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543 lines
12 KiB
Markdown
543 lines
12 KiB
Markdown
---
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title: mbcp.mp_math.plane
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---
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### **class** `Plane3`
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### *method* `__init__(self, a: float, b: float, c: float, d: float)`
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**Description**: 平面方程：ax + by + cz + d = 0
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**Arguments**:
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> - a: x系数
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> - b: y系数
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> - c: z系数
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> - d: 常数项
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def __init__(self, a: float, b: float, c: float, d: float):
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"""
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平面方程：ax + by + cz + d = 0
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Args:
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a: x系数
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b: y系数
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c: z系数
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d: 常数项
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"""
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self.a = a
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self.b = b
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self.c = c
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self.d = d
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```
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</details>
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### *method* `approx(self, other: Plane3) -> bool`
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**Description**: 判断两个平面是否近似相等。
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**Return**: 是否近似相等
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**Arguments**:
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> - other: 另一个平面
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def approx(self, other: 'Plane3') -> bool:
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"""
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判断两个平面是否近似相等。
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Args:
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other: 另一个平面
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Returns:
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是否近似相等
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"""
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if self.a != 0:
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k = other.a / self.a
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return approx(other.b, self.b * k) and approx(other.c, self.c * k) and approx(other.d, self.d * k)
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elif self.b != 0:
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k = other.b / self.b
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return approx(other.a, self.a * k) and approx(other.c, self.c * k) and approx(other.d, self.d * k)
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elif self.c != 0:
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k = other.c / self.c
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return approx(other.a, self.a * k) and approx(other.b, self.b * k) and approx(other.d, self.d * k)
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else:
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return False
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```
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</details>
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### *method* `cal_angle(self, other: Line3 | Plane3) -> AnyAngle`
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**Description**: 计算平面与平面之间的夹角。
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**Return**: 夹角弧度
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**Arguments**:
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> - other: 另一个平面
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**Raises**:
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> - TypeError 不支持的类型
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def cal_angle(self, other: 'Line3 | Plane3') -> 'AnyAngle':
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"""
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计算平面与平面之间的夹角。
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Args:
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other: 另一个平面
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Returns:
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夹角弧度
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Raises:
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TypeError: 不支持的类型
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"""
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if isinstance(other, Line3):
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return self.normal.cal_angle(other.direction).complementary
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elif isinstance(other, Plane3):
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return AnyAngle(math.acos(self.normal @ other.normal / (self.normal.length * other.normal.length)), is_radian=True)
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else:
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raise TypeError(f'Unsupported type: {type(other)}')
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```
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</details>
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### *method* `cal_distance(self, other: Plane3 | Point3) -> float`
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**Description**: 计算平面与平面或点之间的距离。
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**Return**: 距离
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**Arguments**:
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> - other: 另一个平面或点
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**Raises**:
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> - TypeError 不支持的类型
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def cal_distance(self, other: 'Plane3 | Point3') -> float:
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"""
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计算平面与平面或点之间的距离。
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Args:
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other: 另一个平面或点
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Returns:
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距离
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Raises:
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TypeError: 不支持的类型
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"""
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if isinstance(other, Plane3):
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return 0
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elif isinstance(other, Point3):
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return abs(self.a * other.x + self.b * other.y + self.c * other.z + self.d) / (self.a ** 2 + self.b ** 2 + self.c ** 2) ** 0.5
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else:
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raise TypeError(f'Unsupported type: {type(other)}')
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```
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</details>
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### *method* `cal_intersection_line3(self, other: Plane3) -> Line3`
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**Description**: 计算两平面的交线。
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**Return**: 两平面的交线
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**Arguments**:
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> - other: 另一个平面
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def cal_intersection_line3(self, other: 'Plane3') -> 'Line3':
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"""
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计算两平面的交线。
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Args:
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other: 另一个平面
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Returns:
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两平面的交线
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Raises:
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"""
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if self.normal.is_parallel(other.normal):
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raise ValueError('Planes are parallel and have no intersection.')
