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# scipy.interpolate.splprep

`scipy.interpolate.``splprep`(x, w=None, u=None, ub=None, ue=None, k=3, task=0, s=None, t=None, full_output=0, nest=None, per=0, quiet=1)

  Find the B-spline representation of an N-dimensional curve.

  Given a list of N rank-1 arrays, x, which represent a curve in N-dimensional space parametrized by u, find a smooth approximating spline curve g(u). Uses the FORTRAN routine parcur from FITPACK.

  Parameters: | x : array_like A list of sample vector arrays representing the curve. w : array_like, optional Strictly positive rank-1 array of weights the same length as x[0]. The weights are used in computing the weighted least-squares spline fit. If the errors in the x values have standard-deviation given by the vector d, then w should be 1/d. Default is `ones(len(x[0]))`. u : array_like, optional An array of parameter values. If not given, these values are calculated automatically as `M = len(x[0])`, where v[0] = 0 v[i] = v[i-1] + distance(x[i], x[i-1]) u[i] = v[i] / v[M-1] ub, ue : int, optional The end-points of the parameters interval. Defaults to u[0] and u[-1]. k : int, optional Degree of the spline. Cubic splines are recommended. Even values of k should be avoided especially with a small s-value. `1 <= k <= 5`, default is 3. task : int, optional If task==0 (default), find t and c for a given smoothing factor, s. If task==1, find t and c for another value of the smoothing factor, s. There must have been a previous call with task=0 or task=1 for the same set of data. If task=-1 find the weighted least square spline for a given set of knots, t. s : float, optional A smoothing condition. The amount of smoothness is determined by satisfying the conditions: `sum((w * (y - g))**2,axis=0) <= s`, where g(x) is the smoothed interpolation of (x,y). The user can use `s` to control the trade-off between closeness and smoothness of fit. Larger `s` means more smoothing while smaller values of `s` indicate less smoothing. Recommended values of `s` depend on the weights, w. If the weights represent the inverse of the standard-deviation of y, then a good `s` value should be found in the range `(m-sqrt(2*m),m+sqrt(2*m))`, where m is the number of data points in x, y, and w. t : int, optional The knots needed for task=-1. full_output : int, optional If non-zero, then return optional outputs. nest : int, optional An over-estimate of the total number of knots of the spline to help in determining the storage space. By default nest=m/2. Always large enough is nest=m+k+1. per : int, optional If non-zero, data points are considered periodic with period `x[m-1] - x[0]` and a smooth periodic spline approximation is returned. Values of `y[m-1]` and `w[m-1]` are not used. quiet : int, optional Non-zero to suppress messages. This parameter is deprecated; use standard Python warning filters instead.
  Returns: | tck : tuple (t,c,k) a tuple containing the vector of knots, the B-spline coefficients, and the degree of the spline. u : array An array of the values of the parameter. fp : float The weighted sum of squared residuals of the spline approximation. ier : int An integer flag about splrep success. Success is indicated if ier<=0. If ier in [1,2,3] an error occurred but was not raised. Otherwise an error is raised. msg : str A message corresponding to the integer flag, ier.

  See also

  `splrep`, `splev`, `sproot`, `spalde`, `splint`, `bisplrep`, `bisplev`, `UnivariateSpline`, `BivariateSpline`, `BSpline`, `make_interp_spline`

  Notes

  See `splev` for evaluation of the spline and its derivatives. The number of dimensions N must be smaller than 11.

  References

  [R83] | P. Dierckx, “Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing”, 20 (1982) 171-184.

  [R84] | P. Dierckx, “Algorithms for smoothing data with periodic and parametric splines”, report tw55, Dept. Computer Science, K.U.Leuven, 1981.

  [R85] | P. Dierckx, “Curve and surface fitting with splines”, Monographs on Numerical Analysis, Oxford University Press, 1993.

  Examples

  Generate a discretization of a limacon curve in the polar coordinates:

      >>> phi = np.linspace(0, 2.*np.pi, 40)
      >>> r = 0.5 + np.cos(phi)         # polar coords
      >>> x, y = r * np.cos(phi), r * np.sin(phi)    # convert to cartesian

  And interpolate:

      >>> from scipy.interpolate import splprep, splev
      >>> tck, u = splprep([x, y], s=0)
      >>> new_points = splev(u, tck)

  Notice that (i) we force interpolation by using s=0, (ii) the parameterization, `u`, is generated automatically. Now plot the result:

      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots()
      >>> ax.plot(x, y, 'ro')
      >>> ax.plot(new_points[0], new_points[1], 'r-')
      >>> plt.show()

  (Source code)