高中数学 with Python: 二次曲线

Source: Dev.to

Source:

圆锥曲线概述

圆、抛物线、椭圆和双曲线是四种 圆锥曲线——当平面切割双锥体时产生的曲线。
所得到曲线的形状取决于切割平面与锥体轴之间的夹角。

本程序展示的内容

下面的交互式程序可视化一个圆锥、一个切割平面以及由此产生的二维截面。

• 右侧图中的 图例 显示截面的显式方程。
• 使用滑块更改切割角 β，观察曲线在圆、椭圆、抛物线和双曲线之间的转变。

该代码在 AI 的帮助下生成，旨在用于探索，而非作为一步步的课堂练习。

实时笔记本

# Initial setup for displaying Japanese characters in graphs
!pip install -q japanize-matplotlib
import japanize_matplotlib  # Enables proper display of Japanese text in plots

import numpy as np
import matplotlib.pyplot as plt
from ipywidgets import interact, FloatSlider

def draw_conic_with_equation(beta_deg):
    alpha_deg = 45                     # Fixed cone half‑angle
    alpha = np.radians(alpha_deg)
    beta = np.radians(beta_deg)
    d = 0.5                            # Plane offset from the origin

    fig = plt.figure(figsize=(14, 6))

    # --- Left: 3D cone and cutting plane ---
    ax1 = fig.add_subplot(121, projection='3d')
    r = np.linspace(0, 2, 40)
    theta = np.linspace(0, 2 * np.pi, 40)
    R, THETA = np.meshgrid(r, theta)
    X_cone = R * np.cos(THETA)
    Y_cone = R * np.sin(THETA)
    Z_cone_up = R / np.tan(alpha)
    Z_cone_down = -R / np.tan(alpha)

    X_p = np.linspace(-2, 2, 10)
    Y_p = np.linspace(-2, 2, 10)
    XP, YP = np.meshgrid(X_p, Y_p)
    ZP = XP * np.tan(beta) + d

    ax1.plot_surface(X_cone, Y_cone, Z_cone_up, alpha=0.15, color='cyan')
    ax1.plot_surface(X_cone, Y_cone, Z_cone_down, alpha=0.15, color='cyan')
    ax1.plot_surface(XP, YP, ZP, alpha=0.4, color='orange')
    ax1.set_title("3D View")

    # --- Right: 2D cross section ---
    ax2 = fig.add_subplot(122)

    # Expanded equation of the cross section:
    # y^2 = (tan^2(beta) - 1)·x^2 + (2·d·tan(beta))·x + d^2
    x_range = np.linspace(-4, 4, 1000)
    tan_b = np.tan(beta)
    y_sq = (x_range * tan_b + d)**2 - x_range**2

    valid = y_sq >= 0
    x_plot = x_range[valid]
    y_plot = np.sqrt(y_sq[valid])

    # Coefficients for legend display
    A = tan_b**2 - 1
    B = 2 * d * tan_b
    C = d**2
    eq_label = rf'$y^2 = ({A:.2f})x^2 + ({B:.2f})x + {C:.2f}$'

    ax2.plot(x_plot, y_plot, color='blue', linewidth=2, label=eq_label)
    ax2.plot(x_plot, -y_plot, color='blue', linewidth=2)

    # Determine conic type based on angle β
    if beta_deg == 0:
        shape_type = "Circle"
    elif beta_deg < alpha_deg:
        shape_type = "Ellipse"
    elif beta_deg == alpha_deg:
        shape_type = "Parabola"
    else:
        shape_type = "Hyperbola"

    ax2.set_title(f"2D Cross Section: {shape_type}")
    ax2.set_xlabel('x')
    ax2.set_ylabel('y')
    ax2.set_xlim(-3, 3)
    ax2.set_ylim(-3, 3)
    ax2.grid(True)
    ax2.axhline(0, color='black', lw=1)
    ax2.axvline(0, color='black', lw=1)

    # Legend in the lower‑left corner
    ax2.legend(loc='lower left', fontsize=10)

    plt.tight_layout()
    plt.show()

# Interactive slider for the cutting angle β
interact(
    draw_conic_with_equation,
    beta_deg=FloatSlider(min=0, max=85, step=1, value=0, description='Angle β')
)