How to determine the parameters of a parabola passing through given points over a specified interval, if at most one point can be placed on the interval and coverage of at least 70% is required?

To determine the parameters of a parabola passing through given points with the restriction of no more than one point per interval and a coverage requirement of at least 70%, the following approach can be used:

1. Problem formulation:

   • Let there be a set of points ((x_i, y_i)).
   • Find the parameters of the parabola (y = ax^2 + bx + c) that passes through the selected points.
   • No more than one point is placed on each interval.
   • The parabola should cover at least 70% of the points (for example, based on vertical deviation).

2. Algorithm:

   • Divide the domain into intervals.
   • For each interval, select no more than one point to use for approximation.
   • Use the least squares method or another optimization method to fit the parameters (a, b, c), minimizing the error between the parabola and the selected points.
   • Check that the coverage (percentage of points for which the deviation from the parabola does not exceed a specified threshold) is at least 70%.

3. Example implementation in Python:

import numpy as np
from scipy.optimize import curve_fit

# Parabola function
def parabola(x, a, b, c):
    return a * x**2 + b * x + c

# Data
points = np.array([...])  # Array of points (x, y)

# Divide into intervals
intervals = [...]  # List of intervals

# Select points: one per interval
selected_points = []
for interval in intervals:
    pts_in_interval = [p for p in points if interval[0] <= p[0] <= interval[1]]
    if pts_in_interval:
        # Choose a point, e.g., with minimal error or randomly
        selected_points.append(pts_in_interval[0])

x_data = np.array([p[0] for p in selected_points])
y_data = np.array([p[1] for p in selected_points])

# Fit parabola parameters
params, _ = curve_fit(parabola, x_data, y_data)

# Coverage assessment
y_pred = parabola(points[:,0], *params)
errors = np.abs(points[:,1] - y_pred)
threshold = ...  # Set error threshold
coverage = np.mean(errors < threshold)

if coverage >= 0.7:
    print(f"Parabola with parameters {params} covers {coverage*100:.1f}% of points")
else:
    print("The required coverage is not achieved")

Thus, the task reduces to selecting points with constraints and optimizing the parabola parameters to achieve the necessary coverage.