import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
def triangle_ratio(n):
return 1-QQ(2)/(n+1)*sum(QQ(1)/k for k in range(1,n+1))
def parallelogram_ratio(n):
s=sum((1-QQ(1)/2**k)/k for k in range(1,n+2))
return 1-QQ(8)/(3*(n+1))*(s-QQ(1)/((n+1)*2**(n+1)))
print(triangle_ratio(3), parallelogram_ratio(3))
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
def ellipse_ratio(n, digits=100):
C=ComplexBallField(numberdb.bits(digits, losing=80))
pi=C.pi()
def integrand(u, analytic):
s=(pi*u).sin()
c=(pi*u).cos()
return s**4*(u-s*c/pi)**(n-1)
return (1-C(8*n)/3*C.integral(integrand,0,1)).real()
print(ellipse_ratio(3))
The 36 triangle and parallelogram values use exact rational arithmetic. The 18 ellipse values use Arb’s certified complex integration of an entire trigonometric integrand, with 80 guard bits beyond 100 significant digits. The largest measured relative radius is less than $6.1\times10^{-122}$. Every ball is required to be finite, to lie between zero and one, and to have a relative radius below $10^{-110}$.
The polygon formulas are displayed by their author in [4]. An independent exact integration of the signed area contributed by possible hull edges reproduces every rational entry. Its controls are the known three- and four-point means. The integration is repeated after an invertible affine map with determinant $5$ and a translation.
A separate signed-area integral, given for rotationally symmetric distributions on printed page 15 of [5], is evaluated for every ellipse entry at 135 and 175 decimal digits. The known three-point ratio is tested first as a normalization control. The two quadratures agree to absolute tolerance $10^{-130}$ and agree with the Arb computation to $10^{-118}$. Three closed forms at $n=3,4,5$, including the five-point expression in [6], are also checked in 600-bit ball arithmetic. The numerical quadratures are agreement checks; the generator’s Arb enclosures supply the certified digits.
All 54 entries read back from the database were compared with the independent computations. The stored values satisfy all 51 adjacent monotonicity comparisons and the three-point relation to the convex-position probabilities for each of the three affine classes. The attached checks reproduce these comparisons when run with access to the draft.