Knot operations and Bézier extraction

The same curve can be written many ways. Knot insertion and degree elevation change a B-spline’s representation – they add control points – without changing the curve it describes. This tutorial demonstrates that invariance, then extracts the per-element Bézier pieces: the element-local representation that finite-element / isogeometric assembly consumes.

These operations are the workhorses of refinement and analysis; the Spaces, knots & element extraction guide explains the underlying theory (continuity, knot multiplicity, and the SpanwiseElementExtraction operator behind to_beziers).

import matplotlib.pyplot as plt
import numpy as np

from pantr import viz
from pantr.bspline import Bspline, BsplineSpace, BsplineSpace1D

space1d = BsplineSpace1D([0, 0, 0, 1, 2, 3, 3, 3], 2)
control_points = np.array([[0.0, 0.0], [1.0, 2.0], [2.0, -1.0], [3.0, 1.5], [4.0, 0.0]])
curve = Bspline(BsplineSpace([space1d]), control_points)

Knot insertion is geometry-preserving

Inserting knots adds control points and shrinks the control polygon toward the curve, but the curve itself is unchanged (to round-off).

refined = curve.insert_knots([0.5, 1.5, 2.5])
u = np.linspace(0.0, 3.0, 200)
err = np.max(np.abs(np.asarray(curve.evaluate(u)) - np.asarray(refined.evaluate(u))))
print(f"max |curve - refined| after knot insertion = {err:.2e}")

orig_cp = curve.control_points
new_cp = refined.control_points
fig, ax = plt.subplots(figsize=(7, 4), constrained_layout=True)
pts = curve.evaluate(u)
ax.plot(pts[:, 0], pts[:, 1], color="navy", lw=2, label="curve (unchanged)")
ax.plot(orig_cp[:, 0], orig_cp[:, 1], "o--", color="0.7", label="original control net")
ax.plot(new_cp[:, 0], new_cp[:, 1], "s-", color="crimson", alpha=0.6, label="refined control net")
ax.legend()
ax.set_aspect("equal")
ax.set_title("Knot insertion refines the net, not the curve")
plt.show()

max |curve - refined| after knot insertion = 4.44e-16

Degree elevation is geometry-preserving too

elevated = curve.elevate_degree(1)  # quadratic -> cubic
err = np.max(np.abs(np.asarray(curve.evaluate(u)) - np.asarray(elevated.evaluate(u))))
print(f"degree {curve.degree[0]} -> {elevated.degree[0]}, max |curve - elevated| = {err:.2e}")

degree 2 -> 3, max |curve - elevated| = 8.88e-16

Bézier extraction

to_beziers decomposes the spline into one Bézier curve per knot span. We render the elements in alternating colours – this is the element-local view used for isogeometric assembly.

beziers = curve.to_beziers()
scene = viz.Scene()
for i, bez in enumerate(beziers.ravel()):
    scene.add(bez, color=("crimson" if i % 2 else "navy"), show_control_polygon=True)
scene.show()
print(f"{beziers.size} Bézier elements")

Static Scene

Interactive Scene

3 Bézier elements