### 计算两个平面的水平夹角和垂直夹角 在三维空间中，两个平面之间的夹角可以通过它们法向量之间的夹角来计算。假设两个平面 \( P_1 \) 和 \( P_2 \) 的法向量分别为 \( \mathbf{n}_1 = (a_1, b_1, c_1) \) 和 \( \mathbf{n}_2 = (a_2, b_2, c_2) \)，则这两个平面之间的夹角 \( \theta \) 可以通过以下公式计算： \[ \cos(\theta) = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|} \] 其中： - \( \mathbf{n}_1 \cdot \mathbf{n}_2 = a_1a_2 + b_1b_2 + c_1c_2 \) - \( \|\mathbf{n}_1\| = \sqrt{a_1^2 + b_1^2 + c_1^2} \) - \( \|\mathbf{n}_2\| = \sqrt{a_2^2 + b_2^2 + c_2^2} \) 水平夹角和垂直夹角可以分别通过分解法向量到水平面（XY 平面）和垂直方向（Z 轴）来计算[^1]。 以下是 Python 实现代码： ```python import numpy as np def calculate_angle_between_planes(normal_vector1, normal_vector2): """ Calculate the angle between two planes given their normal vectors. Parameters: normal_vector1 (list or tuple): Normal vector of the first plane [a1, b1, c1]. normal_vector2 (list or tuple): Normal vector of the second plane [a2, b2, c2]. Returns: dict: A dictionary containing horizontal and vertical angles in degrees. """ # Convert to numpy arrays for easier calculations n1 = np.array(normal_vector1, dtype=float) n2 = np.array(normal_vector2, dtype=float) # Normalize the vectors n1_normalized = n1 / np.linalg.norm(n1) n2_normalized = n2 / np.linalg.norm(n2) # Calculate the dot product dot_product = np.dot(n1_normalized, n2_normalized) # Calculate the angle between the planes angle_rad = np.arccos(np.clip(dot_product, -1.0, 1.0)) angle_deg = np.degrees(angle_rad) # Decompose into horizontal and vertical components horizontal_angle_rad = np.arctan2( n1_normalized[1] * n2_normalized[0] - n1_normalized[0] * n2_normalized[1], n1_normalized[0] * n2_normalized[0] + n1_normalized[1] * n2_normalized[1] ) horizontal_angle_deg = np.degrees(horizontal_angle_rad) vertical_angle_rad = np.arccos( (n1_normalized[2] * n2_normalized[2]) / (np.sqrt(n1_normalized[2]**2) * np.sqrt(n2_normalized[2]**2)) ) vertical_angle_deg = np.degrees(vertical_angle_rad) return { "angle_between_planes": angle_deg, "horizontal_angle": horizontal_angle_deg, "vertical_angle": vertical_angle_deg } # Example usage normal_vector1 = [1, 0, 0] # Normal vector of plane 1 normal_vector2 = [0, 1, 0] # Normal vector of plane 2 result = calculate_angle_between_planes(normal_vector1, normal_vector2) print("Angle between planes:", result["angle_between_planes"], "degrees") print("Horizontal angle:", result["horizontal_angle"], "degrees") print("Vertical angle:", result["vertical_angle"], "degrees") ``` ### 解释 - `calculate_angle_between_planes` 函数接受两个平面的法向量作为输入，并返回它们之间的夹角、水平夹角和垂直夹角。 - 水平夹角通过投影到 XY 平面计算，而垂直夹角通过投影到 Z 轴计算。 - 使用了 NumPy 库来进行矢量化操作和数学计算。