Python數學可視化——顯函數、隱函數及復雜曲線的交互式繪圖技術

一、引言

在科學計算和數據分析中，函數與方程的可視化是理解數學關系和物理現象的重要工具。本文基于Python的Tkinter和Matplotlib庫，實現一個功能完善的函數與方程可視化工具，支持顯函數、隱函數、特殊曲線（如心形線）及物理場分布（如電勢）的交互式繪圖，并提供安全的表達式解析、圖像保存等功能。

二、核心技術架構

2.1 系統架構與技術選型

• 界面層：使用Tkinter構建GUI，包含類型選擇、表達式輸入、預設函數下拉菜單等控件
• 計算層：
  • 顯函數：通過np.linspace生成采樣點，安全計算函數值
  • 隱函數：基于等高線算法contour繪制等值線
  • 安全機制：通過正則表達式過濾非法字符，限制白名單函數防止代碼注入
• 可視化層：Matplotlib實現圖表渲染，支持動態更新和交互式工具條

2.2 安全表達式解析

def is_valid_expression(expr):"""驗證表達式安全性"""allowed_chars = set("0123456789.+-*/()xy^np_sin_cos_tan_exp_sqrt_log_pi_ ")invalid_chars = set(expr.replace('.', '').replace('_', '')) - allowed_charsif invalid_chars:raise ValueError(f"非法字符: {''.join(invalid_chars)}")# 括號匹配檢查stack = []for char in expr:if char == '(': stack.append(char)elif char == ')':if not stack: raise ValueError("括號不匹配")stack.pop()if stack: raise ValueError("括號不匹配")return Truedef safe_eval(expr, namespace):"""安全執行表達式"""expr = expr.replace('^', '**')  # 替換冪運算符allowed_funcs = {'np': np, 'sin': np.sin, 'cos': np.cos, 'tan': np.tan,'exp': np.exp, 'sqrt': np.sqrt, 'log': np.log, 'pi': np.pi}safe_globals = {"__builtins__": None}safe_locals = {**allowed_funcs, **namespace}compiled_code = compile(expr, '<string>', 'eval')return eval(compiled_code, safe_globals, safe_locals)

三、顯函數可視化

3.1 核心實現

def plot_explicit_function(self, f, x_range, title):"""繪制顯函數"""self.fig.clear()ax = self.fig.add_subplot(111)ax.set_facecolor('white')x = np.linspace(x_range[0], x_range[1], 1000)y = np.array([f(xi) for xi in x])  # 逐點計算防止數組錯誤ax.plot(x, y, 'b-', linewidth=2.5)ax.set_title(title)ax.grid(True, linestyle='--', alpha=0.6)self.optimize_ticks(ax, x_range, (y.min(), y.max()))

3.2 案例演示

案例1：三次函數

# 預設函數定義
self.explicit_presets = {"三次函數": {"func": lambda x: x**3 - 3*x,"expr": "x**3 - 3*x","x_range": (-2.5, 2.5),"title": "三次函數: $y = x^3 - 3x$",}
}

案例2：雙曲線

plot_explicit("1/x", x_range=(-5,5))  # 輸入表達式直接繪制

四、隱函數可視化

4.1 核心實現

def plot_implicit_equation(self, eq, x_range, y_range):"""繪制隱函數F(x,y)=0"""x = np.linspace(x_range[0], x_range[1], 500)y = np.linspace(y_range[0], y_range[1], 500)X, Y = np.meshgrid(x, y)Z = eq(X, Y)self.fig.contour(X, Y, Z, levels=[0], colors='red', linewidths=2.5)self.fig.contourf(X, Y, Z, alpha=0.6)  # 填充色顯示數值分布self.fig.colorbar(label='F(x,y)')

4.2 案例演示

案例1：圓方程

# 預設隱函數
self.implicit_presets["圓"] = {"eq": lambda x, y: x**2 + y**2 - 4,"title": "圓: $x^2 + y^2 = 4$",
}

案例2：笛卡爾葉形線

plot_implicit("x**3 + y**3 - 3*x*y", x_range=(-3,3), y_range=(-3,3))

