import numpy as np
import cv2

def fit_curve(points):
    if len(points) < 3:
        return None, None
    xs = [x for x, y in points]
    ys = [y for x, y in points]
    coeffs = np.polyfit(xs, ys, 2)
    return coeffs, np.poly1d(coeffs)

def draw_trajectory_overlay(frame, coeffs, x_range=(0, 1280)):
    if coeffs is None:
        return frame
    a, b, c = coeffs
    points = []
    for x in range(x_range[0], x_range[1], 10):
        y = int(a * x**2 + b * x + c)
        if 0 <= y < frame.shape[0]:
            points.append((x, y))
    if len(points) >= 2:
        cv2.polylines(frame, [np.array(points, np.int32)], False, (255, 0, 0), 2)
    return frame