Module `mbcp.mp_math.vector`

本模块定义了3维向量的类Vector3，以及一些常用的向量。

class `Vector3`

method `init(self, x: float, y: float, z: float)`

Description: 3维向量

Arguments:

x (float): x轴分量
y (float): y轴分量
z (float): z轴分量

python

def __init__(self, x: float, y: float, z: float):
    self.x = x
    self.y = y
    self.z = z

method `approx(self, other: Vector3, epsilon: float = APPROX) -> bool`

Description: 判断两个向量是否近似相等。

Arguments:

other (Vector3): 另一个向量
epsilon (float): 误差

Return: bool: 是否近似相等

Source code or View on GitHub

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def approx(self, other: 'Vector3', epsilon: float=APPROX) -> bool:
    return all([abs(self.x - other.x) < epsilon, abs(self.y - other.y) < epsilon, abs(self.z - other.z) < epsilon])

method `cal_angle(self, other: Vector3) -> AnyAngle`

Description: 计算两个向量之间的夹角。

TIP

向量夹角计算公式:

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

Arguments:

other (Vector3): 另一个向量

Return: AnyAngle: 夹角

Source code or View on GitHub

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def cal_angle(self, other: 'Vector3') -> 'AnyAngle':
    return AnyAngle(math.acos(self @ other / (self.length * other.length)), is_radian=True)

method `cross(self, other: Vector3) -> Vector3`

Description: 向量积叉乘：v1 x v2 -> v3

TIP

叉乘运算法则为:

v 1 \times v 2 = (v 1_{y} \cdot v 2_{z} - v 1_{z} \cdot v 2_{y}, v 1_{z} \cdot v 2_{x} - v 1_{x} \cdot v 2_{z}, v 1_{x} \cdot v 2_{y} - v 1_{y} \cdot v 2_{x})

转换为行列式形式:

v 1 \times v 2 = | \begin{matrix} i & j & k \\ v 1_{x} & v 1_{y} & v 1_{z} \\ v 2_{x} & v 2_{y} & v 2_{z} \end{matrix} |

Arguments:

other (Vector3): 另一个向量

Return: Vector3: 叉乘结果

Source code or View on GitHub

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def cross(self, other: 'Vector3') -> 'Vector3':
    return Vector3(self.y * other.z - self.z * other.y, self.z * other.x - self.x * other.z, self.x * other.y - self.y * other.x)

method `is_approx_parallel(self, other: Vector3, epsilon: float = APPROX) -> bool`

Description: 判断两个向量是否近似平行。

Arguments:

other (Vector3): 另一个向量
epsilon (float): 误差

Return: bool: 是否近似平行

Source code or View on GitHub

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def is_approx_parallel(self, other: 'Vector3', epsilon: float=APPROX) -> bool:
    return self.cross(other).length < epsilon

method `is_parallel(self, other: Vector3) -> bool`

Description: 判断两个向量是否平行。

Arguments:

other (Vector3): 另一个向量

Return: bool: 是否平行

Source code or View on GitHub

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def is_parallel(self, other: 'Vector3') -> bool:
    return self.cross(other).approx(zero_vector3)

method `normalize(self)`

Description: 将向量归一化。

自体归一化，不返回值。

Source code or View on GitHub

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def normalize(self):
    length = self.length
    self.x /= length
    self.y /= length
    self.z /= length

method `project(self, other: Vector3) -> Vector3`

Description: 计算自向量在另一个向量上的投影向量。

TIP

投影向量计算公式， $p r o j_{v} (u)$ 表示向量 $u$ 在向量 $v$ 上的投影向量:

{proj}_{v} (u) = \frac{u \cdot v}{| v |^{2}} \cdot v

Arguments:

other (Vector3): 另一个向量

Return: Vector3: 投影向量

Source code or View on GitHub

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def project(self, other: 'Vector3') -> 'Vector3':
    return self @ other / other.length * other.unit

@property

method `np_array(self) -> np.ndarray`

Return: np.ndarray: numpy数组

Source code or View on GitHub

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@property
def np_array(self) -> 'np.ndarray':
    return np.array([self.x, self.y, self.z])

@property

method `length(self) -> float`

Description: 向量的模。

Return: float: 模

Source code or View on GitHub

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@property
def length(self) -> float:
    return math.sqrt(self.x ** 2 + self.y ** 2 + self.z ** 2)

@property

method `unit(self) -> Vector3`

Description: 获取该向量的单位向量。

Return: Vector3: 单位向量

Source code or View on GitHub

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@property
def unit(self) -> 'Vector3':
    return self / self.length

method `abs(self)`

Source code or View on GitHub

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def __abs__(self):
    return self.length

@overload

method `self + other: Vector3 => Vector3`

Source code or View on GitHub

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@overload
def __add__(self, other: 'Vector3') -> 'Vector3':
    ...