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direction = self.normal.cross(other.normal)
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x, y, z = (0, 0, 0)
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if self.a != 0 and other.a != 0:
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A = np.array([[self.b, self.c], [other.b, other.c]])
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B = np.array([-self.d, -other.d])
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y, z = np.linalg.solve(A, B)
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elif self.b != 0 and other.b != 0:
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A = np.array([[self.a, self.c], [other.a, other.c]])
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B = np.array([-self.d, -other.d])
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x, z = np.linalg.solve(A, B)
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elif self.c != 0 and other.c != 0:
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A = np.array([[self.a, self.b], [other.a, other.b]])
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B = np.array([-self.d, -other.d])
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x, y = np.linalg.solve(A, B)
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return Line3(Point3(x, y, z), direction)
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```
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</details>
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### *method* `cal_intersection_point3(self, other: Line3) -> Point3`
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**Description**: 计算平面与直线的交点。
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**Return**: 交点
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**Arguments**:
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> - other: 不与平面平行或在平面上的直线
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**Raises**:
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> - ValueError 平面与直线平行或重合
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def cal_intersection_point3(self, other: 'Line3') -> 'Point3':
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"""
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计算平面与直线的交点。
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Args:
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other: 不与平面平行或在平面上的直线
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Returns:
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交点
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Raises:
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ValueError: 平面与直线平行或重合
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"""
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if self.normal @ other.direction == 0:
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raise ValueError('The plane and the line are parallel or coincident.')
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x, y, z = other.get_parametric_equations()
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t = -(self.a * other.point.x + self.b * other.point.y + self.c * other.point.z + self.d) / (self.a * other.direction.x + self.b * other.direction.y + self.c * other.direction.z)
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return Point3(x(t), y(t), z(t))
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```
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</details>
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### *method* `cal_parallel_plane3(self, point: Point3) -> Plane3`
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**Description**: 计算平行于该平面且过指定点的平面。
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**Return**: 所求平面
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**Arguments**:
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> - point: 指定点
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def cal_parallel_plane3(self, point: 'Point3') -> 'Plane3':
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"""
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计算平行于该平面且过指定点的平面。
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Args:
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point: 指定点
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Returns:
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所求平面
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"""
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return Plane3.from_point_and_normal(point, self.normal)
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```
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</details>
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### *method* `is_parallel(self, other: Plane3) -> bool`
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**Description**: 判断两个平面是否平行。
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**Return**: 是否平行
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**Arguments**:
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> - other: 另一个平面
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def is_parallel(self, other: 'Plane3') -> bool:
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"""
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判断两个平面是否平行。
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Args:
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other: 另一个平面
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Returns:
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是否平行
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"""
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return self.normal.is_parallel(other.normal)
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```
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</details>
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### `@property`
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### *method* `normal(self) -> Vector3`
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**Description**: 平面的法向量。
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**Return**: 法向量
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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@property
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def normal(self) -> 'Vector3':
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"""
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平面的法向量。
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Returns:
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法向量
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"""
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return Vector3(self.a, self.b, self.c)
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```
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</details>
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### `@classmethod`
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### *method* `from_point_and_normal(cls, point: Point3, normal: Vector3) -> Plane3`
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**Description**: 工厂函数 由点和法向量构造平面(点法式构造)。
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**Return**: 平面
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**Arguments**:
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> - point: 平面上的一点
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> - normal: 法向量
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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@classmethod
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def from_point_and_normal(cls, point: 'Point3', normal: 'Vector3') -> 'Plane3':
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"""
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工厂函数 由点和法向量构造平面(点法式构造)。
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Args:
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point: 平面上的一点
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normal: 法向量
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Returns:
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平面
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"""
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a, b, c = (normal.x, normal.y, normal.z)
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d = -a * point.x - b * point.y - c * point.z
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return cls(a, b, c, d)
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```
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</details>
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### `@classmethod`
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### *method* `from_three_points(cls, p1: Point3, p2: Point3, p3: Point3) -> Plane3`
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**Description**: 工厂函数 由三点构造平面。
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**Return**: 平面
