笛卡爾坐標系的任意移動與旋轉詳解

1. 笛卡爾坐標系基礎

笛卡爾坐標系用$(x,y)$表示平面內任意點的位置，原點為$(0,0)$。幾何圖形可視為點的集合。

2. 坐標變換原理

2.1 平移變換

將圖形整體移動$(t_x,t_y)$：
$$\{\begin{array}{l}{x}^{\prime }=x+t_x\\ y^{\prime }=y+t_y\end{array}$$

矩陣形式：
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{t}_x\\ t_y\end{array}\right]$$

2.2 旋轉變換

• 繞原點旋轉（逆時針為正方向）：
  $$\{\begin{array}{l}{x}^{\prime }=x\cos \theta ?y\sin \theta \\ y^{\prime }=x\sin \theta +y\cos \theta \end{array}$$

  矩陣形式：
  $$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & ?\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

• 繞任意點$(c_x,c_y)$旋轉：

  1. 平移使旋轉中心到原點
  2. 繞原點旋轉
  3. 平移回原位置
     $$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & ?\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x?c_x\\ y?c_y\end{array}\right]+\left[\begin{array}{c}{c}_x\\ c_y\end{array}\right]$$

3. 組合變換

復雜變換可分解為平移與旋轉的序列操作，矩陣乘法滿足結合律：
$$T_{\text{total}}=T_{\text{translate}}?R_{\text{rotate}}$$

