Ailo0810

in my inequalities above, n = 0, 1, 2, … can be thought of as the number of times the 32-bit signed integer has been overflown. for example, the first ring occurs when it was overflown n = 1 time.

\[

2^{32} n \le \left(\frac{x}{8}\right)^2 + \left(\frac{z}{8}\right)^2 < 2^{32}\left(n + \frac12\right)

\]

$$

2^{32} n \le \left(\frac{x}{8}\right)^2 + \left(\frac{z}{8}\right)^2 < 2^{32}\left(n + \frac12\right)

$$

The issue is in the expression

MathHelper.sqrt((float)(i * i + j * j))

here, i * i + j * j can overflow the 2³¹ - 1 limit of a 32-bit signed integer even before getting converted into float. The next 2³¹ values become negative so sqrt returns NaN.

i and j correspond to the terrain generation coordinates, which for The End are x/8 and z/8.

A 32-bit signed integer can take 2³² unique values before getting wrapped on itself, which can be interpreted as only the first half being positive.

So terrain generates only when

where n = 0, 1, 2, …

The horizontal distance r from the origin at which terrain generates therefore satisfies

Note that x/8 and z/8 in are evaluated as integers, so the strict inequality can be replaced with the exact upper bounds

Interestingly, the area of the nth ring is approximately

which does not depend on n, meaning that all the rings have equal area.