在OpenCV中找到直线和椭圆的交点

Question

我试图找到与直线和椭圆相交的点。我可以如代码中所示手动找到该点，但如何自动找到它。此外，当红点位于椭圆内部时，延伸直线时绿点也应投影到椭圆上。我该如何解决这个问题？代码示例表示赞赏！谢谢！

import cv2
import numpy as np

# Parameters
ellipse_center = np.array([238, 239])
ellipse_axes = np.array([150, 100])  # Semi-major and semi-minor axes
ellipse_angle = np.radians(70)  # Angle of rotation (in radians)

line_point1 = ellipse_center  # Point on the line
line_point2 = np.array([341, 125])  # Another point on the line (moving point)

# Initialize OpenCV window
window_name = "Projecting Point on Rotated Ellipse"
cv2.namedWindow(window_name)
canvas = np.zeros((400, 400, 3), dtype=np.uint8)
    
while True:
    # Clear canvas
    canvas.fill(0)

    # Draw the rotated ellipse
    cv2.ellipse(canvas, tuple(ellipse_center), tuple(ellipse_axes), np.degrees(ellipse_angle), 0, 360, (255, 255, 255), 2)

    # Draw the moving point
    cv2.circle(canvas, (int(line_point2[0]), int(line_point2[1])), 5, (0, 0, 255), -1)

    # Draw the center point of the ellipse
    cv2.circle(canvas, tuple(ellipse_center), 5, (0, 255, 255), -1)

    # Draw the line between moving point and the center of the ellipse
    cv2.line(canvas, (int(line_point2[0]), int(line_point2[1])), tuple(ellipse_center), (0, 0, 255), 1)

    intersection_points = (310, 159) # <- How to find this?
    cv2.circle(canvas, intersection_points, 5, (0, 255, 0), -1)

    # Show canvas
    cv2.imshow(window_name, canvas)

    # Exit if 'q' is pressed
    if cv2.waitKey(1) & 0xFF == ord('q'):
        break

# Close OpenCV window
cv2.destroyAllWindows()

当使用椭圆方程和直线方程时，我得到：

import cv2
import numpy as np
from scipy.optimize import fsolve

# Parameters
ellipse_center = np.array([238, 239])
ellipse_axes = np.array([150, 100])  # Semi-major and semi-minor axes
ellipse_angle = np.radians(70)  # Angle of rotation (in radians)

line_point1 = ellipse_center  # Point on the line
line_point2 = np.array([341, 125])  # Another point on the line (moving point)

# Ellipse equation: (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) is the center of the ellipse
def ellipse_eq(params):
    x, y = params
    h, k = ellipse_center
    a, b = ellipse_axes
    return ((x - h) ** 2 / a ** 2) + ((y - k) ** 2 / b ** 2) - 1

# Line equation: y = mx + c
def line_eq(x):
    m = (line_point2[1] - line_point1[1]) / (line_point2[0] - line_point1[0])
    c = line_point1[1] - m * line_point1[0]
    return m * x + c

# Intersection of the ellipse and the line
def intersection_point():
    x_intercept = fsolve(lambda x: ellipse_eq([x, line_eq(x)]), line_point2[0])
    y_intercept = line_eq(x_intercept)
    return x_intercept[0], y_intercept[0]

# Initialize OpenCV window
window_name = "Projecting Point on Rotated Ellipse"
cv2.namedWindow(window_name)
canvas = np.zeros((400, 400, 3), dtype=np.uint8)

while True:
    # Clear canvas
    canvas.fill(0)

    # Draw the rotated ellipse
    cv2.ellipse(canvas, tuple(ellipse_center), tuple(ellipse_axes), np.degrees(ellipse_angle), 0, 360, (255, 255, 255), 2)

    # Draw the moving point
    cv2.circle(canvas, (int(line_point2[0]), int(line_point2[1])), 5, (0, 0, 255), -1)

    # Draw the center point of the ellipse
    cv2.circle(canvas, tuple(ellipse_center), 5, (0, 255, 255), -1)

    # Draw the line between moving point and the center of the ellipse
    cv2.line(canvas, (int(line_point2[0]), int(line_point2[1])), tuple(ellipse_center), (0, 0, 255), 1)

    # Find and draw the intersection point
    intersection_points = intersection_point()
    cv2.circle(canvas, (int(intersection_points[0]), int(intersection_points[1])), 5, (0, 255, 0), -1)

    # Show canvas
    cv2.imshow(window_name, canvas)

    # Exit if 'q' is pressed
    if cv2.waitKey(1) & 0xFF == ord('q'):
        break

# Close OpenCV window
cv2.destroyAllWindows()

这并不能解决我的问题。我在这里做错了什么？

Answer 1

将直线方程 (y=mx+c) 代入椭圆方程 (x-h)²/a²+(y-k)²/b²=1（y）。这将给出 x 的二次方程，可以通过通常的二次方程公式求解。

二次方为 Ax²+Bx+c=0，其中，在重新排列时，

A=a²m²+b²

B=2a²(c-k)-2b²h

C=b²h²+a²(c-k)²-a²b²

求解 x 并从直线方程中得到 y。

与任何二次方程一样，您可能会遇到具有单根（直线刚刚接触椭圆）和无根（直线错过椭圆）的情况。