四元数是一个代数概念,通常用于描述旋转,特别是在3D建模和游戏中有广泛的应用。
下面就一步一步演示此对象的创建方法,特别要关注双下划线开始和结束的那些特殊方法。
class Quaternion: def __init__(self, w, x, y, z): self.w = w self.x = x self.y = y self.z = z
这是首先定义了__init__初始化方法,通过这个方法,在创建实例时将 分别定义为实数。
>>> q1 = Quaternion(1, 2, 3, 4)>>> q1<__main__.Quaternion at 0x7f4210f483c8>
貌似一个四元数对象定义了。但是,它没有友好的输出。接下来可以通过定义__repr__和__str__,让它对机器或者开发者都更友好。继续在类中增加如下方法:
class Quaternion: def __init__(self, w, x, y, z): self.w = w self.x = x self.y = y self.z = z def __repr__(self): return 'Quaternion({}, {}, {}, {})'.format( self.w, self.x, self.y, self.z) def __str__(self): return 'Q = {:.2f} + {:.2f}i + {:.2f}j + {:.2f}k'.format( self.w, self.x, self.y, self.z)
对于这两方法,在前面的《Python中的5对必知的魔法方法》中已经有介绍,请参考。
>>> q1 # calls q1.__repr__Quaternion(1, 2, 3, 4)>>> print(q1) # calls q1.__str__Q = 1.00 + 2.00i + 3.00j + 4.00k
上述类已经能实现实例化了,但是,该对象还要参与计算,比如加、减法等。这类运算如何实现?
class Quaternion: def __init__(self, w, x, y, z): self.w = w self.x = x self.y = y self.z = z def __repr__(self): return 'Quaternion({}, {}, {}, {})'.format( self.w, self.x, self.y, self.z) def __str__(self): return 'Q = {:.2f} + {:.2f}i + {:.2f}j + {:.2f}k'.format( self.w, self.x, self.y, self.z) def __add__(self, other): w = self.w + other.w x = self.x + other.x y = self.y + other.y z = self.z + other.z return Quaternion(w, x, y, z)
__add__是用于实现对象的加法运算的特殊方法。这样就可以使用+运算符了:
>>> q1 = Quaternion(1, 2, 3, 4)>>> q2 = Quaternion(0, 1, 3, 5)>>> q1 + q2Quaternion(1, 3, 6, 9)
类似地,通过__sub__实现减法运算,不过,这次用一行代码实现。
class Quaternion: def __init__(self, w, x, y, z): self.w = w self.x = x self.y = y self.z = z def __repr__(self): return 'Quaternion({}, {}, {}, {})'.format( self.w, self.x, self.y, self.z) def __str__(self): return 'Q = {:.2f} + {:.2f}i + {:.2f}j + {:.2f}k'.format( self.w, self.x, self.y, self.z) def __add__(self, other): w = self.w + other.w x = self.x + other.x y = self.y + other.y z = self.z + other.z return Quaternion(w, x, y, z) def __sub__(self, other): return Quaternion(*list(map(lambda i, j: i - j, self.__dict__.values(), other.__dict__.values())))
有点酷。这里使用了实例对象的__dict__属性,它以字典形式包含了实例的所有属性,
乘法,如果了解一下线性代数,会感觉有点复杂。其中常见的一个是“点积”,自从Python3.5以后,用@符号调用__matmul__方法实现,对于四元数对象而言不能,就是元素与元素对应相乘。
对于四元数而言——本质就是向量,也可以说是矩阵,其乘法就跟矩阵乘法类似,比如,同样不遵守互换率: 。
class Quaternion: def __init__(self, w, x, y, z): self.w = w self.x = x self.y = y self.z = z def __repr__(self): return 'Quaternion({}, {}, {}, {})'.format( self.w, self.x, self.y, self.z) def __str__(self): return 'Q = {:.2f} + {:.2f}i + {:.2f}j + {:.2f}k'.format( self.w, self.x, self.y, self.z) def __add__(self, other): w = self.w + other.w x = self.x + other.x y = self.y + other.y z = self.z + other.z return Quaternion(w, x, y, z) def __sub__(self, other): return Quaternion(*list(map(lambda i, j: i - j, self.__dict__.values(), other.__dict__.values()))) def __mul__(self, other): if isinstance(other, Quaternion): w = self.w * other.w - self.x * other.x - self.y * other.y - self.z * other.z x = self.w * other.x + self.x * other.w + self.y * other.z - self.z * other.y y = self.w * other.y + self.y * other.w + self.z * other.x - self.x * other.z z = self.w * other.z + self.z * other.w + self.x * other.y - self.y * other.x return Quaternion(w, x, y, z) elif isinstance(other, (int, float)): return Quaternion(*[other * i for i in self.__dict__.values()]) else: raise TypeError('Operation undefined.')