五、特色曲線與物理應用

5.1 心形線（數學藝術）

def plot_heart_curve(self):"""笛卡爾心形線"""eq = lambda x,y: (x**2 + y**2 -1)**3 - x**2*y**3self.plot_implicit_equation(eq, x_range=(-1.5,1.5), y_range=(-1.5,1.5))self.fig.contourf(..., colors='pink', alpha=0.4)  # 填充愛心區域

5.2 電勢分布（物理應用）

def plot_electric_potential(self):"""點電荷電勢分布"""charges = [{"x":-1,"y":0,"q":1}, {"x":1,"y":0,"q":-1}]x = np.linspace(-2.5,2.5,500)y = np.linspace(-2,2,500)X,Y = np.meshgrid(x,y)V = sum(charge['q']/np.sqrt((X-c['x'])**2 + (Y-c['y'])**2) for c in charges)self.fig.contourf(X,Y,V, cmap='coolwarm')  # 溫度映射顯示電勢self.fig.scatter([c['x']], [c['y']], s=300, c=['red','blue'], marker='+-')

六、交互式GUI設計

6.1 界面布局

def __init__(self, root):self.root = rootself.root.geometry("1200x800")# 左側控制面板left_frame = ttk.LabelFrame(root, text="可視化選項")ttk.Radiobutton(left_frame, text="顯函數", variable=self.viz_type, value="explicit")ttk.Radiobutton(left_frame, text="隱函數", variable=self.viz_type, value="implicit")ttk.Radiobutton(left_frame, text="心形線", variable=self.viz_type, value="heart")# 右側繪圖區域self.canvas = FigureCanvasTkAgg(self.fig, master=right_frame)self.toolbar = NavigationToolbar2Tk(self.canvas, toolbar_frame)  # 集成縮放工具

6.2 動態控件更新

def update_controls(self):"""根據選擇類型顯示對應控件"""if self.viz_type.get() == "explicit":self.explicit_frame.pack()self.update_preset_options(self.explicit_presets.keys())elif self.viz_type.get() == "implicit":self.implicit_frame.pack()self.update_preset_options(self.implicit_presets.keys())# 隱藏其他面板

七、高級功能

7.1 圖像保存

def save_image(self):filename = simpledialog.askstring("保存", "文件名")if filename:self.fig.savefig(f"{filename}.png", dpi=150, bbox_inches="tight")messagebox.showinfo("成功", f"保存至: {os.path.abspath(filename)}")

7.2 公式渲染

def get_function_label(self, expr):"""生成LaTeX公式"""expr = expr.replace('np.sin', '\\sin').replace('**', '^')expr = re.sub(r'(\d)/(\d)', r'\\frac{\1}{\2}', expr)  # 自動轉換分數return f"${expr}$"

八、教學與應用場景

8.1 教學場景

• 基礎數學：演示函數圖像變換（平移、縮放、翻轉）
• 解析幾何：對比顯式方程與隱式方程的幾何意義
• 高等數學：展示參數方程（如心形線）與極坐標方程
• 大學物理：可視化電場、磁場等物理場分布

8.2 擴展方向

1. 支持極坐標繪圖
2. 添加導數/積分可視化
3. 集成3D繪圖功能（使用mpl_toolkits.mplot3d）
4. 開發數據導入功能（CSV/Excel）

九、總結

本文實現的函數可視化工具具備以下特點：

1. 安全性：通過表達式過濾和白名單機制防止代碼注入
2. 交互性：支持實時切換函數類型、調整參數、縮放圖表
3. 擴展性：預設函數與自定義輸入結合，方便擴展新類型
4. 專業性：支持LaTeX公式渲染、物理場可視化等專業需求