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Python仿真與動態展示

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from matplotlib.patches import Polygon
import matplotlib# 設置全局字體大小
matplotlib.rcParams.update({'font.size': 12})# 定義三角形頂點 (齊次坐標)
triangle = np.array([[0, 0, 1], [1, 0, 1], [0.5, 1, 1]])# 變換矩陣函數
def translation_matrix(tx, ty):return np.array([[1, 0, tx],[0, 1, ty],[0, 0, 1]])def rotation_matrix(theta):c, s = np.cos(theta), np.sin(theta)return np.array([[c, -s, 0],[s, c, 0],[0, 0, 1]])def rotation_around_point(theta, cx, cy):T1 = translation_matrix(-cx, -cy)R = rotation_matrix(theta)T2 = translation_matrix(cx, cy)return T2 @ R @ T1# 創建三個獨立的畫布
fig1, ax1 = plt.subplots(figsize=(8, 6))
fig2, ax2 = plt.subplots(figsize=(8, 6))
fig3, ax3 = plt.subplots(figsize=(8, 6))# 設置每個畫布的標題和坐標范圍
fig1.suptitle('平移變換演示', fontsize=16)
fig2.suptitle('繞原點旋轉變換演示', fontsize=16)
fig3.suptitle('繞任意點旋轉變換演示', fontsize=16)# 設置坐標范圍
for ax in [ax1, ax2, ax3]:ax.set_xlim(-3, 4)ax.set_ylim(-3, 4)ax.grid(True, linestyle='--', alpha=0.7)ax.set_aspect('equal')ax.set_xlabel('X軸')ax.set_ylabel('Y軸')# 初始化每個畫布的三角形
triangle1 = Polygon(triangle[:, :2], fill=None, edgecolor='blue', linewidth=2.5, alpha=0.9)
triangle2 = Polygon(triangle[:, :2], fill=None, edgecolor='blue', linewidth=2.5, alpha=0.9)
triangle3 = Polygon(triangle[:, :2], fill=None, edgecolor='blue', linewidth=2.5, alpha=0.9)ax1.add_patch(triangle1)
ax2.add_patch(triangle2)
ax3.add_patch(triangle3)# 添加頂點標簽
def add_vertex_labels(ax, vertices):ax.text(vertices[0, 0], vertices[0, 1], 'A', fontsize=14, ha='right', va='bottom', weight='bold', color='darkred', bbox=dict(facecolor='white', alpha=0.7, edgecolor='none', boxstyle='round,pad=0.2'))ax.text(vertices[1, 0], vertices[1, 1], 'B', fontsize=14, ha='left', va='bottom', weight='bold', color='darkgreen', bbox=dict(facecolor='white', alpha=0.7, edgecolor='none', boxstyle='round,pad=0.2'))ax.text(vertices[2, 0], vertices[2, 1], 'C', fontsize=14, ha='center', va='top', weight='bold', color='darkblue', bbox=dict(facecolor='white', alpha=0.7, edgecolor='none', boxstyle='round,pad=0.2'))add_vertex_labels(ax1, triangle)
add_vertex_labels(ax2, triangle)
add_vertex_labels(ax3, triangle)# 創建軌跡線 (每個頂點一種顏色)
trail_length = 50  # 拖尾軌跡長度# 使用漸變色軌跡
traj1_A, = ax1.plot([], [], 'r-', lw=2.0, alpha=1.0, label='A')
traj1_B, = ax1.plot([], [], 'g-', lw=2.0, alpha=1.0, label='B')
traj1_C, = ax1.plot([], [], 'b-', lw=2.0, alpha=1.0, label='C')
ax1.legend(title='頂點軌跡', loc='upper right', fontsize=10)traj2_A, = ax2.plot([], [], 'r-', lw=2.0, alpha=1.0, label='A')
traj2_B, = ax2.plot([], [], 'g-', lw=2.0, alpha=1.0, label='B')
traj2_C, = ax2.plot([], [], 'b-', lw=2.0, alpha=1.0, label='C')
ax2.legend(title='頂點軌跡', loc='upper right', fontsize=10)traj3_A, = ax3.plot([], [], 'r-', lw=2.0, alpha=1.0, label='A')
traj3_B, = ax3.plot([], [], 'g-', lw=2.0, alpha=1.0, label='B')
traj3_C, = ax3.plot([], [], 'b-', lw=2.0, alpha=1.0, label='C')
ax3.legend(title='頂點軌跡', loc='upper right', fontsize=10)# 存儲軌跡數據
traj_data1 = {'A': {'x': [], 'y': []}, 'B': {'x': [], 'y': []}, 'C': {'x': [], 'y': []}}
traj_data2 = {'A': {'x': [], 'y': []}, 'B': {'x': [], 'y': []}, 'C': {'x': [], 'y': []}}
traj_data3 = {'A': {'x': [], 'y': []}, 'B': {'x': [], 'y': []}, 'C': {'x': [], 'y': []}}# 平移動畫更新函數
def update_translation(frame):# 平移參數tx = frame * 0.1ty = frame * 0.05M = translation_matrix(tx, ty)# 應用變換transformed = (M @ triangle.T).T# 更新三角形triangle1.set_xy(transformed[:, :2])# 更新軌跡數據 - 拖尾方式，保留最近50個點# 頂點Atraj_data1['A']['x'].append(transformed[0, 0])traj_data1['A']['y'].append(transformed[0, 1])if len(traj_data1['A']['x']) > trail_length:traj_data1['A']['x'] = traj_data1['A']['x'][-trail_length:]traj_data1['A']['y'] = traj_data1['A']['y'][-trail_length:]# 頂點Btraj_data1['B']['x'].append(transformed[1, 0])traj_data1['B']['y'].append(transformed[1, 1])if len(traj_data1['B']['x']) > trail_length:traj_data1['B']['x'] = traj_data1['B']['x'][-trail_length:]traj_data1['B']['y'] = traj_data1['B']['y'][-trail_length:]# 頂點Ctraj_data1['C']['x'].append(transformed[2, 0])traj_data1['C']['y'].append(transformed[2, 1])if len(traj_data1['C']['x']) > trail_length:traj_data1['C']['x'] = traj_data1['C']['x'][-trail_length:]traj_data1['C']['y'] = traj_data1['C']['y'][-trail_length:]# 更新軌跡線traj1_A.set_data(traj_data1['A']['x'], traj_data1['A']['y'])traj1_B.set_data(traj_data1['B']['x'], traj_data1['B']['y'])traj1_C.set_data(traj_data1['C']['x'], traj_data1['C']['y'])# 更新頂點標簽位置ax1.texts[0].set_position((transformed[0, 0], transformed[0, 1]))ax1.texts[1].set_position((transformed[1, 0], transformed[1, 1]))ax1.texts[2].set_position((transformed[2, 0], transformed[2, 1]))ax1.set_title(f"平移: tx={tx:.1f}, ty={ty:.1f}", fontsize=14)return triangle1, traj1_A, traj1_B, traj1_C# 繞原點旋轉動畫更新函數（360°持續旋轉）
