Basis polynomials

Inputs

This example evaluates five cubic B-spline basis functions, together with their first and second derivatives. n=4 is the highest index, so there are n+1=5 functions; p=3 is their degree. The knot vector has n+p+2=9 entries:

[0, 0, 0, 0, 0.5, 1, 1, 1, 1]

Four repeated knots at each end clamp the basis. The single interior knot at \(u=0.5\) gives two spans with \(C^2\) continuity. The equations are developed in B-spline theory.

Complete script

Run python demos/documentation/basis_polynomials.py, or download the script.

"""Plot cubic B-spline basis functions and their first two derivatives."""
import numpy as np
import matplotlib.pyplot as plt
import nurbspy as nrb

nrb.set_plot_options()

n, p = 4, 3  # Five basis functions, each of degree three.
U = np.array([0., 0., 0., 0., 0.5, 1., 1., 1., 1.])
u = np.linspace(0., 1., 501)
N = nrb.compute_basis_polynomials(n, p, U, u)
dN = nrb.compute_basis_polynomials_derivatives(n, p, U, u, 1)
ddN = nrb.compute_basis_polynomials_derivatives(n, p, U, u, 2)

print("Basis array shape:", N.shape)
print("Basis at u=0.5:", N[:, 250])
print(f"Maximum partition-of-unity error: {np.max(np.abs(N.sum(axis=0) - 1)):.2e}")
print(f"Maximum first-derivative sum: {np.max(np.abs(dN.sum(axis=0))):.2e}")
print(f"Maximum second-derivative sum: {np.max(np.abs(ddN.sum(axis=0))):.2e}")

fig, axes = plt.subplots(1, 3, figsize=(12, 3.6), layout="constrained")
for ax, values, title in zip(axes, [N, dN, ddN],
                             ["Basis functions", "First derivatives", "Second derivatives"]):
    for i, row in enumerate(values):
        ax.plot(u, row, label=f"i={i}")
    ax.axvline(0.5, color="0.7", linestyle=":", linewidth=1)
    ax.set(xlabel="u", ylabel="Value", title=title)
    ax.grid(alpha=0.2)
axes[0].legend(fontsize=10)
plt.show()

Output

N, dN, and ddN all have shape (5, 501). Each row holds one basis function over the parameter samples. At the middle sample:

Basis array shape: (5, 501)
Basis at u=0.5: [0.   0.25 0.5  0.25 0.  ]

The basis values sum to one. Their first and second derivatives sum to zero, up to floating-point roundoff, because they differentiate that constant sum.

Values, slopes, and second derivatives across the two knot spans.

The first and last bases interpolate the ends. At most four cubic bases are nonzero inside a span. To obtain Bernstein polynomials, set p=n and use n+1 zeros followed by n+1 ones.