@overload

method `self + other: Point3 => Point3`

Source code or View on GitHub

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@overload
def __add__(self, other: 'Point3') -> 'Point3':
    ...

method `self + other`

Description: V + P -> P

V + V -> V

Arguments:

other (Vector3 | Point3): 另一个向量或点

Return: Vector3 | Point3: 新的向量或点

Source code or View on GitHub

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def __add__(self, other):
    if isinstance(other, Vector3):
        return Vector3(self.x + other.x, self.y + other.y, self.z + other.z)
    elif isinstance(other, Point3):
        return Point3(self.x + other.x, self.y + other.y, self.z + other.z)
    else:
        raise TypeError(f"unsupported operand type(s) for +: 'Vector3' and '{type(other)}'")

method `self == other`

Description: 判断两个向量是否相等。

Arguments:

other (Vector3): 另一个向量

Return: bool: 是否相等

Source code or View on GitHub

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def __eq__(self, other):
    return approx(self.x, other.x) and approx(self.y, other.y) and approx(self.z, other.z)

method `self + other: Point3 => Point3`

Description: P + V -> P

别去点那边实现了。

Arguments:

other (Point3): 另一个点

Return: Point3: 新的点

Source code or View on GitHub

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def __radd__(self, other: 'Point3') -> 'Point3':
    return Point3(self.x + other.x, self.y + other.y, self.z + other.z)

@overload

method `self - other: Vector3 => Vector3`

Source code or View on GitHub

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@overload
def __sub__(self, other: 'Vector3') -> 'Vector3':
    ...

@overload

method `self - other: Point3 => Point3`

Source code or View on GitHub

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@overload
def __sub__(self, other: 'Point3') -> 'Point3':
    ...

method `self - other`

Description: V - P -> P

V - V -> V

Arguments:

other (Vector3 | Point3): 另一个向量或点

Return: Vector3 | Point3: 新的向量

Source code or View on GitHub

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def __sub__(self, other):
    if isinstance(other, Vector3):
        return Vector3(self.x - other.x, self.y - other.y, self.z - other.z)
    elif isinstance(other, Point3):
        return Point3(self.x - other.x, self.y - other.y, self.z - other.z)
    else:
        raise TypeError(f'unsupported operand type(s) for -: "Vector3" and "{type(other)}"')

method `self - other: Point3`

Description: P - V -> P

Arguments:

other (Point3): 另一个点

Return: Point3: 新的点

Source code or View on GitHub

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def __rsub__(self, other: 'Point3'):
    if isinstance(other, Point3):
        return Point3(other.x - self.x, other.y - self.y, other.z - self.z)
    else:
        raise TypeError(f"unsupported operand type(s) for -: '{type(other)}' and 'Vector3'")

@overload

method `self * other: Vector3 => Vector3`

Source code or View on GitHub

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@overload
def __mul__(self, other: 'Vector3') -> 'Vector3':
    ...

@overload

method `self * other: RealNumber => Vector3`

Source code or View on GitHub

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@overload
def __mul__(self, other: RealNumber) -> 'Vector3':
    ...

