import{_ as s,c as i,o as a,a2 as n}from"./chunks/framework.BV61Qrc0.js";const F=JSON.parse('{"title":"mbcp.mp_math.plane","description":"","frontmatter":{"title":"mbcp.mp_math.plane"},"headers":[],"relativePath":"en/api/mp_math/plane.md","filePath":"en/api/mp_math/plane.md"}'),l={name:"en/api/mp_math/plane.md"},h=n(`

class Plane3 ​

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

平面方程：ax + by + cz + d = 0

Args:

Source code
python
def __init__(self, a: float, b: float, c: float, d: float):
    """
        平面方程：ax + by + cz + d = 0
        Args:
            a:
            b:
            c:
            d:
        """
    self.a = a
    self.b = b
    self.c = c
    self.d = d

def approx(self, other: 'Plane3') -> bool ​

判断两个平面是否近似相等。

Args:

Source code
python
def approx(self, other: 'Plane3') -> bool:
    """
        判断两个平面是否近似相等。
        Args:
            other:

        Returns:
            是否近似相等
        """
    a = 3
    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

def cal_angle(self, other: 'Line3 | Plane3') -> 'AnyAngle' ​

计算平面与平面之间的夹角。

Args:

Source code
python
def cal_angle(self, other: 'Line3 | Plane3') -> 'AnyAngle':
    """
        计算平面与平面之间的夹角。
        Args:
            other: 另一个平面
        Returns:
            夹角弧度
        Raises:
            TypeError: 不支持的类型
        """
    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)}')

def cal_distance(self, other: 'Plane3 | Point3') -> float ​

计算平面与平面或点之间的距离。

Args:

Source code
python
def cal_distance(self, other: 'Plane3 | Point3') -> float:
    """
        计算平面与平面或点之间的距离。
        Args:
            other: 另一个平面或点
        Returns:
            距离
        Raises:
            TypeError: 不支持的类型
        """
    if isinstance(other, Plane3):
        return 0
    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)}')

def cal_intersection_line3(self, other: 'Plane3') -> 'Line3' ​

计算两平面的交线。该方法有问题，待修复。

Args:

Source code
python
def cal_intersection_line3(self, other: 'Plane3') -> 'Line3':
    """
        计算两平面的交线。该方法有问题，待修复。
        Args:
            other: 另一个平面
        Returns:
            交线
        Raises:
        """
    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)

def cal_intersection_point3(self, other: 'Line3') -> 'Point3' ​

计算平面与直线的交点。

Args:

Source code
python
def cal_intersection_point3(self, other: 'Line3') -> 'Point3':
    """
        计算平面与直线的交点。
        Args:
            other: 不与平面平行或在平面上的直线
        Returns:
            交点
        Raises:
            ValueError: 平面与直线平行或重合
        """
    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))

def cal_parallel_plane3(self, point: 'Point3') -> 'Plane3' ​

计算平行于该平面且过指定点的平面。

Args:

Source code
python
def cal_parallel_plane3(self, point: 'Point3') -> 'Plane3':
    """
        计算平行于该平面且过指定点的平面。
        Args:
            point: 指定点
        Returns:
            平面
        """
    return Plane3.from_point_and_normal(point, self.normal)

def is_parallel(self, other: 'Plane3') -> bool ​

判断两个平面是否平行。

Args:

Source code
python
def is_parallel(self, other: 'Plane3') -> bool:
    """
        判断两个平面是否平行。
        Args:
            other: 另一个平面
        Returns:
            是否平行
        """
    return self.normal.is_parallel(other.normal)

@property ​

def normal(self) -> 'Vector3' ​

平面的法向量。

Return:

Source code
python
@property
def normal(self) -> 'Vector3':
    """
        平面的法向量。
        Returns:
            法向量
        """
    return Vector3(self.a, self.b, self.c)

@classmethod ​

def from_point_and_normal(cls, point: 'Point3', normal: 'Vector3') -> 'Plane3' ​

工厂函数 由点和法向量构造平面(点法式构造)。

Args:

Source code
python
@classmethod
def from_point_and_normal(cls, point: 'Point3', normal: 'Vector3') -> 'Plane3':
    """
        工厂函数 由点和法向量构造平面(点法式构造)。
        Args:
            point: 平面上的一点
            normal: 法向量
        Returns:
            平面
        """
    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 ​

def from_three_points(cls, p1: 'Point3', p2: 'Point3', p3: 'Point3') -> 'Plane3' ​

工厂函数 由三点构造平面。

Args:

Source code
python
@classmethod
def from_three_points(cls, p1: 'Point3', p2: 'Point3', p3: 'Point3') -> 'Plane3':
    """
        工厂函数 由三点构造平面。
        Args:
            p1: 点1
            p2: 点2
            p3: 点3
        Returns:
            平面
        """
    v1 = p2 - p1
    v2 = p3 - p1
    normal = v1.cross(v2)
    return cls.from_point_and_normal(p1, normal)

@classmethod ​

def from_two_lines(cls, l1: 'Line3', l2: 'Line3') -> 'Plane3' ​

工厂函数 由两直线构造平面。

Args:

Source code
python
@classmethod
def from_two_lines(cls, l1: 'Line3', l2: 'Line3') -> 'Plane3':
    """
        工厂函数 由两直线构造平面。
        Args:
            l1: 直线1
            l2: 直线2
        Returns:
            平面
        """
    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 ​

def from_point_and_line(cls, point: 'Point3', line: 'Line3') -> 'Plane3' ​

工厂函数 由点和直线构造平面。

Args:

Source code
python
@classmethod
def from_point_and_line(cls, point: 'Point3', line: 'Line3') -> 'Plane3':
    """
        工厂函数 由点和直线构造平面。
        Args:
            point: 面上一点
            line: 面上直线，不包含点
        Returns:
            平面
        """
    return cls.from_point_and_normal(point, line.direction)

def __repr__(self) ​

Source code
python
def __repr__(self):
    return f'Plane3({self.a}, {self.b}, {self.c}, {self.d})'

def __str__(self) ​

Source code
python
def __str__(self):
    s = 'Plane3: '
    if self.a != 0:
        s += f'{sign(self.a, only_neg=True)}{abs(self.a)}x'
    if self.b != 0:
        s += f' {sign(self.b)} {abs(self.b)}y'
    if self.c != 0:
        s += f' {sign(self.c)} {abs(self.c)}z'
    if self.d != 0:
        s += f' {sign(self.d)} {abs(self.d)}'
    return s + ' = 0'

@overload ​

def __and__(self, other: 'Line3') -> 'Point3 | None' ​

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

@overload ​

def __and__(self, other: 'Plane3') -> 'Line3 | None' ​

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

def __and__(self, other) ​

取两平面的交集（人话：交线）

Args:

Source code
python
def __and__(self, other):
    """
        取两平面的交集（人话：交线）
        Args:
            other:
        Returns:
            不平行平面的交线，平面平行返回None
        """
    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)}'")

def __eq__(self, other) -> bool ​

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

def __rand__(self, other: 'Line3') -> 'Point3' ​

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

var k = other.a / self.a ​

var A = np.array([[self.b, self.c], [other.b, other.c]]) ​

var B = np.array([-self.d, -other.d]) ​

var v2 = l2.get_point(1) - l1.point ​

var k = other.b / self.b ​

var A = np.array([[self.a, self.c], [other.a, other.c]]) ​

var B = np.array([-self.d, -other.d]) ​

var k = other.c / self.c ​

var A = np.array([[self.a, self.b], [other.a, other.b]]) ​

var B = np.array([-self.d, -other.d]) ​

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