如果我们有一个二维均匀变量，我们可以像这里讨论的那样在单位圆中生成均匀分布。
我的问题类似，我希望均匀采样两个相交圆的交集区域，其中一个圆总是单位圆，而另一个圆可以自由移动和调整大小，就像这里一样。
u = unifrom2D()
A;B; // Intersection points
for p in allPoints
theta = u.x * (getTheta(A) - getTheta(B)) + getTheta(B)
r = sqrt(u.y + (1- u.y)*length2(lineIntersection(theta)))
p = (r * cos(theta), r * sin(theta))
P(Z < x) = F(x) <=> (x = F^-1(y))
P(Z < F^-1(y)) = F(F^-1(y)) = y <=> (F is monotonous)
P(F(Z) < y) = y
Z 具有所请求的分布 F，那么 F(Z) 是均匀分布的。反过来也是一样的：Z = F^-1(Y),
其中Y在区间[0,1]上均匀分布，满足所需的分布。
如果F的形式为
/ 0, x < a
F(x) = | (F0(x)-F0(a)) / (F0(b)-F0(a)), a <= x <= b
\ 1, b < x
[F(a), F(b)]中均匀选择一个Y0，并设置Z = F0^-1(Y0)。(theta, r)来参数化该线段，其中中心角度theta是从一个线段侧面测量的。当线段的中心角度为alpha时，与从线段开始的角度为theta的扇形相交的线段面积为（对于单位圆，theta在[0, alpha/2]）.F0_theta(theta) = 0.5*(theta - d*(s - d*tan(alpha/2-theta)))
其中 s = AB/2 = sin(alpha/2) 以及 d = dist(M,AB) = cos(alpha/2) （圆心到线段的距离）。（本文不考虑alpha/2 <= theta <= alpha 的对称情况）。
我们需要一个随机变量theta，满足 P(theta < x) = F_theta(x)。无法通过符号计算得出 F_theta 的逆 - 必须通过优化算法（例如牛顿迭代）确定它。
一旦确定了theta，我们需要在以下范围内随机生成半径r：
[r_min, 1], r_min = d/cos(alpha/2-theta).
对于x在[0, 1-r_min]范围内，分布必须是
F0_r(x) = (x+r_min)^2 - r_min^2 = x^2 + 2*x*r_min.
这里可以以符号方式计算其反函数：
F0_r^-1(y) = -r_min + sqrt(r_min^2+y)
这里有一个Python的实现，用于概念证明：
from math import sin,cos,tan,sqrt
from scipy.optimize import newton
# area of segment of unit circle
# alpha: center angle of segment (0 <= alpha <= pi)
def segmentArea(alpha):
return 0.5*(alpha - sin(alpha))
# generate a function that gives the area of a segment of a unit circle
# intersected with a sector of given angle, where the sector starts at one end of the segment.
# The returned function is valid for [0,alpha/2].
# For theta=alpha/2 the returned function gives half of the segment area.
# alpha: center angle of segment (0 <= alpha <= pi)
def segmentAreaByAngle_gen(alpha):
alpha_2 = 0.5*alpha
s,d = sin(alpha_2),cos(alpha_2)
return lambda theta: 0.5*(theta - d*(s - d*tan(alpha_2-theta)))
# generate derivative function generated by segmentAreaByAngle_gen
def segmentAreaByAngleDeriv_gen(alpha):
alpha_2 = 0.5*alpha
d = cos(alpha_2)
return lambda theta: (lambda dr = d/cos(alpha_2-theta): 0.5*(1 - dr*dr))()
# generate inverse of function generated by segmentAreaByAngle_gen
def segmentAreaByAngleInv_gen(alpha):
x0 = sqrt(0.5*segmentArea(alpha)) # initial guess by approximating half of segment with right-angled triangle
return lambda area: newton(lambda theta: segmentAreaByAngle_gen(alpha)(theta) - area, x0, segmentAreaByAngleDeriv_gen(alpha))
# for a segment of the unit circle in canonical position
# (i.e. symmetric to x-axis, on positive side of x-axis)
# generate uniformly distributed random point in upper half
def randomPointInSegmentHalf(alpha):
FInv = segmentAreaByAngleInv_gen(alpha)
areaRandom = random.uniform(0,0.5*segmentArea(alpha))
thetaRandom = FInv(areaRandom)
alpha_2 = 0.5*alpha
d = cos(alpha_2)
rMin = d/cos(alpha_2-thetaRandom)
secAreaRandom = random.uniform(0, 1-rMin*rMin)
rRandom = sqrt(rMin*rMin + secAreaRandom)
return rRandom*cos(alpha_2-thetaRandom), rRandom*sin(alpha_2-thetaRandom)
这个可视化似乎验证了均匀分布（在中心角为pi/2的线段的上半部分）：
import matplotlib.pyplot as plot
segmentPoints = [randomPointInSegmentHalf(pi/2) for _ in range(500)]
plot.scatter(*zip(*segmentPoints))
plot.show()
theta0时，您希望P（点在theta < theta0的区域内）=area(theta0)/area(theta_max)。因此，area(theta0)/area(theta_max)是CDF。 - coproc