How to Approximate Data with a Multi-Segment Cubic Bezier Curve Constrained by Distance and Curvature?

Approximating Data with a Multi-Segment Cubic Bezier Curve with Distance and Curvature Constraints

Problem Statement:

The goal is to approximate given geographical data points with a multi-segment cubic Bezier curve under two constraints:

1. The maximum distance between the curve and the data points cannot exceed a specified tolerance.
2. The curvature of the curve must not exceed a certain sharpness.

Solution:

A two-step solution is proposed:

1. Create a B-Spline Approximation:

   • Use the FITPACK library (accessed through the scipy Python binding) to generate a B-spline that least-squares fits the data points.
   • B-splines allow for specifying smoothness and provide a way to meet the curvature constraint.

2. Convert B-Spline to Bezier Curve:

   • Use a function like the one provided in the solution text to convert the B-spline into a multi-segment Bezier curve.
   • The converted Bezier curve inherits the smoothness and curvature properties of the B-spline.

Code Example:

Here is a Python snippet demonstrating the approach:

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

# Assume the data points are stored in lists x and y.

# Create B-spline approximation
tck, u = interpolate.splprep([x, y], s=3)  # Adjust s parameter for smoothness

# Generate new parameter values for plotting
unew = np.arange(0, 1.01, 0.01)

# Evaluate B-spline at new parameter values
out = interpolate.splev(unew, tck)

# Convert B-spline to Bezier curve
bezier_points = b_spline_to_bezier_series(tck)

# Plot the data points, B-spline, and Bezier curve
plt.figure()
plt.plot(x, y, out[0], out[1], *bezier_points)  # Replace * with individual Bezier curves
plt.show()</code>

Note:

The solution prioritizes smoothness over accuracy. For tighter approximations, it may be necessary to trade off some smoothness to ensure the distance constraint is met.

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