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。
Vector3 __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
def __init__(self, x: float, y: float, z: float):\n """\n 3维向量\n Args:\n x ([`float`](https%3A//docs.python.org/3/library/functions.html#float)): x轴分量\n y (`float`): y轴分量\n z (`float`): z轴分量\n """\n self.x = x\n self.y = y\n self.z = zapprox(self, other: Vector3, epsilon: float = APPROX) -> bool Description: 判断两个向量是否近似相等。
Arguments:
Return: bool: 是否近似相等
def approx(self, other: 'Vector3', epsilon: float=APPROX) -> bool:\n """\n 判断两个向量是否近似相等。\n Args:\n other ([`Vector3`](#class-vector3)): 另一个向量\n epsilon ([`float`](https%3A//docs.python.org/3/library/functions.html#float)): 误差\n\n Returns:\n [`bool`](https%3A//docs.python.org/3/library/functions.html#bool): 是否近似相等\n """\n return all([abs(self.x - other.x) < epsilon, abs(self.y - other.y) < epsilon, abs(self.z - other.z) < epsilon])cal_angle(self, other: Vector3) -> AnyAngle Description: 计算两个向量之间的夹角。
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- other (
Vector3): 另一个向量
Return: AnyAngle: 夹角
def cal_angle(self, other: 'Vector3') -> 'AnyAngle':
"""
计算两个向量之间的夹角。
:::tip
向量夹角计算公式:
$$\\\\theta = \\\\arccos(\\\\frac{v1 \\\\cdot v2}{|v1| \\\\cdot |v2|})$$
:::
Args:
other ([\`Vector3\`](#class-vector3)): 另一个向量
Returns:
[\`AnyAngle\`](./angle#class-anyangle): 夹角
"""
return AnyAngle(math.acos(self @ other / (self.length * other.length)), is_radian=True)cross(self, other: Vector3) -> Vector3 Description: 向量积 叉乘：v1 x v2 -> v3
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- other (
Vector3): 另一个向量
Return: Vector3: 叉乘结果
def cross(self, other: 'Vector3') -> 'Vector3':
"""
向量积 叉乘：v1 x v2 -> v3
:::tip
叉乘运算法则为:
$$ v1 \\\\times v2 = (v1_y \\\\cdot v2_z - v1_z \\\\cdot v2_y, v1_z \\\\cdot v2_x - v1_x \\\\cdot v2_z, v1_x \\\\cdot v2_y - v1_y \\\\cdot v2_x) $$
转换为行列式形式:
$$ v1 \\\\times v2 = \\\\begin{vmatrix} i & j & k \\\\\\\\ v1_x & v1_y & v1_z \\\\\\\\ v2_x & v2_y & v2_z \\\\end{vmatrix} $$
:::
Args:
other ([\`Vector3\`](#class-vector3)): 另一个向量
Returns:
[\`Vector3\`](#class-vector3): 叉乘结果
"""
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)is_approx_parallel(self, other: Vector3, epsilon: float = APPROX) -> bool Description: 判断两个向量是否近似平行。
Arguments:
Return: bool: 是否近似平行
def is_approx_parallel(self, other: 'Vector3', epsilon: float=APPROX) -> bool:
"""
判断两个向量是否近似平行。
Args:
other ([\`Vector3\`](#class-vector3)): 另一个向量
epsilon ([\`float\`](https%3A//docs.python.org/3/library/functions.html#float)): 误差
Returns:
[\`bool\`](https%3A//docs.python.org/3/library/functions.html#bool): 是否近似平行
"""
return self.cross(other).length < epsilonis_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
def is_parallel(self, other: 'Vector3') -> bool:
"""
判断两个向量是否平行。
Args:
other ([\`Vector3\`](#class-vector3)): 另一个向量
Returns:
[\`bool\`](https%3A//docs.python.org/3/library/functions.html#bool): 是否平行
"""
return self.cross(other).approx(zero_vector3)normalize(self) Description: 将向量归一化。
自体归一化，不返回值。
def normalize(self):
"""
将向量归一化。
自体归一化，不返回值。
"""
length = self.length
self.x /= length
self.y /= length
self.z /= lengthnp_array(self) -> np.ndarray Return: np.ndarray: numpy数组
@property
def np_array(self) -> 'np.ndarray':
"""
返回numpy数组
Returns:
[\`np.ndarray\`](https%3A//numpy.org/doc/stable/reference/generated/numpy.ndarray.html): numpy数组
"""
return np.array([self.x, self.y, self.z])length(self) -> float Description: 向量的模。
Return: float: 模
@property
def length(self) -> float:
"""
向量的模。
Returns:
[\`float\`](https%3A//docs.python.org/3/library/functions.html#float): 模
"""
return math.sqrt(self.x ** 2 + self.y ** 2 + self.z ** 2)unit(self) -> Vector3 Description: 获取该向量的单位向量。
Return: Vector3: 单位向量
@property
def unit(self) -> 'Vector3':
"""
获取该向量的单位向量。
Returns:
[\`Vector3\`](#class-vector3): 单位向量
"""
return self / self.length__abs__(self) def __abs__(self):
