# scipy.interpolate.insert¶

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.

Parameters
x (u)array_like

A 1-D point at which to insert a new knot(s). 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.

Returns
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.

Notes

Based on algorithms from  and .

Manipulating the tck-tuples directly is not recommended. In new code, prefer using the `BSpline` objects.

References

1

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

2

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

Examples

You can insert knots into a B-spline.

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

A knot is inserted:

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

Some knots are inserted:

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