p587.py

# 
# Solution to Project Euler problem 587

import itertools, math


# Start by defining the coordinate system in a convenient way. The position and scale of the diagram don't
# matter because we only care about the ratio of areas, not the absolute areas. So, let the bottom left
# of the diagram be the origin (x = 0, y = 0), and let each circle to have a radius of 1.
# 
# The leftmost circle is centered at (1, 1), and its equation is (x - 1)^2 + (y - 1)^2 = 1.
# The diagonal line has slope = s = 1 / n (for any positive n), and the line's equation is y = s * x.
# From basic geometry, the area of the blue L-section is 1 - pi / 4.
# 
# Let's find the x-coordinate where the diagonal line intersects the first circle.
# Take the equation of the circle and substitute y = s * x for the line:
# 
#   (x - 1)^2 + (s*x - 1)^2 = 1.
#   (x^2 - 2x + 1) + (s^2 x^2 - 2s*x + 1) = 1.
#   (1 + s^2)x^2 + (-2 - 2s)x + 1 = 0.
# 
# We can apply the quadratic formula with a = 1 + s^2, b = -2 - 2s, c = 1. There are two solutions for x,
# and we only want the smaller value. Thus, let X = (-b - sqrt(b^2 - 4ac)) / (2a). Or equivalently
# with more numerical stability (using the Citardauq formula), X = (2c) / (-b + sqrt(b^2 - 4ac)).
# 
# The orange concave triangle can be divided into two parts by a vertical line:
# 
# - The left part is a proper triangle, whose area is easily seen as x * y / 2 = X^2 * s / 2.
# 
# - The right part is the region between the circle and the baseline. Let's re-express
#   the circle's equation in terms of y, and only keep the lower semicircle:
#   
#     (x - 1)^2 + (y - 1)^2 = 1.
#     (y - 1)^2 = 1 - (x - 1)^2.
#     y - 1 = -sqrt(1 - (x - 1)^2).
#     y = 1 - sqrt(1 - (x - 1)^2).
#     y = 1 - sqrt(1 - (x^2 - 2x + 1)).
#     y = 1 - sqrt(2x - x^2).
#   
#   Now, the indefinite integral of f(x) = 1 - sqrt(2x - x^2) with respect to x
#   is F(x) = (x - 1) - [sqrt(2x - x^2) * (x - 1) + asin(x - 1)] / 2.
#   Finding this integral is not obvious, but verifying it is a fairly straightforward
#   mechanical procedure involving differentiation and simplification.
#   
#   The area of the right part is the integral of f(x) for x from X to 1, because the start is
#   the x-coordinate where line meets the circle, and the end is where the circle meets the baseline.
#   Hence the area is equal to F(1) - F(X).
# 
# All in all, for any given n, the area of the orange concave triangle is X^2 * s / 2 + F(1) - F(X).
# The rest of the algorithm is a brute-force search with n = 1, 2, 3, ... until the ratio condition is met.
# 
# Additional notes:
# - Intuitively, as n increases and the slope gets smaller, the area of the orange concave triangle should strictly
#   decrease. This statement is in fact true, but proving it involves a big pile of differentiation and algebra.
#   0. We need to show that X (which is the x-coordinate of the line-circle intersection) increases with n.
#      We'd differentiate X with respect to n, and get an expression that is always positive for any positive n.
#   1. Because X increases with n, the area of the right part, with its always-positive integrand, must decrease.
#   2. As for the left part, we'd differentiate X^2 * s / 2 with respect to n, and get a huge messy formula.
#      It turns out this formula is negative for all n > 1. Hence the area of this triangle also decreases with n.
#   After we prove that increasing n leads to decreasing orange area, we could use
#   binary search to find the minimum value of n needed to meet the ratio requirement.
# - The use of floating-point arithmetic, for basic arithmetic operations (+ - * /) and irrational functions (sqrt,
#   asin) alike, is inherently difficult or impossible to prove the correctness of. Furthermore, the algorithms
#   for irrational functions are hard to understand and beyond the scope of this problem, and the error bounds for
#   all operations are difficult to reason about.
#   It should be possible to solve this particular problem using only integer arithmetic in a provably correct way.
#   The basic idea would be to round the result of each operation both down and up to an integer fraction,
#   keep track of pessimistic intervals that are guaranteed to contain the true value, accept a comparison only
#   if the intervals don't overlap, and recompute everything at a higher precision if a comparison is inconclusive.
#   Note: Because it doesn't seem easy to compute pi and asin(), it might be better to
#   approximate integrals directly using the Darboux definition of lower and upper sums.
def compute():
	# The indefinite integral of (1 - sqrt(2x - x^2)) dx.
	def integral(x):
		t = x - 1.0
		return t - (math.sqrt(x * (2.0 - x)) * t + math.asin(t)) / 2.0
	
	lsectionarea = 1.0 - math.pi / 4.0
	for i in itertools.count(1):
		slope = 1.0 / i
		a = slope**2 + 1.0
		b = -2.0 * (slope + 1.0)
		c = 1.0
		x = (2.0 * c) / (-b + math.sqrt(b * b - 4 * a * c))
		concavetrianglearea = (x**2 * slope / 2) + (integral(1.0) - integral(x))
		if concavetrianglearea / lsectionarea < 0.001:
			return str(i)


if __name__ == "__main__":
	print(compute())