to I sort an array by the nth column ? (★★☆)

    # Author: Steve Tjoa
    
    Z = np.random.randint(0,10,(3,3))
    print(Z)
    print(Z[Z[:,1].argsort()])
    

44. How to tell if a given 2D array has null columns ? (★★☆)

    # Author: Warren Weckesser
    
    Z = np.random.randint(0,3,(3,10))
    print((~Z.any(axis=0)).any())
    

45. Find the nearest value from a given value in an array (★★☆)

    Z = np.random.uniform(0,1,10)
    z = 0.5
    m = Z.flat[np.abs(Z - z).argmin()]
    print(m)
    

46. Consider a given vector, how to add 1 to each element indexed by a second vector (be careful with repeated indices) ? (★★★)

    # Author: Brett Olsen
    
    Z = np.ones(10)
    I = np.random.randint(0,len(Z),20)
    Z += np.bincount(I, minlength=len(Z))
    print(Z)
    

47. How to accumulate elements of a vector (X) to an array (F) based on an index list (I) ? (★★★)

    # Author: Alan G Isaac
    
    X = [1,2,3,4,5,6]
    I = [1,3,9,3,4,1]
    F = np.bincount(I,X)
    print(F)
    

48. Considering a (w,h,3) image of (dtype=ubyte), compute the number of unique colors (★★★)

    # Author: Nadav Horesh
    
    w,h = 16,16
    I = np.random.randint(0,2,(h,w,3)).astype(np.ubyte)
    F = I[...,0]*256*256 + I[...,1]*256 +I[...,2]
    n = len(np.unique(F))
    print(np.unique(I))
    

49. Considering a four dimensions array, how to get sum over the last two axis at once ? (★★★)

    A = np.random.randint(0,10,(3,4,3,4))
    sum = A.reshape(A.shape[:-2] + (-1,)).sum(axis=-1)
    print(sum)
    

50. Considering a one-dimensional vector D, how to compute means of subsets of D using a vector S of same size describing subset indices ? (★★★)

    # Author: Jaime Fernández del Río
    
    D = np.random.uniform(0,1,100)
    S = np.random.randint(0,10,100)
    D_sums = np.bincount(S, weights=D)
    D_counts = np.bincount(S)
    D_means = D_sums / D_counts
    print(D_means)
    

51. How to get the diagonal of a dot product ? (★★★)

    # Author: Mathieu Blondel
    
    # Slow version
    np.diag(np.dot(A, B))
    
    # Fast version
    np.sum(A * B.T, axis=1)
    
    # Faster version
    np.einsum("ij,ji->i", A, B).
    

52. Consider the vector [1, 2, 3, 4, 5], how to build a new vector with 3 consecutive zeros interleaved between each value ? (★★★)

    # Author: Warren Weckesser
    
    Z = np.array([1,2,3,4,5])
    nz = 3
    Z0 = np.zeros(len(Z) + (len(Z)-1)*(nz))
    Z0[::nz+1] = Z
    print(Z0)
    

53. Consider an array of dimension (5,5,3), how to mulitply it by an array with dimensions (5,5) ? (★★★)

    A = np.ones((5,5,3))
    B = 2*np.ones((5,5))
    print(A * B[:,:,None])
    

54. How to swap two rows of an array ? (★★★)

    # Author: Eelco Hoogendoorn
    
    A = np.arange(25).reshape(5,5)
    A[[0,1]] = A[[1,0]]
    print(A)
    

55. Consider a set of 10 triplets describing 10 triangles (with shared vertices), find the set of unique line segments composing all the triangles (★★★)

    # Author: Nicolas P. Rougier
    
    faces = np.random.randint(0,100,(10,3))
    F = np.roll(faces.repeat(2,axis=1),-1,axis=1)
    F = F.reshape(len(F)*3,2)
    F = np.sort(F,axis=1)
    G = F.view( dtype=[('p0',F.dtype),('p1',F.dtype)] )
    G = np.unique(G)
    print(G)
    

56. Given an array C that is a bincount, how to produce an array A such that np.bincount(A) == C ? (★★★)

    # Author: Jaime Fernández del Río
    
    C = np.bincount([1,1,2,3,4,4,6])
    A = np.repeat(np.arange(len(C)), C)
    print(A)
    

