Module mbcp.mp_math.plane

本模块定义了三维空间中的平面类

class Plane3

method __init__(self, a: float, b: float, c: float, d: float)

Description: 平面方程：ax + by + cz + d = 0

Arguments:

• a (float): x系数
• b (float): y系数
• c (float): z系数
• d (float): 常数项

Source code or View on GitHub

def __init__(self, a: float, b: float, c: float, d: float):
    self.a = a
    self.b = b
    self.c = c
    self.d = d

method approx(self, other: Plane3) -> bool

Description: 判断两个平面是否近似相等。

Arguments:

• other (Plane3): 另一个平面

Return: bool: 是否近似相等

Source code or View on GitHub

def approx(self, other: 'Plane3') -> bool:
    if self.a != 0:
        k = other.a / self.a
        return approx(other.b, self.b * k) and approx(other.c, self.c * k) and approx(other.d, self.d * k)
    elif self.b != 0:
        k = other.b / self.b
        return approx(other.a, self.a * k) and approx(other.c, self.c * k) and approx(other.d, self.d * k)
    elif self.c != 0:
        k = other.c / self.c
        return approx(other.a, self.a * k) and approx(other.b, self.b * k) and approx(other.d, self.d * k)
    else:
        return False

method cal_angle(self, other: Line3 | Plane3) -> AnyAngle

Description: 计算平面与平面之间的夹角。

TIP

平面间夹角计算公式:

$$\theta =\arccos (\frac{n1\cdot n2}{|n1|\cdot |n2|})$$

其中 $n1$ 和 $n2$ 分别为两个平面的法向量

TIP

平面与直线夹角计算公式:

$$\theta =\arccos (\frac{n\cdot d}{|n|\cdot |d|})$$

其中 $n$ 为平面的法向量，$d$ 为直线的方向向量

Arguments:

• other (Line3 | Plane3): 另一个平面或直线

Return: AnyAngle: 夹角

Raises:

• TypeError 不支持的类型

Source code or View on GitHub

def cal_angle(self, other: 'Line3 | Plane3') -> 'AnyAngle':
    if isinstance(other, Line3):
        return self.normal.cal_angle(other.direction).complementary
    elif isinstance(other, Plane3):
        return AnyAngle(math.acos(self.normal @ other.normal / (self.normal.length * other.normal.length)), is_radian=True)
    else:
        raise TypeError(f'Unsupported type: {type(other)}')

method cal_distance(self, other: Plane3 | Point3) -> float

Description: 计算平面与平面或点之间的距离。

TIP

平面和平面之间的距离计算公式: 暂未实现

• 平行 = 0
• 相交 = 0
• 不平行 = $\frac{|\overrightarrow{P_1{P}_2}\cdot \overrightarrow{n}|}{|\overrightarrow{n}|}$ 其中，$P_1$和$P_2$分别为两个平面上的点，$\overrightarrow{n}$为平面的法向量。

TIP

平面和点之间的距离计算公式:

$$\frac{|\overrightarrow{P_{1}P}\cdot \overrightarrow{n}|}{|\overrightarrow{n}|}$$

其中，$P_1$为平面上的点，$P$为点，$\overrightarrow{n}$为平面的法向量。

Arguments:

• other (Plane3 | Point3): 另一个平面或点

Return: float: 距离

Raises:

• TypeError 不支持的类型

Source code or View on GitHub

def cal_distance(self, other: 'Plane3 | Point3') -> float:
    if isinstance(other, Plane3):
        raise NotImplementedError('Not implemented yet.')
    elif isinstance(other, Point3):
        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
    else:
        raise TypeError(f'Unsupported type: {type(other)}')

method cal_intersection_line3(self, other: Plane3) -> Line3

Description: 计算两平面的交线。

TIP

计算两平面交线的一般步骤:

1. 求两平面的法向量的叉乘得到方向向量

$$d=n1\times n2$$

2. 寻找直线上的一点，依次假设$x=0$, $y=0$, $z=0$，并代入两平面方程求出合适的点 直线最终可用参数方程或点向式表示

$$\{\begin{array}{l}x=x_0+dt\\ y=y_0+dt\\ z=z_0+dt\end{array}$$

或

$$\frac{x-x_0}{m}=\frac{y-y_0}{n}=\frac{z-z_0}{p}$$

Arguments:

• other (Plane3): 另一个平面

Return: Line3: 交线

Raises:

• ValueError 平面平行且无交线

Source code or View on GitHub

def cal_intersection_line3(self, other: 'Plane3') -> 'Line3':
    if self.normal.is_parallel(other.normal):
        raise ValueError('Planes are parallel and have no intersection.')
    direction = self.normal.cross(other.normal)
    x, y, z = (0, 0, 0)
    if self.a != 0 and other.a != 0:
        A = np.array([[self.b, self.c], [other.b, other.c]])
        B = np.array([-self.d, -other.d])
        y, z = np.linalg.solve(A, B)
    elif self.b != 0 and other.b != 0:
        A = np.array([[self.a, self.c], [other.a, other.c]])
        B = np.array([-self.d, -other.d])
        x, z = np.linalg.solve(A, B)
    elif self.c != 0 and other.c != 0:
        A = np.array([[self.a, self.b], [other.a, other.b]])
        B = np.array([-self.d, -other.d])
        x, y = np.linalg.solve(A, B)
    return Line3(Point3(x, y, z), direction)

method cal_intersection_point3(self, other: Line3) -> Point3

Description: 计算平面与直线的交点。

TIP

计算平面与直线交点的一般步骤:

