Python数学可视化：显函数、隐函数及复杂曲线的交互式绘图

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.widgets import Slider, RadioButtons, Button
import matplotlib.gridspec as gridspec
from matplotlib import cm
import sympy as sp
from scipy.optimize import fsolve

# 创建图形和布局
fig = plt.figure(figsize=(14, 10))
fig.suptitle('数学函数交互式可视化', fontsize=16, fontweight='bold')

# 使用GridSpec创建复杂布局
gs = gridspec.GridSpec(3, 3, height_ratios=[1, 0.1, 0.1], width_ratios=[1, 0.03, 0.2])
ax = plt.subplot(gs[0, 0])
ax_param = plt.subplot(gs[2, 0])
ax_radio = plt.subplot(gs[0, 2])
ax_color = plt.subplot(gs[0, 1])
ax_reset = plt.subplot(gs[2, 2])

# 设置坐标轴
ax.set_xlim(-5, 5)
ax.set_ylim(-5, 5)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.grid(True, linestyle='--', alpha=0.7)
ax.set_aspect('equal', adjustable='box')
ax.set_facecolor('#f0f0f0')

# 初始函数参数
initial_params = {
    'amplitude': 1.0,
    'frequency': 1.0,
    'phase': 0.0,
    'a': 1.0,
    'b': 1.0,
    'c': 0.5,
    'n': 3,
    'density': 100,
    'color': 'blue'
}

# 定义函数类型
function_types = ['显函数', '隐函数', '参数方程', '极坐标', '3D曲面']
current_function = function_types[0]

# 显函数示例
def explicit_function(x, params):
    a = params['amplitude']
    f = params['frequency']
    p = params['phase']
    return a * np.sin(f * x + p)

# 隐函数示例 (椭圆)
def implicit_function(x, y, params):
    a = params['a']
    b = params['b']
    return (x/a)**2 + (y/b)**2 - 1

# 参数方程示例 (圆)
def parametric_function(t, params):
    a = params['a']
    b = params['b']
    return a * np.cos(t), b * np.sin(t)

# 极坐标函数示例 (玫瑰线)
def polar_function(theta, params):
    n = params['n']
    return np.cos(n * theta)

# 3D函数示例 (双曲面)
def three_d_function(x, y, params):
    a = params['a']
    b = params['b']
    c = params['c']
    return (x**2/a**2) + (y**2/b**2) - (c**2)

# 绘制函数
def plot_function(params, func_type):
    ax.clear()
    ax.set_xlim(-5, 5)
    ax.set_ylim(-5, 5)
    ax.set_xlabel('X')
    ax.set_ylabel('Y')
    ax.grid(True, linestyle='--', alpha=0.7)
    ax.set_aspect('equal', adjustable='box')
    ax.set_facecolor('#f0f0f0')
    
    color = params['color']
    density = int(params['density'])
    
    if func_type == '显函数':
        x = np.linspace(-5, 5, 1000)
        y = explicit_function(x, params)
        ax.plot(x, y, color=color, linewidth=2)
        ax.set_title(f'显函数: y = {params["amplitude"]:.1f}·sin({params["frequency"]:.1f}x + {params["phase"]:.1f})')
    
    elif func_type == '隐函数':
        x = np.linspace(-5, 5, density)
        y = np.linspace(-5, 5, density)
        X, Y = np.meshgrid(x, y)
        Z = implicit_function(X, Y, params)
        contour = ax.contour(X, Y, Z, [0], colors=color, linewidths=2)
        ax.set_title(f'隐函数: (x/{params["a"]:.1f})² + (y/{params["b"]:.1f})² = 1')
    
    elif func_type == '参数方程':
        t = np.linspace(0, 2*np.pi, density)
        x, y = parametric_function(t, params)
        ax.plot(x, y, color=color, linewidth=2)
        ax.set_title(f'参数方程: x = {params["a"]:.1f}·cos(t), y = {params["b"]:.1f}·sin(t)')
    
    elif func_type == '极坐标':
        theta = np.linspace(0, 2*np.pi, density)
        r = polar_function(theta, params)
        x = r * np.cos(theta)
        y = r * np.sin(theta)
        ax.plot(x, y, color=color, linewidth=2)
        ax.set_title(f'极坐标: r = cos({params["n"]}θ)')
    
    elif func_type == '3D曲面':
        # 为3D图创建新的坐标轴
        ax.remove()
        ax_3d = fig.add_subplot(gs[0, 0], projection='3d')
        
        x = np.linspace(-5, 5, density)
        y = np.linspace(-5, 5, density)
        X, Y = np.meshgrid(x, y)
        Z = three_d_function(X, Y, params)
        
        # 创建3D曲面图
        surface = ax_3d.plot_surface(X, Y, Z, cmap=cm.viridis, 
                                    rstride=1, cstride=1, alpha=0.8, 
                                    edgecolor='none', antialiased=True)
        
