笛卡尔坐标系的任意移动与旋转详解

1. 笛卡尔坐标系基础

笛卡尔坐标系用$(x,y)$表示平面内任意点的位置，原点为$(0,0)$。几何图形可视为点的集合。

2. 坐标变换原理

2.1 平移变换

将图形整体移动$(t_x,t_y)$：
$$\{\begin{array}{l}{x}^{\prime }=x+t_x\\ y^{\prime }=y+t_y\end{array}$$

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

2.2 旋转变换

• 绕原点旋转（逆时针为正方向）：
  $$\{\begin{array}{l}{x}^{\prime }=x\cos \theta -y\sin \theta \\ y^{\prime }=x\sin \theta +y\cos \theta \end{array}$$

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

• 绕任意点$(c_x,c_y)$旋转：

  1. 平移使旋转中心到原点
  2. 绕原点旋转
  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. 组合变换

复杂变换可分解为平移与旋转的序列操作，矩阵乘法满足结合律：
$$T_{\text{total}}=T_{\text{translate}}\cdot R_{\text{rotate}}$$

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Python仿真与动态展示

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from matplotlib.patches import Polygon
import matplotlib# 设置全局字体大小
matplotlib.rcParams.update({'font.size': 12})# 定义三角形顶点 (齐次坐标)
triangle = np.array([[0, 0, 1], [1, 0, 1], [0.5, 1, 1]])# 变换矩阵函数
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# 创建三个独立的画布
fig1, ax1 = plt.subplots(figsize=(8, 6))
fig2, ax2 = plt.subplots(figsize=(8, 6))
fig3, ax3 = plt.subplots(figsize=(8, 6))# 设置每个画布的标题和坐标范围
fig1.suptitle('平移变换演示', fontsize=16)
fig2.suptitle('绕原点旋转变换演示', fontsize=16)
fig3.suptitle('绕任意点旋转变换演示', fontsize=16)# 设置坐标范围
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轴')ax.set_ylabel('Y轴')# 初始化每个画布的三角形
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)# 添加顶点标签
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)# 创建轨迹线 (每个顶点一种颜色)
trail_length = 50  # 拖尾轨迹长度# 使用渐变色轨迹
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='顶点轨迹', 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='顶点轨迹', 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='顶点轨迹', loc='upper right', fontsize=10)# 存储轨迹数据
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': []}}# 平移动画更新函数
def update_translation(frame):# 平移参数tx = frame * 0.1ty = frame * 0.05M = translation_matrix(tx, ty)# 应用变换transformed = (M @ triangle.T).T# 更新三角形triangle1.set_xy(transformed[:, :2])# 更新轨迹数据 - 拖尾方式，保留最近50个点# 顶点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:]# 顶点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:]# 顶点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:]# 更新轨迹线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'])# 更新顶点标签位置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# 绕原点旋转动画更新函数（360°持续旋转）
def update_rotation_origin(frame):# 旋转参数 - 持续旋转theta = frame * 0.05  # 每帧增加0.05弧度# 计算旋转矩阵M = rotation_matrix(theta)# 应用变换transformed = (M @ triangle.T).T# 更新三角形triangle2.set_xy(transformed[:, :2])# 更新轨迹数据 - 拖尾方式，保留最近50个点# 顶点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:]# 顶点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:]# 顶点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:]# 更新轨迹线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'])# 更新顶点标签位置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]))# 显示当前角度（转换为度数）degrees = np.degrees(theta) % 360ax2.set_title(f"绕原点旋转: {degrees:.0f}°", fontsize=14)return triangle2, traj2_A, traj2_B, traj2_C# 绕任意点旋转动画更新函数（360°持续旋转）
def update_rotation_arbitrary(frame):# 旋转参数 - 持续旋转theta = frame * 0.05  # 每帧增加0.05弧度cx, cy = 1.5, 0.5  # 旋转中心# 标记旋转中心（只在第一帧标记）if frame == 0:ax3.plot(cx, cy, 'ro', markersize=10, alpha=0.9)ax3.text(cx, cy+0.4, '旋转中心', ha='center', va='bottom', fontsize=12, weight='bold', color='red',bbox=dict(facecolor='white', alpha=0.7, edgecolor='none', boxstyle='round,pad=0.2'))# 计算绕任意点旋转的变换矩阵M = rotation_around_point(theta, cx, cy)# 应用变换transformed = (M @ triangle.T).T# 更新三角形triangle3.set_xy(transformed[:, :2])# 更新轨迹数据 - 拖尾方式，保留最近50个点# 顶点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:]# 顶点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:]# 顶点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:]# 更新轨迹线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'])# 更新顶点标签位置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]))# 显示当前角度（转换为度数）degrees = np.degrees(theta) % 360ax3.set_title(f"绕点({cx},{cy})旋转: {degrees:.0f}°", fontsize=14)return triangle3, traj3_A, traj3_B, traj3_C# 创建三个独立的动画
ani1 = FuncAnimation(fig1, update_translation, frames=30, interval=50, blit=True)
# 旋转动画设置为无限循环
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)# 调整窗口位置避免重叠
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()

动画说明：

1. 平移阶段（0-30帧）：三角形沿向量$(2,1.5)$移动
   

2. 绕原点旋转（31-60帧）：三角形绕$(0,0)$逆时针旋转360°
   

3. 绕任意点旋转（61-90帧）：三角形绕红点$(1.5,0.8)$逆时针旋转360°
   

关键数学原理：

• 平移是向量加法
• 旋转是矩阵乘法
• 绕任意点旋转 = 平移至原点 → 旋转 → 平移回原位

通过组合基本变换，可实现复杂刚体运动仿真，广泛应用于机器人学、计算机图形学等领域。

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