模組 `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)`

説明: 平面方程：ax + by + cz + d = 0

變數説明:

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

python

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`

説明: 判断两个平面是否近似相等。

變數説明:

other (Plane3): 另一个平面

返回: bool: 是否近似相等

源碼或於GitHub上查看

python

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`

説明: 计算平面与平面之间的夹角。

TIP

平面间夹角计算公式:

θ = \arccos (\frac{n 1 \cdot n 2}{| n 1 | \cdot | n 2 |})

其中 $n 1$ 和 $n 2$ 分别为两个平面的法向量

TIP

平面与直线夹角计算公式:

θ = \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

變數説明:

other (Line3 | Plane3): 另一个平面或直线

返回: AnyAngle: 夹角

抛出:

TypeError 不支持的类型

源碼或於GitHub上查看

python

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`

説明: 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 = $\frac{| \vec{P_{1} P_{2}} \cdot \vec{n} |}{| \vec{n} |}$ 其中， $P_{1}$ 和 $P_{2}$ 分别为两个平面上的点， $\vec{n}$ 为平面的法向量。

TIP

平面和点之间的距离计算公式:

\frac{| \vec{P_{1} P} \cdot \vec{n} |}{| \vec{n} |}

其中， $P_{1}$ 为平面上的点， $P$ 为点， $\vec{n}$ 为平面的法向量。

變數説明:

other (Plane3 | Point3): 另一个平面或点

返回: float: 距离

抛出:

TypeError 不支持的类型

源碼或於GitHub上查看

python

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`

説明: 计算两平面的交线。

TIP

计算两平面交线的一般步骤:

求两平面的法向量的叉乘得到方向向量

d = n 1 \times n 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{cases} x = x_{0} + d t \\ y = y_{0} + d t \\ z = z_{0} + d t \end{cases}

或

\frac{x - x_{0}}{m} = \frac{y - y_{0}}{n} = \frac{z - z_{0}}{p}

變數説明:

other (Plane3): 另一个平面

返回: Line3: 交线

抛出:

ValueError 平面平行且无交线

源碼或於GitHub上查看

python

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`

説明: 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:

求直线的参数方程
代入平面方程，解出t
代入直线参数方程，求出交点

變數説明:

other (Line3): 直线

返回: Point3: 交点

抛出:

ValueError 平面与直线平行或重合

源碼或於GitHub上查看

python

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`

説明: 计算平行于该平面且过指定点的平面。

變數説明:

point (Point3): 指定点

返回: Plane3: 平面

源碼或於GitHub上查看

python

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

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

説明: 判断两个平面是否平行。

變數説明:

other (Plane3): 另一个平面

返回: bool: 是否平行

源碼或於GitHub上查看

python

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

@property

method `normal(self) -> Vector3`

説明: 平面的法向量。

返回: Vector3: 法向量

源碼或於GitHub上查看

python

@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`

説明: 工厂函数由点和法向量构造平面(点法式构造)。

變數説明:

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

返回: Plane3: 平面

源碼或於GitHub上查看

python

@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`

説明: 工厂函数由三点构造平面。

變數説明:

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

返回: 平面

源碼或於GitHub上查看

python

@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`

説明: 工厂函数由两直线构造平面。

變數説明:

l1 (Line3): 直线
l2 (Line3): 直线

返回: 平面

源碼或於GitHub上查看

python

@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`

説明: 工厂函数由点和直线构造平面。

變數説明:

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

返回: 平面

源碼或於GitHub上查看

python

@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`

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python

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

@overload

method `self & other: Plane3 => Line3 | None`

源碼或於GitHub上查看

python

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

method `self & other`

説明: 取两平面的交集（人话：交线）

變數説明:

other (Line3 | Plane3): 另一个平面或直线

返回: Line3 | Point3 | None: 交集

抛出:

TypeError 不支持的类型

源碼或於GitHub上查看

python

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`

説明: 判断两个平面是否等价。

變數説明:

other (Plane3): 另一个平面

返回: bool: 是否等价

源碼或於GitHub上查看

python

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

method `self & other: Line3 => Point3`

源碼或於GitHub上查看

python

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

模組 mbcp.mp_math.plane ​

class Plane3 ​

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

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

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

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

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

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

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

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

method normal(self) -> Vector3 ​

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

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

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

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

method self & other: Line3 => Point3 | None ​

method self & other: Plane3 => Line3 | None ​

method self & other ​

method self == other => bool ​

method self & other: Line3 => Point3 ​

模組 `mbcp.mp_math.plane`

class `Plane3`

method `init(self, a: float, b: float, c: float, d: float)`

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

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

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

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

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

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

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

method `normal(self) -> Vector3`

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

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

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

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

method `self & other: Line3 => Point3 | None`

method `self & other: Plane3 => Line3 | None`

method `self & other`

method `self == other => bool`

method `self & other: Line3 => Point3`