57. How to compute averages using a sliding window over an array ? (★★★)

    # Author: Jaime Fernández del Río
    
    def moving_average(a, n=3) :
        ret = np.cumsum(a, dtype=float)
        ret[n:] = ret[n:] - ret[:-n]
        return ret[n - 1:] / n
    Z = np.arange(20)
    print(moving_average(Z, n=3))
    

58. Consider a one-dimensional array Z, build a two-dimensional array whose first row is (Z[0],Z[1],Z[2]) and each subsequent row is shifted by 1 (last row should be (Z[-3],Z[-2],Z[-1]) (★★★)

    # Author: Joe Kington / Erik Rigtorp
    from numpy.lib import stride_tricks
    
    def rolling(a, window):
        shape = (a.size - window + 1, window)
        strides = (a.itemsize, a.itemsize)
        return stride_tricks.as_strided(a, shape=shape, strides=strides)
    Z = rolling(np.arange(10), 3)
    print(Z)
    

59. How to negate a boolean, or to change the sign of a float inplace ? (★★★)

    # Author: Nathaniel J. Smith
    
    Z = np.random.randint(0,2,100)
    np.logical_not(arr, out=arr)
    
    Z = np.random.uniform(-1.0,1.0,100)
    np.negative(arr, out=arr)
    

60. Consider 2 sets of points P0,P1 describing lines (2d) and a point p, how to compute distance from p to each line i (P0[i],P1[i]) ? (★★★)

    def distance(P0, P1, p):
        T = P1 - P0
        L = (T**2).sum(axis=1)
        U = -((P0[:,0]-p[...,0])*T[:,0] + (P0[:,1]-p[...,1])*T[:,1]) / L
        U = U.reshape(len(U),1)
        D = P0 + U*T - p
        return np.sqrt((D**2).sum(axis=1))
    
    P0 = np.random.uniform(-10,10,(10,2))
    P1 = np.random.uniform(-10,10,(10,2))
    p  = np.random.uniform(-10,10,( 1,2))
    print(distance(P0, P1, p))
    

61. Consider 2 sets of points P0,P1 describing lines (2d) and a set of points P, how to compute distance from each point j (P[j]) to each line i (P0[i],P1[i]) ? (★★★)

    # Author: Italmassov Kuanysh
    # based on distance function from previous question
    P0 = np.random.uniform(-10, 10, (10,2))
    P1 = np.random.uniform(-10,10,(10,2))
    p = np.random.uniform(-10, 10, (10,2))
    print np.array([distance(P0,P1,p_i) for p_i in p])
    

62. Consider an arbitrary array, write a function that extract a subpart with a fixed shape and centered on a given element (pad with a fill value when necessary) (★★★)

    # Author: Nicolas Rougier
    
    Z = np.random.randint(0,10,(10,10))
    shape = (5,5)
    fill  = 0
    position = (1,1)
    
    R = np.ones(shape, dtype=Z.dtype)*fill
    P  = np.array(list(position)).astype(int)
    Rs = np.array(list(R.shape)).astype(int)
    Zs = np.array(list(Z.shape)).astype(int)
    
    R_start = np.zeros((len(shape),)).astype(int)
    R_stop  = np.array(list(shape)).astype(int)
    Z_start = (P-Rs//2)
    Z_stop  = (P+Rs//2)+Rs%2
    
    R_start = (R_start - np.minimum(Z_start,0)).tolist()
    Z_start = (np.maximum(Z_start,0)).tolist()
    R_stop = np.maximum(R_start, (R_stop - np.maximum(Z_stop-Zs,0))).tolist()
    Z_stop = (np.minimum(Z_stop,Zs)).tolist()
    
    r = [slice(start,stop) for start,stop in zip(R_start,R_stop)]
    z = [slice(start,stop) for start,stop in zip(Z_start,Z_stop)]
    R[r] = Z[z]
    print(Z)
    print(R)
    