def update_rotation_origin(frame):# 旋轉參數 - 持續旋轉theta = frame * 0.05  # 每幀增加0.05弧度# 計算旋轉矩陣M = rotation_matrix(theta)# 應用變換transformed = (M @ triangle.T).T# 更新三角形triangle2.set_xy(transformed[:, :2])# 更新軌跡數據 - 拖尾方式，保留最近50個點# 頂點Atraj_data2['A']['x'].append(transformed[0, 0])traj_data2['A']['y'].append(transformed[0, 1])if len(traj_data2['A']['x']) > trail_length:traj_data2['A']['x'] = traj_data2['A']['x'][-trail_length:]traj_data2['A']['y'] = traj_data2['A']['y'][-trail_length:]# 頂點Btraj_data2['B']['x'].append(transformed[1, 0])traj_data2['B']['y'].append(transformed[1, 1])if len(traj_data2['B']['x']) > trail_length:traj_data2['B']['x'] = traj_data2['B']['x'][-trail_length:]traj_data2['B']['y'] = traj_data2['B']['y'][-trail_length:]# 頂點Ctraj_data2['C']['x'].append(transformed[2, 0])traj_data2['C']['y'].append(transformed[2, 1])if len(traj_data2['C']['x']) > trail_length:traj_data2['C']['x'] = traj_data2['C']['x'][-trail_length:]traj_data2['C']['y'] = traj_data2['C']['y'][-trail_length:]# 更新軌跡線traj2_A.set_data(traj_data2['A']['x'], traj_data2['A']['y'])traj2_B.set_data(traj_data2['B']['x'], traj_data2['B']['y'])traj2_C.set_data(traj_data2['C']['x'], traj_data2['C']['y'])# 更新頂點標簽位置ax2.texts[0].set_position((transformed[0, 0], transformed[0, 1]))ax2.texts[1].set_position((transformed[1, 0], transformed[1, 1]))ax2.texts[2].set_position((transformed[2, 0], transformed[2, 1]))# 顯示當前角度（轉換為度數）degrees = np.degrees(theta) % 360ax2.set_title(f"繞原點旋轉: {degrees:.0f}°", fontsize=14)return triangle2, traj2_A, traj2_B, traj2_C# 繞任意點旋轉動畫更新函數（360°持續旋轉）
def update_rotation_arbitrary(frame):# 旋轉參數 - 持續旋轉theta = frame * 0.05  # 每幀增加0.05弧度cx, cy = 1.5, 0.5  # 旋轉中心# 標記旋轉中心（只在第一幀標記）if frame == 0:ax3.plot(cx, cy, 'ro', markersize=10, alpha=0.9)ax3.text(cx, cy+0.4, '旋轉中心', ha='center', va='bottom', fontsize=12, weight='bold', color='red',bbox=dict(facecolor='white', alpha=0.7, edgecolor='none', boxstyle='round,pad=0.2'))# 計算繞任意點旋轉的變換矩陣M = rotation_around_point(theta, cx, cy)# 應用變換transformed = (M @ triangle.T).T# 更新三角形triangle3.set_xy(transformed[:, :2])# 更新軌跡數據 - 拖尾方式，保留最近50個點# 頂點Atraj_data3['A']['x'].append(transformed[0, 0])traj_data3['A']['y'].append(transformed[0, 1])if len(traj_data3['A']['x']) > trail_length:traj_data3['A']['x'] = traj_data3['A']['x'][-trail_length:]traj_data3['A']['y'] = traj_data3['A']['y'][-trail_length:]# 頂點Btraj_data3['B']['x'].append(transformed[1, 0])traj_data3['B']['y'].append(transformed[1, 1])if len(traj_data3['B']['x']) > trail_length:traj_data3['B']['x'] = traj_data3['B']['x'][-trail_length:]traj_data3['B']['y'] = traj_data3['B']['y'][-trail_length:]# 頂點Ctraj_data3['C']['x'].append(transformed[2, 0])traj_data3['C']['y'].append(transformed[2, 1])if len(traj_data3['C']['x']) > trail_length:traj_data3['C']['x'] = traj_data3['C']['x'][-trail_length:]traj_data3['C']['y'] = traj_data3['C']['y'][-trail_length:]# 更新軌跡線traj3_A.set_data(traj_data3['A']['x'], traj_data3['A']['y'])traj3_B.set_data(traj_data3['B']['x'], traj_data3['B']['y'])traj3_C.set_data(traj_data3['C']['x'], traj_data3['C']['y'])# 更新頂點標簽位置ax3.texts[0].set_position((transformed[0, 0], transformed[0, 1]))ax3.texts[1].set_position((transformed[1, 0], transformed[1, 1]))ax3.texts[2].set_position((transformed[2, 0], transformed[2, 1]))# 顯示當前角度（轉換為度數）degrees = np.degrees(theta) % 360ax3.set_title(f"繞點({cx},{cy})旋轉: {degrees:.0f}°", fontsize=14)return triangle3, traj3_A, traj3_B, traj3_C# 創建三個獨立的動畫
ani1 = FuncAnimation(fig1, update_translation, frames=30, interval=50, blit=True)
# 旋轉動畫設置為無限循環
ani2 = FuncAnimation(fig2, update_rotation_origin, frames=200, interval=50, blit=True, repeat=True)
ani3 = FuncAnimation(fig3, update_rotation_arbitrary, frames=200, interval=50, blit=True, repeat=True)# 調整窗口位置避免重疊
fig1.canvas.manager.window.wm_geometry("+100+100")
fig2.canvas.manager.window.wm_geometry("+600+100")
fig3.canvas.manager.window.wm_geometry("+1100+100")plt.tight_layout()
plt.show()

動畫說明：

1. 平移階段（0-30幀）：三角形沿向量$(2,1.5)$移動
   

2. 繞原點旋轉（31-60幀）：三角形繞$(0,0)$逆時針旋轉360°
   

3. 繞任意點旋轉（61-90幀）：三角形繞紅點$(1.5,0.8)$逆時針旋轉360°
   

關鍵數學原理：

• 平移是向量加法
• 旋轉是矩陣乘法
• 繞任意點旋轉 = 平移至原點 → 旋轉 → 平移回原位

通過組合基本變換，可實現復雜剛體運動仿真，廣泛應用于機器人學、計算機圖形學等領域。

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