生成两个平方和==1的随机数

Question

嗨，我想生成两个随机数，使得它们的平方之和等于1。

我写了这段代码。它们的平方之和可能不完全等于1，但应该接近0.999。我也使用了if(math.isclose(abs(gene_value_1)**2 + abs(gene_value_2)**2, 1))，但它不起作用。

gene_value_1 = random.uniform(0, 1)
gene_value_2 = random.uniform(0, 1)
if(abs(gene_value_1)**2 + abs(gene_value_2)**2) == 1:
    print(added)

我想要生成两个随机数，它们的平方之和几乎等于1。

Answer 1

您的要求实际上不是两个不同的数字。它适用于平方和等于1的一对数字。

如果是x**2 + y**2 = 1，那么y完全由x：y = sqrt(1 - x**2)决定。

gene_value_1 = random.uniform(0, 1)
gene_value_2 = math.sqrt(1.0 - gene_value_1**2)

正如对问题和答案的评论中所提到的，以这种方式得到的分布对于这两个数字是不一致的。由于gene_value_1和gene_value_2在均匀情况下将描述笛卡尔空间中的单位圆，所以您可以这样做。

angle = random.uniform(0, 2 * math.pi)
gene_value_1 = math.cos(angle)
gene_value_2 = math.sin(angle)

Answer 2

我完全赞扬Mark Dickinson's suggestion使用三角函数生成两个平方和为1的随机数。这是最重要的三角恒等式之一，有时被称为正弦曲线和余弦的毕达哥拉斯公式。

正弦^2(θ)+cos^2(θ)=1

因此，我们在区间theta中随机采样[-pi, pi]，并取其sine和cosine。这给了我们两个数字，当它独立地平方，然后求和，等于1。

因此，实现应该类似于：

def squared_unity():
    r = np.random.uniform(-np.pi, np.pi)
    r1, r2 = np.cos(r), np.sin(r)
    # sanity check
    sq_sum = np.sum(np.square([r1, r2]))
    print("the two random numbers are: {}, {}".format(round(r1, 4), round(r2, 4)))
    print("sum of the squares of them is: {}".format(round(sq_sum, 4)))

In [172]: for i in range(10): 
     ...:     squared_unity() 

the two random numbers are: -0.4232, 0.906
sum of the squares of them is: 1.0
the two random numbers are: -0.6432, 0.7657
sum of the squares of them is: 1.0
the two random numbers are: -0.9854, 0.1701
sum of the squares of them is: 1.0
the two random numbers are: 0.6192, -0.7852
sum of the squares of them is: 1.0
the two random numbers are: 0.613, 0.7901
sum of the squares of them is: 1.0
the two random numbers are: 0.3289, -0.9444
sum of the squares of them is: 1.0
the two random numbers are: -0.6289, -0.7775
sum of the squares of them is: 1.0
the two random numbers are: 0.5851, 0.811
sum of the squares of them is: 1.0
the two random numbers are: -0.9515, 0.3076
sum of the squares of them is: 1.0
the two random numbers are: 0.992, -0.1258
sum of the squares of them is: 1.0

Answer 3

产生两个随机数，其平方之和几乎等于1。

假设几乎相等意味着一些小的三角洲

delta = 0.00000000001
gene_value_1 = random.uniform(0,1)
gene_value_2 = math.sqrt(1.0 - gene_value_1**2)
gene_value_2 = random.uniform(gene_value_2-delta,gene_value_2+delta)