method `self * other: int | float | Vector3 => Vector3`

Description: 数组运算非点乘。点乘使用@，叉乘使用cross。

Arguments:

other (Vector3 | float): 另一个向量或数

Return: Vector3: 数组运算结果

Source code or View on GitHub

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def __mul__(self, other: 'int | float | Vector3') -> 'Vector3':
    if isinstance(other, Vector3):
        return Vector3(self.x * other.x, self.y * other.y, self.z * other.z)
    elif isinstance(other, (float, int)):
        return Vector3(self.x * other, self.y * other, self.z * other)
    else:
        raise TypeError(f"unsupported operand type(s) for *: 'Vector3' and '{type(other)}'")

method `self * other: RealNumber => Vector3`

Source code or View on GitHub

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def __rmul__(self, other: 'RealNumber') -> 'Vector3':
    return self.__mul__(other)

method `self @ other: Vector3 => RealNumber`

Description: 点乘。

Arguments:

other (Vector3): 另一个向量

Return: float: 点乘结果

Source code or View on GitHub

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def __matmul__(self, other: 'Vector3') -> 'RealNumber':
    return self.x * other.x + self.y * other.y + self.z * other.z

method `self / other: RealNumber => Vector3`

Source code or View on GitHub

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def __truediv__(self, other: RealNumber) -> 'Vector3':
    return Vector3(self.x / other, self.y / other, self.z / other)

method `- self => Vector3`

Description: 取负。

Return: Vector3: 负向量

Source code or View on GitHub

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def __neg__(self) -> 'Vector3':
    return Vector3(-self.x, -self.y, -self.z)

var `zero_vector3`

Description: 零向量
Type: Vector3
Default: Vector3(0, 0, 0)

var `x_axis`

Description: x轴单位向量
Type: Vector3
Default: Vector3(1, 0, 0)

var `y_axis`

Description: y轴单位向量
Type: Vector3
Default: Vector3(0, 1, 0)

var `z_axis`

Description: z轴单位向量
Type: Vector3
Default: Vector3(0, 0, 1)

Module mbcp.mp_math.vector ​

class Vector3 ​

method __init__(self, x: float, y: float, z: float) ​

method approx(self, other: Vector3, epsilon: float = APPROX) -> bool ​

method cal_angle(self, other: Vector3) -> AnyAngle ​

method cross(self, other: Vector3) -> Vector3 ​

method is_approx_parallel(self, other: Vector3, epsilon: float = APPROX) -> bool ​

method is_parallel(self, other: Vector3) -> bool ​

method normalize(self) ​

method project(self, other: Vector3) -> Vector3 ​

method np_array(self) -> np.ndarray ​

method length(self) -> float ​

method unit(self) -> Vector3 ​

method __abs__(self) ​

method self + other: Vector3 => Vector3 ​

method self + other: Point3 => Point3 ​

method self + other ​

method self == other ​

method self + other: Point3 => Point3 ​

method self - other: Vector3 => Vector3 ​

method self - other: Point3 => Point3 ​

method self - other ​

method self - other: Point3 ​

method self * other: Vector3 => Vector3 ​

method self * other: RealNumber => Vector3 ​

method self * other: int | float | Vector3 => Vector3 ​

method self * other: RealNumber => Vector3 ​

method self @ other: Vector3 => RealNumber ​

method self / other: RealNumber => Vector3 ​

method - self => Vector3 ​

var zero_vector3 ​

var x_axis ​

var y_axis ​

var z_axis ​

Module `mbcp.mp_math.vector`

class `Vector3`

method `init(self, x: float, y: float, z: float)`

method `approx(self, other: Vector3, epsilon: float = APPROX) -> bool`

method `cal_angle(self, other: Vector3) -> AnyAngle`

method `cross(self, other: Vector3) -> Vector3`

method `is_approx_parallel(self, other: Vector3, epsilon: float = APPROX) -> bool`

method `is_parallel(self, other: Vector3) -> bool`

method `normalize(self)`

method `project(self, other: Vector3) -> Vector3`

method `np_array(self) -> np.ndarray`

method `length(self) -> float`

method `unit(self) -> Vector3`

method `abs(self)`

method `self + other: Vector3 => Vector3`

method `self + other: Point3 => Point3`

method `self + other`

method `self == other`

method `self + other: Point3 => Point3`

method `self - other: Vector3 => Vector3`

method `self - other: Point3 => Point3`

method `self - other`

method `self - other: Point3`

method `self * other: Vector3 => Vector3`

method `self * other: RealNumber => Vector3`

method `self * other: int | float | Vector3 => Vector3`

method `self * other: RealNumber => Vector3`

method `self @ other: Vector3 => RealNumber`

method `self / other: RealNumber => Vector3`

method `- self => Vector3`

var `zero_vector3`

var `x_axis`

var `y_axis`

var `z_axis`