在__mul__方法中,如果other引用一个四元数对象,那么就会计算Hamilton积,并返回一个新的对象;如果other是一个标量(比如整数),就会与四元数对象中的每个元素相乘。
如前所述,四元数的乘法不遵循交换律,但是,如果执行2 * q1这样的操作,按照上面的方式,会报错——在上面的__mul__方法中解决了q1 * 2的运算,而一般我们认为这两个计算是相同的。为此,定义了·rmul·来解决此问题:
class Quaternion: def __init__(self, w, x, y, z): self.w = w self.x = x self.y = y self.z = z def __repr__(self): return 'Quaternion({}, {}, {}, {})'.format( self.w, self.x, self.y, self.z) def __str__(self): return 'Q = {:.2f} + {:.2f}i + {:.2f}j + {:.2f}k'.format( self.w, self.x, self.y, self.z) def __add__(self, other): w = self.w + other.w x = self.x + other.x y = self.y + other.y z = self.z + other.z return Quaternion(w, x, y, z) def __sub__(self, other): return Quaternion(*list(map(lambda i, j: i - j, self.__dict__.values(), other.__dict__.values()))) def __mul__(self, other): if isinstance(other, Quaternion): w = self.w * other.w - self.x * other.x - self.y * other.y - self.z * other.z x = self.w * other.x + self.x * other.w + self.y * other.z - self.z * other.y y = self.w * other.y + self.y * other.w + self.z * other.x - self.x * other.z z = self.w * other.z + self.z * other.w + self.x * other.y - self.y * other.x return Quaternion(w, x, y, z) elif isinstance(other, (int, float)): return Quaternion(*[other * i for i in self.__dict__.values()]) else: raise TypeError('Operation undefined.') def __rmul__(self, other): if isinstance(other, (int, float)): return self.__mul__(other) else: raise TypeError('Operation undefined.')
比较两个四元数是否相等,可以通过定义__eq__方法来实现。
class Quaternion: def __init__(self, w, x, y, z): self.w = w self.x = x self.y = y self.z = z def __repr__(self): return 'Quaternion({}, {}, {}, {})'.format( self.w, self.x, self.y, self.z) def __str__(self): return 'Q = {:.2f} + {:.2f}i + {:.2f}j + {:.2f}k'.format( self.w, self.x, self.y, self.z) def __add__(self, other): w = self.w + other.w x = self.x + other.x y = self.y + other.y z = self.z + other.z return Quaternion(w, x, y, z) def __sub__(self, other): return Quaternion(*list(map(lambda i, j: i - j, self.__dict__.values(), other.__dict__.values()))) def __mul__(self, other): if isinstance(other, Quaternion): w = self.w * other.w - self.x * other.x - self.y * other.y - self.z * other.z x = self.w * other.x + self.x * other.w + self.y * other.z - self.z * other.y y = self.w * other.y + self.y * other.w + self.z * other.x - self.x * other.z z = self.w * other.z + self.z * other.w + self.x * other.y - self.y * other.x return Quaternion(w, x, y, z) elif isinstance(other, (int, float)): return Quaternion(*[other * i for i in self.__dict__.values()]) else: raise TypeError('Operation undefined.') def __rmul__(self, other): if isinstance(other, (int, float)): return self.__mul__(other) else: raise TypeError('Operation undefined.') def __eq__(self, other): r = list(map(lambda i, j: abs(i) == abs(j), self.__dict__.values(), other.__dict__.values())) return sum(r) == len(r)
下面的方法,也可以接续到前面的类中,不过就不是特殊放方法了。
from math import sqrtdef norm(self): return sqrt(sum([i**2 for i in self.__dict__.values()))def conjugate(self): x, y, z = -self.x, -self.y, -self.z return Quaterion(self.w, x, y, z)def normalize(self): norm = self.norm() return Quaternion(*[i / norm for in self.__dict__.values()])def inverse(self): qconj = self.conjugate() norm = self.norm() return Quaternion(*[i / norm for i in qconj.__dict__.values()])
用这些方法实现了各自常见的操作。
通过本文的这个示例,可以更深刻理解双下划线的方法为编程带来的便捷。
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