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**Arguments**:
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> - p1: 点1
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> - p2: 点2
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> - p3: 点3
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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@classmethod
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def from_three_points(cls, p1: 'Point3', p2: 'Point3', p3: 'Point3') -> 'Plane3':
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"""
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工厂函数 由三点构造平面。
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Args:
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p1: 点1
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p2: 点2
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p3: 点3
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Returns:
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平面
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"""
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v1 = p2 - p1
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v2 = p3 - p1
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normal = v1.cross(v2)
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return cls.from_point_and_normal(p1, normal)
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```
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</details>
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### `@classmethod`
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### *method* `from_two_lines(cls, l1: Line3, l2: Line3) -> Plane3`
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**Description**: 工厂函数 由两直线构造平面。
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**Return**: 平面
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**Arguments**:
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> - l1: 直线1
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> - l2: 直线2
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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@classmethod
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def from_two_lines(cls, l1: 'Line3', l2: 'Line3') -> 'Plane3':
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"""
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工厂函数 由两直线构造平面。
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Args:
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l1: 直线1
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l2: 直线2
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Returns:
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平面
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"""
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v1 = l1.direction
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v2 = l2.point - l1.point
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if v2 == zero_vector3:
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v2 = l2.get_point(1) - l1.point
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return cls.from_point_and_normal(l1.point, v1.cross(v2))
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```
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</details>
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### `@classmethod`
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### *method* `from_point_and_line(cls, point: Point3, line: Line3) -> Plane3`
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**Description**: 工厂函数 由点和直线构造平面。
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**Return**: 平面
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**Arguments**:
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> - point: 面上一点
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> - line: 面上直线，不包含点
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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@classmethod
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def from_point_and_line(cls, point: 'Point3', line: 'Line3') -> 'Plane3':
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"""
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工厂函数 由点和直线构造平面。
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Args:
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point: 面上一点
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line: 面上直线，不包含点
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Returns:
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平面
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"""
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return cls.from_point_and_normal(point, line.direction)
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```
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</details>
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### `@overload`
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### *method* `__and__(self, other: Line3) -> Point3 | None`
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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@overload
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def __and__(self, other: 'Line3') -> 'Point3 | None':
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...
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```
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</details>
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### `@overload`
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### *method* `__and__(self, other: Plane3) -> Line3 | None`
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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@overload
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def __and__(self, other: 'Plane3') -> 'Line3 | None':
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...
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```
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</details>
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### *method* `__and__(self, other)`
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**Description**: 取两平面的交集（人话：交线）
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**Return**: 不平行平面的交线，平面平行返回None
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**Arguments**:
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> - other:
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def __and__(self, other):
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"""
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取两平面的交集（人话：交线）
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Args:
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other:
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Returns:
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不平行平面的交线，平面平行返回None
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"""
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if isinstance(other, Plane3):
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if self.normal.is_parallel(other.normal):
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return None
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return self.cal_intersection_line3(other)
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elif isinstance(other, Line3):
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if self.normal @ other.direction == 0:
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return None
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return self.cal_intersection_point3(other)
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else:
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raise TypeError(f"unsupported operand type(s) for &: 'Plane3' and '{type(other)}'")
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```
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</details>
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### *method* `__eq__(self, other) -> bool`
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def __eq__(self, other) -> bool:
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return self.approx(other)
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```
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</details>
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### *method* `__rand__(self, other: Line3) -> Point3`
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<details>
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<summary> <b>Source code</b> </summary>
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```python
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def __rand__(self, other: 'Line3') -> 'Point3':
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return self.cal_intersection_point3(other)
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```
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</details>
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