return self.length@overload
self + other: Vector3 => Vector3 @overload
def __add__(self, other: 'Vector3') -> 'Vector3':
...@overload
self + other: Point3 => Point3 @overload
def __add__(self, other: 'Point3') -> 'Point3':
...self + other Description: V + P -> P
V + V -> V
Arguments:
Return: Vector3 | Point3: 新的向量或点
def __add__(self, other):
"""
V + P -> P
V + V -> V
Args:
other ([\`Vector3\`](#class-vector3) | [\`Point3\`](./point#class-point3)): 另一个向量或点
Returns:
[\`Vector3\`](#class-vector3) | [\`Point3\`](./point#class-point3): 新的向量或点
"""
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)}'")__eq__(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
def __eq__(self, other):
"""
判断两个向量是否相等。
Args:
other ([\`Vector3\`](#class-vector3)): 另一个向量
Returns:
[\`bool\`](https%3A//docs.python.org/3/library/functions.html#bool): 是否相等
"""
return approx(self.x, other.x) and approx(self.y, other.y) and approx(self.z, other.z)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
def __radd__(self, other: 'Point3') -> 'Point3':
"""
P + V -> P
别去点那边实现了。
Args:
other ([\`Point3\`](./point#class-point3)): 另一个点
Returns:
[\`Point3\`](./point#class-point3): 新的点
"""
return Point3(self.x + other.x, self.y + other.y, self.z + other.z)@overload
self - other: Vector3 => Vector3 @overload
def __sub__(self, other: 'Vector3') -> 'Vector3':
...@overload
self - other: Point3 => Point3 @overload
def __sub__(self, other: 'Point3') -> 'Point3':
...self - other Description: V - P -> P
V - V -> V
Arguments:
Return: Vector3 | Point3: 新的向量
def __sub__(self, other):
"""
V - P -> P
V - V -> V
Args:
other ([\`Vector3\`](#class-vector3) | [\`Point3\`](./point#class-point3)): 另一个向量或点
Returns:
[\`Vector3\`](#class-vector3) | [\`Point3\`](./point#class-point3): 新的向量
"""
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)}"')self - other: Point3 Description: P - V -> P
Arguments:
- other (
Point3): 另一个点
Return: Point3: 新的点
def __rsub__(self, other: 'Point3'):
"""
P - V -> P
Args:
other ([\`Point3\`](./point#class-point3)): 另一个点
Returns:
[\`Point3\`](./point#class-point3): 新的点
"""
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
self * other: Vector3 => Vector3 @overload
def __mul__(self, other: 'Vector3') -> 'Vector3':
...@overload
self * other: RealNumber => Vector3 @overload
def __mul__(self, other: RealNumber) -> 'Vector3':
...self * other: int | float | Vector3 => Vector3 Description: 数组运算 非点乘。点乘使用@，叉乘使用cross。
Arguments:
Return: Vector3: 数组运算结果
def __mul__(self, other: 'int | float | Vector3') -> 'Vector3':
"""
数组运算 非点乘。点乘使用@，叉乘使用cross。
Args:
other ([\`Vector3\`](#class-vector3) | [\`float\`](https%3A//docs.python.org/3/library/functions.html#float)): 另一个向量或数
Returns:
[\`Vector3\`](#class-vector): 数组运算结果
"""
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)}'")self * other: RealNumber => Vector3 def __rmul__(self, other: 'RealNumber') -> 'Vector3':
return self.__mul__(other)self @ other: Vector3 => RealNumber Description: 点乘。
Arguments:
- other (
Vector3): 另一个向量
Return: float: 点乘结果
def __matmul__(self, other: 'Vector3') -> 'RealNumber':
"""
点乘。
Args:
other ([\`Vector3\`](#class-vector3)): 另一个向量
Returns:
[\`float\`](https%3A//docs.python.org/3/library/functions.html#float): 点乘结果
"""
return self.x * other.x + self.y * other.y + self.z * other.zself / other: RealNumber => Vector3 def __truediv__(self, other: RealNumber) -> 'Vector3':
return Vector3(self.x / other, self.y / other, self.z / other)- self => Vector3 Description: 取负。
Return: Vector3: 负向量
def __neg__(self) -> 'Vector3':
"""
取负。
Returns:
[\`Vector3\`](#class-vector3): 负向量
"""
return Vector3(-self.x, -self.y, -self.z)zero_vector3 Description: 零向量
Type: Vector3
Default: Vector3(0, 0, 0)
x_axis Description: x轴单位向量
Type: Vector3
Default: Vector3(1, 0, 0)
y_axis Description: y轴单位向量
Type: Vector3
Default: Vector3(0, 1, 0)
z_axis Description: z轴单位向量
Type: Vector3
Default: Vector3(0, 0, 1)