This page was generated from doc/euclidean/natural-properties.ipynb. Interactive online version: .

Properties of Natural Splines

The most important property of natural splines is that they are \(C^2\) continuous, which means that the second derivatives match at section borders.

import splines

vertices = [
    (0, 0),
    (1, 0),
    (2, 1),
    (3, 1),
]

We use the class splines.Natural …

s = splines.Natural(vertices)

… and a plotting function from helper.py:

from helper import plot_spline_2d

plot_spline_2d(s)

def plot_natural(*args, **kwargs):
    plot_spline_2d(splines.Natural(*args, **kwargs), chords=False)

A downside of natural splines is that they don’t provide local control. Changing only a single control point potentially influences the whole curve.

plot_natural([
    (0, 0),
    (0.5, 0),
    (2, -1),
    (3, 2),
    (1, 3),
    (-2, 2),
])
plot_natural([
    (0, 0),
    (0.5, 0),
    (2, -0.5),
    (3, 2),
    (1, 3),
    (-2, 2),
])

By default, natural end conditions are used, but alternatively, the end tangents can be clamped to given values.

plot_natural(vertices, endconditions='natural')
plot_natural(vertices, endconditions=[[0, 0], 'natural'])
plot_natural(vertices, endconditions=[[1, -1], 'natural'])
plot_natural(vertices, endconditions=[[2, -2], 'natural'])

plot_natural(vertices, endconditions='closed')

plot_natural(vertices, endconditions='closed', alpha=0.5)