Smooth spline representation of an arbitrary contour, f(length) --> x,y

Question

Suppose I have a set of x,y coordinates that mark points along contour. Is there a way that I can build a spline representation of the contour that I can evaluate at a parti

Answer 1

You want to use a parametric spline, where instead of interpolating y from the x values, you set up a new parameter, t, and interpolate both y and x from the values of t, using univariate splines for both. How you assign t values to each point affects the result, and using distance, as your question suggest, may be a good idea:

from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
import scipy.interpolate

x = np.array([ 2.,  1.,  1.,  2.,  2.,  4.,  4.,  3.])
y = np.array([ 1.,  2.,  3.,  4.,  2.,  3.,  2.,  1.])
plt.plot(x,y, label='poly')

t = np.arange(x.shape[0], dtype=float)
t /= t[-1]
nt = np.linspace(0, 1, 100)
x1 = scipy.interpolate.spline(t, x, nt)
y1 = scipy.interpolate.spline(t, y, nt)
plt.plot(x1, y1, label='range_spline')

t = np.zeros(x.shape)
t[1:] = np.sqrt((x[1:] - x[:-1])**2 + (y[1:] - y[:-1])**2)
t = np.cumsum(t)
t /= t[-1]
x2 = scipy.interpolate.spline(t, x, nt)
y2 = scipy.interpolate.spline(t, y, nt)
plt.plot(x2, y2, label='dist_spline')

plt.legend(loc='best')
plt.show()

Answer 2

Here is an example using splprep and splev:

import numpy as np
import scipy.interpolate
from matplotlib.pyplot import plot

# x,y coordinates of contour points, not monotonically increasing
x = np.array([2.,  1.,  1.,  2.,  2.,  4.,  4.,  3.])
y = np.array([1.,  2.,  3.,  4.,  2.,  3.,  2.,  1.])

# f: X --> Y might not be a 1:1 correspondence
plot(x, y, '-o')

# get the cumulative distance along the contour
dist = np.sqrt((x[:-1] - x[1:])**2 + (y[:-1] - y[1:])**2)
dist_along = np.concatenate(([0], dist.cumsum()))

# build a spline representation of the contour
spline, u = scipy.interpolate.splprep([x, y], u=dist_along, s=0)

# resample it at smaller distance intervals
interp_d = np.linspace(dist_along[0], dist_along[-1], 50)
interp_x, interp_y = scipy.interpolate.splev(interp_d, spline)
plot(interp_x, interp_y, '-o')