63. Consider an array Z = [1,2,3,4,5,6,7,8,9,10,11,12,13,14], how to generate an array R = [[1,2,3,4], [2,3,4,5], [3,4,5,6], ..., [11,12,13,14]] ? (★★★)

    # Author: Stefan van der Walt
    
    Z = np.arange(1,15,dtype=uint32)
    R = stride_tricks.as_strided(Z,(11,4),(4,4))
    print(R)
    

64. Compute a matrix rank (★★★)

    # Author: Stefan van der Walt
    
    Z = np.random.uniform(0,1,(10,10))
    U, S, V = np.linalg.svd(Z) # Singular Value Decomposition
    rank = np.sum(S > 1e-10)
    

65. How to find the most frequent value in an array ?

    Z = np.random.randint(0,10,50)
    print(np.bincount(Z).argmax())
    

66. Extract all the contiguous 3x3 blocks from a random 10x10 matrix (★★★)

    # Author: Chris Barker
    
    Z = np.random.randint(0,5,(10,10))
    n = 3
    i = 1 + (Z.shape[0]-3)
    j = 1 + (Z.shape[1]-3)
    C = stride_tricks.as_strided(Z, shape=(i, j, n, n), strides=Z.strides + Z.strides)
    print(C)
    

67. Create a 2D array subclass such that Z[i,j] == Z[j,i] (★★★)

    # Author: Eric O. Lebigot
    # Note: only works for 2d array and value setting using indices
    
    class Symetric(np.ndarray):
        def __setitem__(self, (i,j), value):
            super(Symetric, self).__setitem__((i,j), value)
            super(Symetric, self).__setitem__((j,i), value)
    
    def symetric(Z):
        return np.asarray(Z + Z.T - np.diag(Z.diagonal())).view(Symetric)
    
    S = symetric(np.random.randint(0,10,(5,5)))
    S[2,3] = 42
    print(S)
    

68. Consider a set of p matrices wich shape (n,n) and a set of p vectors with shape (n,1). How to compute the sum of of the p matrix products at once ? (result has shape (n,1)) (★★★)

    # Author: Stefan van der Walt
    
    p, n = 10, 20
    M = np.ones((p,n,n))
    V = np.ones((p,n,1))
    S = np.tensordot(M, V, axes=[[0, 2], [0, 1]])
    print(S)
    
    # It works, because:
    # M is (p,n,n)
    # V is (p,n,1)
    # Thus, summing over the paired axes 0 and 0 (of M and V independently),
    # and 2 and 1, to remain with a (n,1) vector.
    

69. Consider a 16x16 array, how to get the block-sum (block size is 4x4) ? (★★★)

    # Author: Robert Kern
    
    Z = np.ones(16,16)
    k = 4
    S = np.add.reduceat(np.add.reduceat(Z, np.arange(0, Z.shape[0], k), axis=0),
                                           np.arange(0, Z.shape[1], k), axis=1)
    

70. How to implement the Game of Life using numpy arrays ? (★★★)

    # Author: Nicolas Rougier
    
    def iterate(Z):
        # Count neighbours
        N = (Z[0:-2,0:-2] + Z[0:-2,1:-1] + Z[0:-2,2:] +
             Z[1:-1,0:-2]                + Z[1:-1,2:] +
             Z[2:  ,0:-2] + Z[2:  ,1:-1] + Z[2:  ,2:])
    
        # Apply rules
        birth = (N==3) & (Z[1:-1,1:-1]==0)
        survive = ((N==2) | (N==3)) & (Z[1:-1,1:-1]==1)
        Z[...] = 0
        Z[1:-1,1:-1][birth | survive] = 1
        return Z
    
    Z = np.random.randint(0,2,(50,50))
    for i in range(100): Z = iterate(Z)
    

71. How to get the n largest values of an array (★★★)

    Z = np.arange(10000)
    np.random.shuffle(Z)
    n = 5
    
    # Slow
    print (Z[np.argsort(Z)[-n:]])
    
    # Fast
    print (Z[np.argpartition(-Z,n)[:n]])
    

72. Given an arbitrary number of vectors, build the cartesian product (every combinations of every item) (★★★)

    # Author: Stefan Van der Walt
    
    def cartesian(arrays):
        arrays = [np.asarray(a) for a in arrays]
        shape = (len(x) for x in arrays)
    
        ix = np.indices(shape, dtype=int)
        ix = ix.reshape(len(arrays), -1).T
    
        for n, arr in enumerate(arrays):
            ix[:, n] = arrays[n][ix[:, n]]
    
        return ix
    
    print (cartesian(([1, 2, 3], [4, 5], [6, 7])))
    