1. 求直线的参数方程
2. 代入平面方程，解出t
3. 代入直线参数方程，求出交点

Arguments:

• other (Line3): 直线

Return: Point3: 交点

Raises:

• ValueError 平面与直线平行或重合

Source code or View on GitHub

def cal_intersection_point3(self, other: 'Line3') -> 'Point3':
    if self.normal @ other.direction == 0:
        raise ValueError('The plane and the line are parallel or coincident.')
    x, y, z = other.get_parametric_equations()
    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)
    return Point3(x(t), y(t), z(t))

method cal_parallel_plane3(self, point: Point3) -> Plane3

Description: 计算平行于该平面且过指定点的平面。

Arguments:

• point (Point3): 指定点

Return: Plane3: 平面

Source code or View on GitHub

def cal_parallel_plane3(self, point: 'Point3') -> 'Plane3':
    return Plane3.from_point_and_normal(point, self.normal)

method is_parallel(self, other: Plane3) -> bool

Description: 判断两个平面是否平行。

Arguments:

• other (Plane3): 另一个平面

Return: bool: 是否平行

Source code or View on GitHub

def is_parallel(self, other: 'Plane3') -> bool:
    return self.normal.is_parallel(other.normal)

@property

method normal(self) -> Vector3

Description: 平面的法向量。

Return: Vector3: 法向量

Source code or View on GitHub

@property
def normal(self) -> 'Vector3':
    return Vector3(self.a, self.b, self.c)

@classmethod

method from_point_and_normal(cls, point: Point3, normal: Vector3) -> Plane3

Description: 工厂函数 由点和法向量构造平面(点法式构造)。

Arguments:

• point (Point3): 平面上一点
• normal (Vector3): 法向量

Return: Plane3: 平面

Source code or View on GitHub

@classmethod
def from_point_and_normal(cls, point: 'Point3', normal: 'Vector3') -> 'Plane3':
    a, b, c = (normal.x, normal.y, normal.z)
    d = -a * point.x - b * point.y - c * point.z
    return cls(a, b, c, d)

@classmethod

method from_three_points(cls, p1: Point3, p2: Point3, p3: Point3) -> Plane3

Description: 工厂函数 由三点构造平面。

Arguments:

• p1 (Point3): 点1
• p2 (Point3): 点2
• p3 (Point3): 点3

Return: 平面

Source code or View on GitHub

@classmethod
def from_three_points(cls, p1: 'Point3', p2: 'Point3', p3: 'Point3') -> 'Plane3':
    v1 = p2 - p1
    v2 = p3 - p1
    normal = v1.cross(v2)
    return cls.from_point_and_normal(p1, normal)

@classmethod

method from_two_lines(cls, l1: Line3, l2: Line3) -> Plane3

Description: 工厂函数 由两直线构造平面。

Arguments:

• l1 (Line3): 直线
• l2 (Line3): 直线

Return: 平面

Source code or View on GitHub

@classmethod
def from_two_lines(cls, l1: 'Line3', l2: 'Line3') -> 'Plane3':
    v1 = l1.direction
    v2 = l2.point - l1.point
    if v2 == zero_vector3:
        v2 = l2.get_point(1) - l1.point
    return cls.from_point_and_normal(l1.point, v1.cross(v2))

@classmethod

method from_point_and_line(cls, point: Point3, line: Line3) -> Plane3

Description: 工厂函数 由点和直线构造平面。

Arguments:

• point (Point3): 平面上一点
• line (Line3): 直线

Return: 平面

Source code or View on GitHub

@classmethod
def from_point_and_line(cls, point: 'Point3', line: 'Line3') -> 'Plane3':
    return cls.from_point_and_normal(point, line.direction)

@overload

method self & other: Line3 => Point3 | None

Source code or View on GitHub

@overload
def __and__(self, other: 'Line3') -> 'Point3 | None':
    ...

@overload

method self & other: Plane3 => Line3 | None

Source code or View on GitHub

@overload
def __and__(self, other: 'Plane3') -> 'Line3 | None':
    ...

method self & other

Description: 取两平面的交集（人话：交线）

Arguments:

• other (Line3 | Plane3): 另一个平面或直线

Return: Line3 | Point3 | None: 交集

Raises:

• TypeError 不支持的类型

Source code or View on GitHub

def __and__(self, other):
    if isinstance(other, Plane3):
        if self.normal.is_parallel(other.normal):
            return None
        return self.cal_intersection_line3(other)
    elif isinstance(other, Line3):
        if self.normal @ other.direction == 0:
            return None
        return self.cal_intersection_point3(other)
    else:
        raise TypeError(f"unsupported operand type(s) for &: 'Plane3' and '{type(other)}'")

method self == other => bool

Description: 判断两个平面是否等价。

Arguments:

• other (Plane3): 另一个平面

Return: bool: 是否等价

Source code or View on GitHub

def __eq__(self, other) -> bool:
    return self.approx(other)

method self & other: Line3 => Point3

Source code or View on GitHub

def __rand__(self, other: 'Line3') -> 'Point3':
    return self.cal_intersection_point3(other)