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3b80460 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | """Reproduce independent checks of positional endpoint-distance formulas.
Run: python3 verify_positional.py
Quadrature checks the endpoint formula independently; it is not a proof.
Opaque examples have separate elementary convex-hull proofs stated below.
No sampled line-intersection count is promoted to a rigorous global bound.
"""
from pathlib import Path
import json
import math
import platform
import numpy as np
import scipy
from scipy.integrate import quad
from positional_overlap import pair_overlap,positional_certificate
ROOT=Path(__file__).resolve().parent
def projection_overlap(theta,first,second):
n=np.array([math.cos(theta),math.sin(theta)])
u=np.asarray(first,dtype=float)@n
v=np.asarray(second,dtype=float)@n
return max(0.,min(float(u.max()),float(v.max()))-
max(float(u.min()),float(v.min())))
def independent_quadrature(first,second):
points=np.vstack([first,second]).astype(float)
critical=[0.,math.pi]
for i in range(4):
for j in range(i+1,4):
difference=points[j]-points[i]
if np.linalg.norm(difference)>0:
critical.append((math.atan2(difference[1],difference[0])+math.pi/2)%math.pi)
critical=sorted(set(critical))
values=[]
errors=[]
for a,b in zip(critical[:-1],critical[1:]):
if b-a<=1e-15:
continue
v,e=quad(lambda theta:projection_overlap(theta,first,second),a,b,
epsabs=1e-12,epsrel=1e-12)
values.append(v)
errors.append(e)
return math.fsum(values),math.fsum(errors)
def checks():
analytic=[
("parallel unit segments, separation 1",[[0,0],[1,0]],[[0,1],[1,1]],2*(math.sqrt(2)-1)),
("crossing orthogonal diameters of length 2",[[-1,0],[1,0]],[[0,-1],[0,1]],8-4*math.sqrt(2)),
("right-triangle adjacent legs",[[0,0],[1,0]],[[0,0],[0,1]],2-math.sqrt(2)),
("collinear shared endpoint",[[0,0],[1,0]],[[1,0],[2,0]],0.),
("collinear disjoint segments",[[0,0],[1,0]],[[2,0],[3,0]],0.),
("degenerate isolated point",[[0,0],[0,0]],[[1,0],[2,1]],0.)]
analytic_records=[]
for label,first,second,expected in analytic:
actual=pair_overlap(first,second)
quadrature,error=independent_quadrature(first,second)
assert abs(actual-expected)<1e-10
assert abs(actual-quadrature)<1e-10
analytic_records.append({"case":label,"expected":expected,"endpoint_formula":actual,
"independent_quadrature":quadrature,
"quadrature_reported_error":error})
rejected=False
try:
pair_overlap([[0,0],[2,0]],[[1,0],[3,0]])
except ValueError:
rejected=True
assert rejected
rng=np.random.default_rng(20261029)
random_records=[]
for number in range(100):
first=rng.uniform(-3,3,size=(2,2))
second=rng.uniform(-3,3,size=(2,2))
actual=pair_overlap(first,second)
quadrature,error=independent_quadrature(first,second)
symmetry=max(abs(actual-pair_overlap(second,first)),
abs(actual-pair_overlap(first[::-1],second)),
abs(actual-pair_overlap(first,second[::-1])))
assert abs(actual-quadrature)<2e-9
assert symmetry<2e-9
random_records.append({"index":number,"first":first.tolist(),"second":second.tolist(),
"endpoint_formula":actual,"independent_quadrature":quadrature,
"absolute_discrepancy":abs(actual-quadrature),
"quadrature_reported_error":error,"symmetry_error":symmetry})
square=[([-1,-1],[1,-1]),([1,-1],[1,1]),([1,1],[-1,1]),([-1,1],[-1,-1])]
square_record=positional_certificate(square,certified_max_hits=2)
square_record["independent_geometric_facts"]=(
"The square boundary surrounds the unit disk, so every line meeting the disk meets the boundary. "
"A line not containing an edge and not through a vertex meets a convex polygon boundary at most twice; the exceptional lines have zero line measure.")
root2=math.sqrt(2)
cross=[([-root2,0],[root2,0]),([0,-root2],[0,root2])]
cross_record=positional_certificate(cross,certified_max_hits=2)
cross_record["independent_geometric_facts"]=(
"The cross is connected. Its convex hull is the diamond |x|+|y|<=sqrt(2), which contains the unit disk. "
"Every linear projection of a connected set is an interval, so containment of the disk in its convex hull makes it opaque. Two segments give the automatic almost-everywhere hit bound 2.")
triangle_vertices=np.array([[0.,2.],[-math.sqrt(3),-1.],[math.sqrt(3),-1.]])
triangle=[(triangle_vertices[i],triangle_vertices[(i+1)%3]) for i in range(3)]
triangle_record=positional_certificate(triangle,certified_max_hits=2)
triangle_record["independent_geometric_facts"]=(
"The equilateral triangle is circumscribed about the unit disk; its boundary is opaque and a generic line hits it at most twice.")
for record in [square_record,cross_record,triangle_record]:
assert record["forest_necessary_lower_bound"]<=record["total_segment_length"]+1e-10
assert record["conditional_pair_necessary_lower_bound"]<=record["total_segment_length"]+1e-10
assert abs(square_record["pair_energy"]-8)<1e-10
assert abs(triangle_record["pair_energy"]-6*math.sqrt(3))<1e-10
assert abs(cross_record["pair_energy"]-(8*math.sqrt(2)-8))<1e-10
fractional=[]
for n in [3,4,6,12,24,96,384]:
half_perimeter=n*math.tan(math.pi/n)
fractional.append({"circumscribed_regular_polygon_sides":n,
"half_density_cost":half_perimeter,
"excess_above_pi":half_perimeter-math.pi})
assert all(fractional[i]["half_density_cost"]>fractional[i+1]["half_density_cost"]
for i in range(len(fractional)-1))
assert fractional[-1]["excess_above_pi"]<1e-4
return {"scope":"Necessary positional certificates, not an opacity decision procedure and not a solution of the minimum opaque-barrier problem.",
"arithmetic":"Exact formulas evaluated in float64; numerical tests do not replace the analytic proofs.",
"runtime":{"python":platform.python_version(),"numpy":np.__version__,"scipy":scipy.__version__},
"summary":{"all_assertions_passed":True,"analytic_cases":len(analytic_records),
"random_cases":len(random_records),"positive_collinear_overlap_rejected":rejected,
"max_random_quadrature_discrepancy":max(r["absolute_discrepancy"] for r in random_records),
"max_random_symmetry_discrepancy":max(r["symmetry_error"] for r in random_records),
"actual_opaque_examples":3},
"analytic_cases":analytic_records,"random_cases":random_records,
"actual_opaque_examples":{"square_boundary":square_record,"cross":cross_record,"triangle_boundary":triangle_record},
"fractional_no_go":{"analytic_statement":
"Half of the unit-circle boundary measure has cost pi and weighted intersection count 1 on every non-tangent line through the disk. Consequently line-count covering constraints allowing fractional density cannot imply a bound strictly greater than pi. For a regular circumscribed N-gon, half boundary density has cost N*tan(pi/N), tending to pi. This is a fractional relaxation, not a physical barrier.",
"polygon_diagnostics":fractional}}
if __name__=="__main__":
result=checks()
(ROOT/"POSITIONAL_VERIFICATION.json").write_text(json.dumps(result,indent=2)+"\n")
print(json.dumps(result["summary"],indent=2))
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