73. How to create a record array from a regular array ? (★★★)

    Z = np.array([("Hello", 2.5, 3),
                  ("World", 3.6, 2)])
    R = np.core.records.fromarrays(Z.T,
                                   names='col1, col2, col3',
                                   formats = 'S8, f8, i8')
    

74. Comsider a large vector Z, compute Z to the power of 3 using 3 different methods (★★★)

    Author: Ryan G.
    
    x = np.random.rand(5e7)
    
    %timeit np.power(x,3)
    1 loops, best of 3: 574 ms per loop
    
    %timeit x*x*x
    1 loops, best of 3: 429 ms per loop
    
    %timeit np.einsum('i,i,i->i',x,x,x)
    1 loops, best of 3: 244 ms per loop
    

75. Consider two arrays A and B of shape (8,3) and (2,2). How to find rows of A that contain elements of each row of B regardless of the order of the elements in B ? (★★★)

    # Author: Gabe Schwartz
    
    A = np.random.randint(0,5,(8,3))
    B = np.random.randint(0,5,(2,2))
    
    C = (A[..., np.newaxis, np.newaxis] == B)
    rows = (C.sum(axis=(1,2,3)) >= B.shape[1]).nonzero()[0]
    print(rows)
    

76. Considering a 10x3 matrix, extract rows with unequal values (e.g. [2,2,3]) (★★★)

    # Author: Robert Kern
    
    Z = np.random.randint(0,5,(10,3))
    E = np.logical_and.reduce(Z[:,1:] == Z[:,:-1], axis=1)
    U = Z[~E]
    print(Z)
    print(U)
    

77. Convert a vector of ints into a matrix binary representation (★★★)

    # Author: Warren Weckesser
    
    I = np.array([0, 1, 2, 3, 15, 16, 32, 64, 128])
    B = ((I.reshape(-1,1) & (2**np.arange(8))) != 0).astype(int)
    print(B[:,::-1])
    
    # Author: Daniel T. McDonald
    
    I = np.array([0, 1, 2, 3, 15, 16, 32, 64, 128], dtype=np.uint8)
    print(np.unpackbits(I[:, np.newaxis], axis=1))
    

78. Given a two dimensional array, how to extract unique rows ? (★★★)

    # Author: Jaime Fernández del Río
    
    Z = np.random.randint(0,2,(6,3))
    T = np.ascontiguousarray(Z).view(np.dtype((np.void, Z.dtype.itemsize * Z.shape[1])))
    _, idx = np.unique(T, return_index=True)
    uZ = Z[idx]
    print(uZ)
    

79. Considering 2 vectors A & B, write the einsum equivalent of inner, outer, sum, and mul function (★★★)

    # Author: Alex Riley
    # Make sure to read: http://ajcr.net/Basic-guide-to-einsum/
    
    np.einsum('i->', A)       # np.sum(A)
    np.einsum('i,i->i', A, B) # A * B
    np.einsum('i,i', A, B)    # np.inner(A, B)
    np.einsum('i,j', A, B)    # np.outer(A, B)
    

80. Considering a path described by two vectors (X,Y), how to sample it using equidistant samples (★★★) ?

    # Author: Bas Swinckels
    
    phi = np.arange(0, 10*np.pi, 0.1)
    a = 1
    x = a*phi*np.cos(phi)
    y = a*phi*np.sin(phi)
    
    dr = (np.diff(x)**2 + np.diff(y)**2)**.5 # segment lengths
    r = np.zeros_like(x)
    r[1:] = np.cumsum(dr)                # integrate path
    r_int = np.linspace(0, r.max(), 200) # regular spaced path
    x_int = np.interp(r_int, r, x)       # integrate path
    y_int = np.interp(r_int, r, y)