該工具可廣泛應用于數學教學、工程仿真、科學研究等領域，幫助用戶快速建立數學表達式與圖形之間的直觀聯系。

十、完整代碼

import tkinter as tk
from tkinter import ttk, messagebox, simpledialog
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.backends.backend_tkagg import FigureCanvasTkAgg, NavigationToolbar2Tk
from matplotlib.figure import Figure
import matplotlib.patches as patches
from matplotlib import ticker
from matplotlib.colors import ListedColormap
import re
import os# 設置matplotlib支持中文顯示
plt.rcParams["font.family"] = ["SimHei","WenQuanYi Micro Hei","Heiti TC","Arial Unicode MS",
]
plt.rcParams["axes.unicode_minus"] = False  # 解決負號顯示問題class FunctionVisualizer:def __init__(self, root):self.root = rootself.root.title("函數與方程可視化工具")self.root.geometry("1200x800")self.root.minsize(1000, 700)# 設置主題顏色self.bg_color = "#f5f5f5"self.frame_color = "#ffffff"self.button_color = "#3b82f6"self.button_text_color = "#ffffff"# 預設函數分組（顯函數/隱函數）self.explicit_presets = {"三次函數": {"func": lambda x: x**3 - 3 * x,"expr": "x**3 - 3*x","x_range": (-2.5, 2.5),"title": "三次函數: $y = x^3 - 3x$",},"雙曲線": {"func": lambda x: 1 / x,"expr": "1/x","x_range": (-5, 5),"title": "雙曲線: $y = \\frac{1}{x}$",},"指數函數": {"func": lambda x: np.exp(x),"expr": "np.exp(x)","x_range": (-3, 3),"title": "指數函數: $y = e^x$",},}self.implicit_presets = {"圓": {"eq": lambda x, y: x**2 + y**2 - 4,"expr": "x**2 + y**2 - 4","x_range": (-3, 3),"y_range": (-3, 3),"title": "圓: $x^2 + y^2 = 4$",},"橢圓": {"eq": lambda x, y: x**2 / 4 + y**2 / 9 - 1,"expr": "x**2/4 + y**2/9 - 1","x_range": (-3, 3),"y_range": (-4, 4),"title": "橢圓: $\\frac{x^2}{4} + \\frac{y^2}{9} = 1$",},"雙曲線(隱式)": {"eq": lambda x, y: x**2 - y**2 - 1,"expr": "x**2 - y**2 - 1","x_range": (-3, 3),"y_range": (-3, 3),"title": "雙曲線: $x^2 - y^2 = 1$",},"笛卡爾葉形線": {"eq": lambda x, y: x**3 + y**3 - 3 * x * y,"expr": "x**3 + y**3 - 3*x*y","x_range": (-3, 3),"y_range": (-3, 3),"title": "笛卡爾葉形線: $x^3 + y^3 = 3xy$",},}# 創建主框架main_frame = ttk.Frame(self.root, padding=10)main_frame.pack(fill=tk.BOTH, expand=True)# 創建左側控制面板left_frame = ttk.LabelFrame(main_frame, text="函數與方程可視化選項", padding=10, width=375)left_frame.pack(side=tk.LEFT, fill=tk.Y, padx=(0, 10))left_frame.pack_propagate(False)  # 固定寬度# 創建右側繪圖區域right_frame = ttk.Frame(main_frame)right_frame.pack(side=tk.RIGHT, fill=tk.BOTH, expand=True)# 創建繪圖區域和工具欄容器self.plot_frame = ttk.Frame(right_frame)self.plot_frame.pack(fill=tk.BOTH, expand=True, padx=5, pady=5)# 初始化繪圖區域self.fig = Figure(figsize=(8, 6), dpi=100)self.canvas = FigureCanvasTkAgg(self.fig, master=self.plot_frame)self.canvas.get_tk_widget().pack(fill=tk.BOTH, expand=True)# 添加工具欄self.toolbar_frame = ttk.Frame(right_frame, height=40)self.toolbar_frame.pack(fill=tk.X, padx=5, pady=(0, 5))self.toolbar = NavigationToolbar2Tk(self.canvas, self.toolbar_frame)self.toolbar.update()# 添加控制選項self.create_controls(left_frame)# 初始顯示self.plot_predefined_function()def