scipy.interpolate.insert(x, tck, m=1, per=0)[source]#

Insert knots into a B-spline.

Given the knots and coefficients of a B-spline representation, create a new B-spline with a knot inserted m times at point x. This is a wrapper around the FORTRAN routine insert of FITPACK.

x (u)float

A knot value at which to insert a new knot. If tck was returned from splprep, then the parameter values, u should be given.

tcka BSpline instance or a tuple

If tuple, then it is expected to be a tuple (t,c,k) containing the vector of knots, the B-spline coefficients, and the degree of the spline.

mint, optional

The number of times to insert the given knot (its multiplicity). Default is 1.

perint, optional

If non-zero, the input spline is considered periodic.

BSpline instance or a tuple

A new B-spline with knots t, coefficients c, and degree k. t(k+1) <= x <= t(n-k), where k is the degree of the spline. In case of a periodic spline (per != 0) there must be either at least k interior knots t(j) satisfying t(k+1)<t(j)<=x or at least k interior knots t(j) satisfying x<=t(j)<t(n-k). A tuple is returned iff the input argument tck is a tuple, otherwise a BSpline object is constructed and returned.


Based on algorithms from [1] and [2].

Manipulating the tck-tuples directly is not recommended. In new code, prefer using the BSpline objects, in particular BSpline.insert_knot method.



W. Boehm, “Inserting new knots into b-spline curves.”, Computer Aided Design, 12, p.199-201, 1980.


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


You can insert knots into a B-spline.

>>> from scipy.interpolate import splrep, insert
>>> import numpy as np
>>> x = np.linspace(0, 10, 5)
>>> y = np.sin(x)
>>> tck = splrep(x, y)
>>> tck[0]
array([ 0.,  0.,  0.,  0.,  5., 10., 10., 10., 10.])

A knot is inserted:

>>> tck_inserted = insert(3, tck)
>>> tck_inserted[0]
array([ 0.,  0.,  0.,  0.,  3.,  5., 10., 10., 10., 10.])

Some knots are inserted:

>>> tck_inserted2 = insert(8, tck, m=3)
>>> tck_inserted2[0]
array([ 0.,  0.,  0.,  0.,  5.,  8.,  8.,  8., 10., 10., 10., 10.])