        # 添加等高线
        ax_3d.contour(X, Y, Z, 10, zdir='z', offset=-5, cmap=cm.viridis, alpha=0.5)
        
        ax_3d.set_xlim(-5, 5)
        ax_3d.set_ylim(-5, 5)
        ax_3d.set_zlim(-5, 5)
        ax_3d.set_xlabel('X')
        ax_3d.set_ylabel('Y')
        ax_3d.set_zlabel('Z')
        ax_3d.set_title(f'3D曲面: (x²/{params["a"]:.1f}²) + (y²/{params["b"]:.1f}²) - {params["c"]:.1f}² = 0')
        ax_3d.grid(True, linestyle='--', alpha=0.7)
        fig.colorbar(surface, ax=ax_3d, shrink=0.5, aspect=10)
        
        return ax_3d
    
    return ax

# 创建滑块
slider_amp = Slider(ax=plt.axes([0.1, 0.05, 0.3, 0.03]), label='振幅', valmin=0.1, valmax=3.0, valinit=initial_params['amplitude'])
slider_freq = Slider(ax=plt.axes([0.1, 0.01, 0.3, 0.03]), label='频率', valmin=0.1, valmax=5.0, valinit=initial_params['frequency'])
slider_phase = Slider(ax=plt.axes([0.5, 0.05, 0.3, 0.03]), label='相位', valmin=-np.pi, valmax=np.pi, valinit=initial_params['phase'])
slider_a = Slider(ax=plt.axes([0.5, 0.01, 0.3, 0.03]), label='参数 a', valmin=0.1, valmax=3.0, valinit=initial_params['a'])
slider_b = Slider(ax=plt.axes([0.1, 0.05, 0.3, 0.03]), label='参数 b', valmin=0.1, valmax=3.0, valinit=initial_params['b'])
slider_c = Slider(ax=plt.axes([0.1, 0.01, 0.3, 0.03]), label='参数 c', valmin=0.1, valmax=3.0, valinit=initial_params['c'])
slider_n = Slider(ax=plt.axes([0.5, 0.05, 0.3, 0.03]), label='参数 n', valmin=1, valmax=10, valinit=initial_params['n'], valstep=1)
slider_density = Slider(ax=plt.axes([0.5, 0.01, 0.3, 0.03]), label='密度', valmin=20, valmax=500, valinit=initial_params['density'], valstep=10)

# 创建颜色选择滑块
ax_color = plt.axes([0.92, 0.25, 0.03, 0.5])
slider_color = Slider(ax=ax_color, label='颜色', valmin=0, valmax=1, valinit=0, 
                     orientation='vertical', color='#ff0000')

# 创建单选按钮
radio = RadioButtons(ax_radio, function_types, active=0)

# 重置按钮
reset_button = Button(ax_reset, '重置参数')

# 初始绘图
plot_function(initial_params, current_function)

# 更新函数
def update(val):
    params = {
        'amplitude': slider_amp.val,
        'frequency': slider_freq.val,
        'phase': slider_phase.val,
        'a': slider_a.val,
        'b': slider_b.val,
        'c': slider_c.val,
        'n': slider_n.val,
        'density': slider_density.val,
        'color': plt.cm.hsv(slider_color.val)
    }
    
    # 根据颜色滑块值设置颜色
    slider_color.ax.set_facecolor(plt.cm.hsv(slider_color.val))
    
    # 根据当前函数类型更新显示
    if current_function == '显函数':
        slider_amp.ax.set_visible(True)
        slider_freq.ax.set_visible(True)
        slider_phase.ax.set_visible(True)
        slider_a.ax.set_visible(False)
        slider_b.ax.set_visible(False)
        slider_c.ax.set_visible(False)
        slider_n.ax.set_visible(False)
    elif current_function == '隐函数':
        slider_amp.ax.set_visible(False)
        slider_freq.ax.set_visible(False)
        slider_phase.ax.set_visible(False)
        slider_a.ax.set_visible(True)
        slider_b.ax.set_visible(True)
        slider_c.ax.set_visible(False)
        slider_n.ax.set_visible(False)
    elif current_function == '参数方程':
        slider_amp.ax.set_visible(False)
        slider_freq.ax.set_visible(False)
        slider_phase.ax.set_visible(False)
        slider_a.ax.set_visible(True)
        slider_b.ax.set_visible(True)
        slider_c.ax.set_visible(False)
        slider_n.ax.set_visible(False)
    elif current_function == '极坐标':
        slider_amp.ax.set_visible(False)
        slider_freq.ax.set_visible(False)
        slider_phase.ax.set_visible(False)
        slider_a.ax.set_visible(False)
        slider_b.ax.set_visible(False)
        slider_c.ax.set_visible(False)
        slider_n.ax.set_visible(True)
    elif current_function == '3D曲面':
        slider_amp.ax.set_visible(False)
        slider_freq.ax.set_visible(False)
        slider_phase.ax.set_visible(False)
        slider_a.ax.set_visible(True)
        slider_b.ax.set_visible(True)
        slider_c.ax.set_visible(True)
        slider_n.ax.set_visible(False)
    
    plot_function(params, current_function)
    fig.canvas.draw_idle()

# 函数类型选择
def select_function(label):
    global current_function
    current_function = label
    update(None)

# 重置函数
def reset(event):
    slider_amp.reset()
    slider_freq.reset()
    slider_phase.reset()
    slider_a.reset()
    slider_b.reset()
    slider_c.reset()
    slider_n.reset()
    slider_density.reset()
    slider_color.reset()

# 连接事件
slider_amp.on_changed(update)
slider_freq.on_changed(update)
slider_phase.on_changed(update)
slider_a.on_changed(update)
slider_b.on_changed(update)
slider_c.on_changed(update)
slider_n.on_changed(update)
slider_density.on_changed(update)
slider_color.on_changed(update)
radio.on_clicked(select_function)
reset_button.on_clicked(reset)

# 初始更新
update(None)

plt.tight_layout(rect=[0, 0.1, 0.95, 0.95])
plt.subplots_adjust(bottom=0.15)
plt.show()

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numpy matplotlib scipy sympy