create_controls(self, parent):"""創建控制選項"""ttk.Label(parent, text="選擇可視化類型:", font=("SimHei", 10, "bold")).pack(anchor=tk.W, pady=(0, 10))# 可視化類型選擇self.viz_type = tk.StringVar(value="explicit")types = [("顯函數", "explicit"),("隱函數", "implicit"),("心形線", "heart"),("電勢分布", "potential"),]for text, value in types:ttk.Radiobutton(parent,text=text,variable=self.viz_type,value=value,command=self.update_controls,).pack(anchor=tk.W, padx=5, pady=2)# 預設函數下拉菜單（動態更新選項）self.preset_frame = ttk.LabelFrame(parent, text="預設函數", padding=10)self.preset_frame.pack(fill=tk.X, pady=10)# 動態選項變量self.preset_functions = tk.StringVar()self.preset_combobox = ttk.Combobox(self.preset_frame, textvariable=self.preset_functions, width=30)self.preset_combobox.pack(fill=tk.X, pady=5)ttk.Button(self.preset_frame,text="繪制預設函數",command=self.plot_predefined_function,).pack(fill=tk.X, pady=5)# 顯函數輸入self.explicit_frame = ttk.LabelFrame(parent, text="顯函數輸入", padding=10)self.explicit_frame.pack(fill=tk.X, pady=10)ttk.Label(self.explicit_frame, text="函數表達式 (例如 x**2):").pack(anchor=tk.W)self.explicit_entry = ttk.Entry(self.explicit_frame, width=30)self.explicit_entry.insert(0, "x**3 - 3*x")self.explicit_entry.pack(fill=tk.X, pady=5)ttk.Label(self.explicit_frame, text="X范圍 (min,max):").pack(anchor=tk.W)self.x_range_entry = ttk.Entry(self.explicit_frame, width=30)self.x_range_entry.insert(0, "-2.5,2.5")self.x_range_entry.pack(fill=tk.X, pady=5)ttk.Button(self.explicit_frame, text="繪制顯函數", command=self.plot_explicit).pack(fill=tk.X, pady=5)# 隱函數輸入self.implicit_frame = ttk.LabelFrame(parent, text="隱函數輸入", padding=10)self.implicit_frame.pack(fill=tk.X, pady=10)ttk.Label(self.implicit_frame, text="方程表達式 (例如 x**2 + y**2 - 4):").pack(anchor=tk.W)self.implicit_entry = ttk.Entry(self.implicit_frame, width=30)self.implicit_entry.insert(0, "x**3 + y**3 - 3*x*y")self.implicit_entry.pack(fill=tk.X, pady=5)ttk.Label(self.implicit_frame, text="X范圍 (min,max):").pack(anchor=tk.W)self.implicit_x_range_entry = ttk.Entry(self.implicit_frame, width=30)self.implicit_x_range_entry.insert(0, "-3,3")self.implicit_x_range_entry.pack(fill=tk.X, pady=5)ttk.Label(self.implicit_frame, text="Y范圍 (min,max):").pack(anchor=tk.W)self.implicit_y_range_entry = ttk.Entry(self.implicit_frame, width=30)self.implicit_y_range_entry.insert(0, "-3,3")self.implicit_y_range_entry.pack(fill=tk.X, pady=5)ttk.Button(self.implicit_frame, text="繪制隱函數", command=self.plot_implicit).pack(fill=tk.X, pady=5)# 保存圖像按鈕ttk.Button(parent, text="保存圖像", command=self.save_image).pack(side=tk.BOTTOM, pady=10)# 初始更新控件狀態self.update_controls()def update_controls(self):"""更新控件狀態"""viz_type = self.viz_type.get()# 隱藏所有輸入面板self.preset_frame.pack_forget()self.explicit_frame.pack_forget()self.implicit_frame.pack_forget()# 顯示對應面板if viz_type == "explicit":self.explicit_frame.pack(fill=tk.X, pady=10)self.update_preset_options(self.explicit_presets.keys())  # 顯函數預設elif viz_type == "implicit":self.implicit_frame.pack(fill=tk.X, pady=10)self.update_preset_options(self.implicit_presets.keys())  # 隱函數預設elif viz_type == "heart":self.plot_heart_curve()elif viz_type == "potential":self.plot_electric_potential()# 顯示預設框架self.preset_frame.pack(fill=tk.X, pady=10)def update_preset_options(self, options=None):"""動態更新預設函數選項"""if options is None:options = []self.preset_combobox["values"] = list(options)if options:self.preset_functions.set(list(options)[0])  # 默認選擇第一個def plot_predefined_function(self):"""繪制預設函數"""viz_type = self.viz_type.get()selected = self.preset_functions.get()self.fig.clear()ax = self.fig.add_subplot(111)ax.set_facecolor("white")self.fig.set_facecolor("white")if viz_type == "explicit" and selected in self.explicit_presets:data = self.explicit_presets[selected]self.plot_explicit_function(f=data["func"], x_range=data["x_range"], title=data["title"])# 更新顯函數輸入框self.explicit_entry.delete(0, tk.END)self.explicit_entry.insert(0, data["expr"])self.x_range_entry.delete(0, tk.END)self.x_range_entry.insert(0, f"{data['x_range'][0]},{data['x_range'][1]}")elif viz_type == "implicit" and selected in self.implicit_presets:data = self.implicit_presets[selected]self.plot_implicit_equation(eq=data["eq"],x_range=data["x_range"],y_range=data["y_range"],title=data["title"],)# 更新隱函數輸入框self.implicit_entry.delete(0, tk.END)self.implicit_entry.insert(0, data["expr"])self.implicit_x_range_entry.delete(0, tk.END)self.implicit_x_range_entry.insert(0, f"{data['x_range'][0]},{data['x_range'][1]}")self.implicit_y_range_entry.delete(0, tk.END)self.implicit_y_range_entry.insert(0, f"{data['y_range'][0]},{data['y_range'][1]}")self.canvas.draw()def is_valid_expression(self, expr):"""驗證表達式是否為有效的數學表達式"""# 允許的字符：數字、運算符、函數名、xy變量、小數點、括號、空格allowed_chars = set("0123456789.+-*/()xy^np_sin_cos_tan_exp_sqrt_log_pi_ ")# 移除所有允許的字符，檢查是否還有剩余cleaned = expr.replace('.', '').replace('_', '')invalid_chars = set(cleaned) - allowed_charsif invalid_chars:raise ValueError(f"非法字符: {''.join(invalid_chars)}")# 檢查括號匹配stack = []for char in expr:if char == '(':stack.append(char)elif char == ')':if not stack:raise ValueError("括號不匹配：缺少左括號")stack.pop()if stack:raise ValueError("括號不匹配：缺少右括號")return Truedef safe_eval(self, expr, namespace):"""安全地執行表達式計算"""try:self.is_valid_expression(expr)# 替換常見函數別名expr = expr.replace('^', '**')  # 替換^為**# 白名單函數和變量allowed_funcs = {'np': np,'sin': np.sin,'cos': np.cos,'tan': np.tan,'exp': np.exp,'sqrt': np.sqrt,'log': np.log,'pi': np.pi,'arctan2': np.arctan2,}# 創建安全命名空間safe_globals = {"__builtins__": None}safe_locals = {**allowed_funcs, **namespace}# 使用編譯后的代碼提高安全性compiled_code = compile(expr, '<string>', 'eval')return eval(compiled_code, safe_globals, safe_locals)except Exception as e:raise ValueError(f"表達式錯誤: {str(e)}")def plot_explicit(self):"""繪制用戶輸入的顯函數"""try:func_str = self.explicit_entry.get().strip()x_range_str = self.x_range_entry.get().strip()if not func_str or not x_range_str:raise ValueError("請輸入函數表達式和X范圍")# 解析x范圍x_min, x_max = map(float, x_range_str.split(","))if x_min >= x_max:raise ValueError("X范圍的最小值必須小于最大值")# 生成x值x_vals = np.linspace(x_min, x_max, 1000)# 安全計算y值（逐個點計算，避免數組錯誤）y_vals = np.zeros_like(x_vals)for i, x in enumerate(x_vals):y_vals[i] = self.safe_eval(func_str, {'x': x})# 繪制函數self.plot_explicit_function(f=lambda x: y_vals,x_range=(x_min, x_max),title=f"顯函數: $y = {self.get_function_label(func_str)}$",)self.canvas.draw()except Exception as e:messagebox.showerror("錯誤", f"繪制顯函數時出錯: {str(e)}")def plot_implicit(self):"""繪制用戶輸入的隱函數（修復網格點數不匹配問題）"""try:eq_str = self.implicit_entry.get().strip()x_range_str = self.implicit_x_range_entry.get().strip()y_range_str = self.implicit_y_range_entry.get().strip()if not eq_str or not x_range_str or not y_range_str:raise ValueError("請輸入完整的方程表達式和范圍")# 解析范圍x_min, x_max = map(float, x_range_str.split(","))y_min, y_max = map(float, y_range_str.split(","))if x_min >= x_max or y_min >= y_max:raise ValueError("范圍的最小值必須小于最大值")# 創建向量化的方程函數（直接處理數組輸入）eq = lambda X, Y: self.safe_eval(eq_str, {'x': X, 'y': Y})# 調用隱函數繪圖函數，使用默認分辨率500（與函數內部一致）self.plot_implicit_equation(eq=eq,x_range=(x_min, x_max),y_range=(y_min, y_max),title=f"隱函數: ${self.get_function_label(eq_str)} = 0$",)self.canvas.draw()except Exception as e:messagebox.showerror("錯誤", f"繪制隱函數時出錯: {str(e)}")except Exception as e:messagebox.showerror("錯誤", f"繪制隱函數時出錯: {str(e)}")def plot_explicit_function(self, f, x_range=(-5, 5), title="顯函數圖像"):"""繪制顯函數 y = f(x) 的圖像參數:f: 函數對象x_range: x軸范圍title: 圖像標題"""self.fig.clear()ax = self.fig.add_subplot(111)# 設置背景為白色ax.set_facecolor("white")self.fig.set_facecolor("white")# 創建網格和樣式ax.grid(True, linestyle="--", alpha=0.6)ax.spines["left"].set_position("zero")ax.spines["bottom"].set_position("zero")ax.spines["right"].set_visible(False)ax.spines["top"].set_visible(False)# 生成數據x = np.linspace(x_range[0], x_range[1], 1000)try:y = f(x)except Exception as e:messagebox.showerror("函數錯誤", f"計算函數值時出錯: {str(e)}")return# 繪制函數曲線ax.plot(x, y, "b-", linewidth=2.5)# 設置標題和標簽ax.set_title(title, fontsize=16, pad=20)ax.set_xlabel("x", fontsize=12, labelpad=-10, x=1.02)ax.set_ylabel("y", fontsize=12, labelpad=-20, y=1.02, rotation=0)# 優化坐標軸刻度self.optimize_ticks(ax, x_range, (np.min(y), np.max(y)))self.fig.tight_layout()def plot_implicit_equation(self,eq,x_range=(-3, 3),y_range=(-3, 3),resolution=500,levels=[0],cmap="viridis",title="隱函數圖像",):"""繪制隱函數 F(x, y) = 0 的圖像參數:eq: 函數對象x_range, y_range: 繪圖范圍resolution: 網格分辨率levels: 繪制等高線的值cmap: 顏色映射title: 圖像標題"""self.fig.clear()ax = self.fig.add_subplot(111)# 設置背景為白色ax.set_facecolor("white")self.fig.set_facecolor("white")# 創建網格x = np.linspace(x_range[0], x_range[1], resolution)y = np.linspace(y_range[0], y_range[1], resolution)X, Y = np.meshgrid(x, y)# 計算方程值try:Z = eq(X, Y)except Exception as e:messagebox.showerror("方程錯誤", f"計算方程值時出錯: {str(e)}")return# 繪制等高線 (隱函數曲線)contour = ax.contour(X, Y, Z, levels=levels, colors="red", linewidths=2.5)# 添加填充色顯示方程值的變化 (只在需要時)if len(levels) > 1:ax.contourf(X, Y, Z, levels=np.linspace(Z.min(), Z.max(), 100), cmap=cmap, alpha=0.6)# 添加顏色條cbar = self.fig.colorbar(contour)cbar.set_label("F(x, y)", rotation=270, labelpad=20)# 設置網格和樣式ax.grid(True, linestyle="--", alpha=0.4)ax.set_aspect("equal")# 設置標題和標簽ax.set_title(title, fontsize=16, pad=20)ax.set_xlabel("x", fontsize=12)ax.set_ylabel("y", fontsize=12)# 添加零線ax.axhline(0, color="black", linewidth=0.8, alpha=0.7)ax.axvline(0, color="black", linewidth=0.8, alpha=0.7)# 優化坐標軸刻度self.optimize_ticks(ax, x_range, y_range)self.fig.tight_layout()def optimize_ticks(self, ax, x_range, y_range):"""優化坐標軸刻度，避免刻度過于密集"""x_min, x_max = x_rangey_min, y_max = y_range# 根據數據范圍自動設置刻度x_span = x_max - x_miny_span = y_max - y_min# 設置合理的刻度間隔x_major_locator = ticker.MaxNLocator(nbins=7)y_major_locator = ticker.MaxNLocator(nbins=7)ax.xaxis.set_major_locator(x_major_locator)ax.yaxis.set_major_locator(y_major_locator)def plot_heart_curve(self):"""繪制心形線"""self.fig.clear()# 創建圖像和子圖ax1 = self.fig.add_subplot(111)ax1.set_aspect("equal")ax1.set_title("心形線: $(x^2+y^2-1)^3 - x^2y^3 = 0$", fontsize=14)# 設置背景為白色ax1.set_facecolor("white")self.fig.set_facecolor("white")# 定義心形線方程def heart_eq(x, y):return (x**2 + y**2 - 1) ** 3 - x**2 * y**3# 生成網格x = np.linspace(-1.5, 1.5, 500)y = np.linspace(-1.5, 1.5, 500)X, Y = np.meshgrid(x, y)Z = heart_eq(X, Y)# 繪制心形線contour = ax1.contour(X, Y, Z, levels=[0], colors="red", linewidths=3)# 填充顏色ax1.contourf(X, Y, Z, levels=[-1000, 0], colors=["pink"], alpha=0.4)# 添加網格和樣式ax1.grid(True, linestyle="--", alpha=0.3)ax1.set_xlim(-1.5, 1.5)ax1.set_ylim(-1.5, 1.5)# 優化坐標軸刻度self.optimize_ticks(ax1, (-1.5, 1.5), (-1.5, 1.5))self.fig.tight_layout()self.canvas.draw()def plot_electric_potential(self):"""可視化點電荷系統的電勢分布"""self.fig.clear()ax = self.fig.add_subplot(111)# 設置背景為白色ax.set_facecolor("white")self.fig.set_facecolor("white")# 定義兩個點電荷的位置和電荷量charges = [{"x": -1, "y": 0, "q": 1},  # 正電荷{"x": 1, "y": 0, "q": -1},  # 負電荷]# 創建網格x = np.linspace(-2.5, 2.5, 500)y = np.linspace(-2, 2, 500)X, Y = np.meshgrid(x, y)# 計算電勢 (k=1)V = np.zeros_like(X)for charge in charges:r = np.sqrt((X - charge["x"]) ** 2 + (Y - charge["y"]) ** 2)V += charge["q"] / r# 避免除以零V = np.nan_to_num(V, posinf=10, neginf=-10)# 繪制電勢等高線 (使用contourf創建填充等高線)levels = np.linspace(-10, 10, 21)contourf = ax.contourf(X, Y, V, levels=levels, cmap="coolwarm", alpha=0.8)contour = ax.contour(X, Y, V, levels=levels, colors="k", linewidths=0.5)ax.clabel(contour, inline=True, fontsize=8)# 繪制電荷位置for charge in charges:color = "red" if charge["q"] > 0 else "blue"marker = "+" if charge["q"] > 0 else "_"ax.scatter(charge["x"], charge["y"], s=300, c=color, marker=marker, linewidths=2)ax.text(charge["x"],charge["y"] + 0.2,f"{charge['q']}q",ha="center",fontsize=12,weight="bold",)# 設置標題和標簽ax.set_title("兩個點電荷的電勢分布", fontsize=16, pad=20)ax.set_xlabel("x (m)", fontsize=12)ax.set_ylabel("y (m)", fontsize=12)# 添加網格和樣式ax.set_aspect("equal")ax.grid(True, linestyle="--", alpha=0.4)# 添加坐標軸ax.axhline(0, color="k", linewidth=0.8, alpha=0.7)ax.axvline(0, color="k", linewidth=0.8, alpha=0.7)# 添加物理公式ax.text(1.5,1.8,r"$V = \sum \frac{kq_i}{r_i}$",fontsize=14,bbox=dict(facecolor="white", alpha=0.8),)# 添加顏色條cbar = self.fig.colorbar(contourf, label="電勢 (V)")# 優化坐標軸刻度self.optimize_ticks(ax, (-2.5, 2.5), (-2, 2))self.fig.tight_layout()self.canvas.draw()def get_function_label(self, func_str):"""生成函數的LaTeX標簽"""# 安全檢查，防止惡意代碼if any(word in func_str.lower() for word in ["import", "os", "sys", "subprocess"]):raise ValueError("檢測到不安全的代碼")# 直接使用原始字符串，不再進行轉義safe_str = func_str# 替換常見的數學函數replacements = {r'np\.sin\(([^)]+)\)': r'\sin(\1)',r'np\.cos\(([^)]+)\)': r'\cos(\1)',r'np\.tan\(([^)]+)\)': r'\tan(\1)',r'np\.exp\(([^)]+)\)': r'\exp(\1)',r'np\.sqrt\(([^)]+)\)': r'\sqrt{\1}',r'np\.log\(([^)]+)\)': r'\ln(\1)',r'np\.pi': r'\pi',r'\*\*': r'^',r'\*': r'\cdot',}# 應用所有替換，捕獲可能的正則表達式錯誤for pattern, replacement in replacements.items():try:safe_str = re.sub(pattern, replacement, safe_str)except re.error as e:continue  # 跳過有問題的替換# 處理分數 - 更穩健的方法if '/' in safe_str:# 只替換不包含字母的分數表達式if re.search(r'\d+\.?\d*/\d+\.?\d*', safe_str):parts = safe_str.split('/')if len(parts) == 2:numerator = parts[0].strip()denominator = parts[1].strip()safe_str = r'\frac{' + numerator + '}{' + denominator + '}'return safe_strdef save_image(self):"""保存當前圖像"""try:filename = simpledialog.askstring("保存圖像", "請輸入文件名:", initialvalue="function_plot.png")if filename:if not filename.endswith(".png"):filename += ".png"self.fig.savefig(filename, dpi=150, bbox_inches="tight")messagebox.showinfo("成功", f"圖像已保存至: {os.path.abspath(filename)}")except Exception as e:messagebox.showerror("保存錯誤", f"保存圖像時出錯: {e}")def main():root = tk.Tk()# 設置樣式style = ttk.Style()style.configure("TFrame", background="#f5f5f5")style.configure("TLabelframe", background="#ffffff", relief="sunken")style.configure("TLabelframe.Label", background="#ffffff", font=("SimHei", 10, "bold"))style.configure("TButton", padding=5)# 嘗試設置中文字體try:plt.rcParams["font.family"] = ["SimHei"]except:try:plt.rcParams["font.family"] = ["WenQuanYi Micro Hei"]except:try:plt.rcParams["font.family"] = ["Heiti TC"]except:try:plt.rcParams["font.family"] = ["Arial Unicode MS"]except:plt.rcParams["font.family"] = ["DejaVu Sans", "sans-serif"]print("警告: 未找到中文字體，圖表文字可能無法正確顯示")app = FunctionVisualizer(root)root.mainloop()if __name__ == "__main__":main()