关于椭圆的方程(有Python画的动图)
flyfish
几何定义
椭圆是平面上所有到两个固定点(焦点)的距离之和为常数的点的集合。这两个固定点叫做焦点。
解析几何描述
设椭圆的两个焦点为 F 1 F_1 F1 和 F 2 F_2 F2,焦距(两焦点之间的距离的一半)为 c c c,长轴的半长轴为 a a a,短轴的半短轴为 b b b,椭圆上任意一点到这两个焦点的距离之和是一个常数 2 a 2a 2a。如果椭圆的中心在原点,长轴平行于 x x x 轴,则椭圆的标准方程为: x 2 a 2 + y 2 b 2 = 1 \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 a2x2+b2y2=1如果长轴平行于 y y y 轴,只需交换 a a a 和 b b b 的位置: x 2 b 2 + y 2 a 2 = 1 \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 b2x2+a2y2=1
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.animation import FuncAnimation, PillowWriterdef plot_ellipse_with_moving_point(a, b, num_frames=100, interval=50):# 椭圆方程参数theta = np.linspace(0, 2 * np.pi, num_frames)x = a * np.cos(theta)y = b * np.sin(theta)# 焦点位置c = np.sqrt(a**2 - b**2)F1 = (-c, 0)F2 = (c, 0)# 创建图形fig, ax = plt.subplots(figsize=(8, 6))ax.plot(x, y, label=f'Ellipse: $\\frac{{x^2}}{{{a}^2}} + \\frac{{y^2}}{{{b}^2}} = 1$')ax.scatter(*F1, color='red')ax.scatter(*F2, color='red')ax.text(F1[0], F1[1], 'F1', fontsize=12, ha='right')ax.text(F2[0], F2[1], 'F2', fontsize=12, ha='left')ax.axhline(0, color='black', linewidth=0.5)ax.axvline(0, color='black', linewidth=0.5)ax.grid(color='gray', linestyle='--', linewidth=0.5)ax.set_aspect('equal', adjustable='box')ax.set_title('Ellipse with Moving Point')ax.set_xlabel('x')ax.set_ylabel('y')ax.legend()# 初始化点 P 和连接线point, = ax.plot([], [], 'bo')line1, = ax.plot([], [], 'gray', linestyle='dotted')line2, = ax.plot([], [], 'gray', linestyle='dotted')# 初始化函数def init():point.set_data([], [])line1.set_data([], [])line2.set_data([], [])return point, line1, line2# 更新函数def update(frame):P = (a * np.cos(theta[frame]), b * np.sin(theta[frame]))point.set_data([P[0]], [P[1]])line1.set_data([F1[0], P[0]], [F1[1], P[1]])line2.set_data([F2[0], P[0]], [F2[1], P[1]])return point, line1, line2# 创建动画ani = FuncAnimation(fig, update, frames=num_frames, init_func=init, interval=interval, blit=True)# 保存动画ani.save('ellipse_with_moving_point.gif', writer=PillowWriter(fps=20))plt.show()# 参数
a = 5
b = 3
plot_ellipse_with_moving_point(a, b)