Find the B-spline representation of an N-D curve.
Given a list of N rank-1 arrays, `x`, which represent a curve in N-D 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 `x0`. 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(x0))``. u : array_like, optional An array of parameter values. If not given, these values are calculated automatically as ``M = len(x0)``, where
v0 = 0
vi = vi-1 + distance(`xi`, `xi-1`)
ui = vi / vM-1
ub, ue : int, optional The end-points of the parameters interval. Defaults to u0 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 ``xm-1 - x0`` and a smooth periodic spline approximation is returned. Values of ``ym-1`` and ``wm-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.
The number of coefficients in the `c` array is ``k+1`` less then the number of knots, ``len(t)``. This is in contrast with `splrep`, which zero-pads the array of coefficients to have the same length as the array of knots. These additional coefficients are ignored by evaluation routines, `splev` and `BSpline`.
References ---------- .. 1 P. Dierckx, 'Algorithms for smoothing data with periodic and parametric splines, Computer Graphics and Image Processing', 20 (1982) 171-184. .. 2 P. Dierckx, 'Algorithms for smoothing data with periodic and parametric splines', report tw55, Dept. Computer Science, K.U.Leuven, 1981. .. 3 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_points0, new_points1, 'r-') >>> plt.show()