How to Approximate Data with a Multi-Segment Cubic Bézier Curve Considering Distance and Curvature Constraints?

Approximating Data with a Multi-Segment Cubic Bézier Curve: Incorporating Distance and Curvature Constraints

Problem:
The author seeks an algorithm for approximating given geo data using a multi-segment cubic Bézier curve with two constraints:

1. The Bézier curve must not deviate from the data points by more than a specified distance.
2. The Bézier curve must exhibit curvature within a specified tolerance.

Solution:

The author discovered a solution involving the conversion of a B-Spline that approximates the data in a least squares sense to a multi-segment Bézier curve using the FITPACK library and the Python binding from scipy. The B-Spline representation offers advantages in smoothness control and providing a way to specify the desired smoothness of the approximation.

Algorithm (Simplified):

1. Using the FITPACK library, generate a B-Spline that closely approximates the given geo data in a least square sense.
2. Convert the generated B-Spline into a multi-segment cubic Bézier curve using the provided b_spline_to_bezier_series function.
3. Adjust the smoothness parameter s in splprep to find a good fit that satisfies both distance and curvature constraints.

Implementation:

<code class="python">import matplotlib.pyplot as plt
import numpy as np
from scipy import interpolate

# Read data into lists x and y
tck, u = interpolate.splprep([x, y], s=3)  # Generate B-Spline with smoothness parameter s=3

# Convert B-Spline to Bézier curve
bezier_curves = b_spline_to_bezier_series(tck)

# Evaluate and plot the Bézier curve
unew = np.arange(0, 1.01, 0.01)
out = interpolate.splev(unew, tck)
plt.figure()
plt.plot(x, y, out[0], out[1])
plt.show()</code>

By adjusting the smoothness parameter s, the user can find a curve that satisfies the desired distance and curvature constraints. The provided b_spline_to_bezier_series function converts the B-Spline back into a multi-segment cubic Bézier curve for further analysis or manipulation.

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