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Properties of Kochanek–Bartels Splines

Kochanek–Bartels splines (a.k.a. TCB splines) are interpolating cubic polynomial splines, with three user-defined parameters per vertex (of course they can also be chosen to be the same three values for the whole spline), which can be used to change the shape and velocity of the spline.

These three parameters are called \(T\) for “tension”, \(C\) for “continuity” and \(B\) for “bias”. With the default values of \(C = 0\) and \(B = 0\), a Kochanek–Bartels spline is identical to a cardinal spline. If the “tension” parameter also has its default value \(T = 0\), it is also identical to a Catmull–Rom spline.

TODO: comparison of T with “tension” parameter of cardinal splines

import splines

helper.py

from helper import plot_spline_2d

import matplotlib.pyplot as plt
import numpy as np

def plot_tcb(*tcb, ax=None):
    """Plot four TCB examples."""
    if ax is None:
        ax = plt.gca()
    vertices = [
        (-2.5, 0),
        (-1, 1.5),
        (0, 0.1),
        (1, 1.5),
        (2.5, 0),
        (1, -1.5),
        (0, -0.1),
        (-1, -1.5),
    ]
    for idx, tcb in zip([1, 7, 3, 5], tcb):
        all_tcb = np.zeros((len(vertices), 3))
        all_tcb[idx] = tcb
        s = splines.KochanekBartels(
            vertices, tcb=all_tcb, endconditions='closed')
        label = ', '.join(
            f'{name} = {value}'
            for name, value in zip('TCB', tcb)
            if value)
        plot_spline_2d(s, chords=False, label=label, ax=ax)
    plot_spline_2d(
        splines.KochanekBartels(vertices, endconditions='closed'),
        color='lightgrey', chords=False, ax=ax)
    lines = [l for l in ax.get_lines() if not l.get_label().startswith('_')]
    # https://matplotlib.org/tutorials/intermediate/legend_guide.html#multiple-legends-on-the-same-axes
    ax.add_artist(ax.legend(
        handles=lines[:2], bbox_to_anchor=(0, 0., 0.5, 1),
        loc='center', numpoints=3))
    ax.legend(
        handles=lines[2:], bbox_to_anchor=(0.5, 0., 0.5, 1),
        loc='center', numpoints=3)

Tension

plot_tcb((0.5, 0, 0), (1, 0, 0), (-0.5, 0, 0), (-1, 0, 0))

Continuity

TODO: When \(C_i = 0\), we are back at a Catmull–Rom spline. When \(C_i = -1\), we get a tangent like in a piecewise linear curve. When \(C_i = 1\), we get some weird “inverse corners”.

plot_tcb((0, -0.5, 0), (0, -1, 0), (0, 0.5, 0), (0, 1, 0))

\(T = 1\) and \(C = -1\): similar shape (a.k.a. “image”), different timing:

plot_tcb((1, 0, 0), (0, -1, 0), (0.5, 0, 0), (0, -0.5, 0))

shape in “corners” is similar, but speed is different! with re-parameterization (TODO: link), it doesn’t make too much of a difference.

A value of \(C=-1\) on adjacent vertices leads to linear segments:

vertices1 = [(0, 0), (1, 1), (0, 2), (2, 2), (3, 1), (2, 0)]
s1 = splines.KochanekBartels(vertices1, tcb=(0, -1, 0), endconditions='closed')
plot_spline_2d(s1, chords=False)

Bias

“overshoot”: -1 is full “undershoot”

plot_tcb((0, 0, 0.5), (0, 0, 1), (0, 0, -0.5), (0, 0, -1))

Bias \(-1\) followed by \(+1\) can be used to achieve linear segments between two control points:

vertices2 = [(0, 0), (1.5, 0), (1, 1), (0, 0.5)]
tcb2 = [(0, 0, -1), (0, 0, 1), (0, 0, -1), (0, 0, 1)]
s2 = splines.KochanekBartels(vertices2, tcb=tcb2, endconditions='closed')
plot_spline_2d(s2, chords=False)

A sequence of \(B=-1\), \(C=-1\) and \(B=+1\) can be used to get two adjacent linear segments:

vertices3 = [(0, 0), (1, 0), (0, 0.5)]
tcb3 = [(0, 0, -1), (0, -1, 0), (0, 0, 1)]
s3 = splines.KochanekBartels(vertices3, tcb=tcb3, endconditions='closed')
plot_spline_2d(s3, chords=False)

Combinations

accumulated tension and continuity vs. opposite T and C:

plot_tcb((1, -1, 0), (-1, 1, 0), (-1, -1, 0), (1, 1, 0))

plot_tcb((1, 0, 1), (-1, 0, 1), (0, -1, 1), (0, 1, -1))

TODO: expain non-intuitive cases