input: saved_eval/hfar_smoke_n7_s41_rational.json  (n = 7)
E1 ok: all 7 vertices exactly on S^2 with z > 0
E2 g(Q) simple (exact): True; signed doubled area 50.4741 (> 0 means counterclockwise)
E2 origin strictly left of every edge line of g(Q) (exact): True
E2 winding number of g(Q) about the origin (exact): 1
E2 => Q simple, R star-shaped wrt N with N interior to the kernel, Q in the open hemisphere about N: True
E3 sign det(v_{i-1}, v_i, v_{i+1}) for i = 0..6: [1, 1, 1, 1, 1, 1, -1]
E3 exact sign of the sine numerator s_i: [1, 1, 1, 1, 1, 1, -1]
  v0: l_0 = [0.67335478479400745, 0.67335478479400745]  tau_0 = [2.4991842494810621, 2.4991842494810621]  interior angle (deg) ~ 36.8073
  v1: l_1 = [0.40124477194257263, 0.40124477194257263]  tau_1 = [0.63729613833873411, 0.63729613833873411]  interior angle (deg) ~ 143.4856
  v2: l_2 = [0.64889397183648845, 0.64889397183648845]  tau_2 = [0.1887704297732751, 0.1887704297732751]  interior angle (deg) ~ 169.1843
  v3: l_3 = [1.3714958257882395, 1.3714958257882395]  tau_3 = [2.2985210132155913, 2.2985210132155913]  interior angle (deg) ~ 48.3044
  v4: l_4 = [0.25862474038714273, 0.25862474038714273]  tau_4 = [0.64798088713340452, 0.64798088713340452]  interior angle (deg) ~ 142.8734
  v5: l_5 = [1.5171891376549216, 1.5171891376549216]  tau_5 = [2.4958587638007419, 2.4958587638007419]  interior angle (deg) ~ 36.9978
  v6: l_6 = [1.5031146923460084, 1.5031146923460084]  tau_6 = [-2.9866016632008798, -2.9866016632008798]  interior angle (deg) ~ 351.1197
Gauss-Bonnet: sum tau = [5.7810098185419294, 5.7810098185419294], Area(fan about N) = [0.50217548863765737, 0.50217548863765737], sum tau + Area - 2pi = [-7.5338294331276316e-90, 1.3430168590715263e-89] (contains 0: True)
cut at v0: crossing pairs [(2, 5), (2, 6)] ; certified-disjoint pairs 13 ; undecided [] ; adjacent ok True
   end point w_n = ([-0.1158189121308535, -0.1158189121308535], [0.35412750147448768, 0.35412750147448768])
   CROSS segments (2,5): signs (o1,o2,o3,o4) = +--+ ; o = [[0.69705345675206676, 0.69705345675206676], [-0.036585521363144899, -0.036585521363144899], [-0.50218468368672387, -0.50218468368672387], [0.23145429442848772, 0.23145429442848772]]
   distances of endpoints from the other line: [1.074218, -0.056381, -0.330997, 0.152555] ; certified lower bound 0.056381
   crossing at fraction [0.68451199931727202, 0.68451199931727202] of segment 2 and [0.95013143732202376, 0.95013143732202376] of segment 5
   CROSS segments (2,6): signs (o1,o2,o3,o4) = -++- ; o = [[-0.036585521363144899, -0.036585521363144899], [0.58112920806938673, 0.58112920806938673], [0.40527716882276332, 0.40527716882276332], [-0.21243756060976823, -0.21243756060976823]]
   distances of endpoints from the other line: [-0.056381, 0.895569, 0.269625, -0.141332] ; certified lower bound 0.056381
   crossing at fraction [0.65609115261841722, 0.65609115261841722] of segment 2 and [0.059227212206441102, 0.059227212206441102] of segment 6
cut at v1: crossing pairs [(1, 4), (1, 5)] ; certified-disjoint pairs 13 ; undecided [] ; adjacent ok True
   end point w_n = ([-0.14203782935043696, -0.14203782935043696], [0.14250858372632097, 0.14250858372632097])
   CROSS segments (1,4): signs (o1,o2,o3,o4) = +--+ ; o = [[0.69705345675206676, 0.69705345675206676], [-0.036585521363144899, -0.036585521363144899], [-0.50218468368672387, -0.50218468368672387], [0.23145429442848772, 0.23145429442848772]]
   distances of endpoints from the other line: [1.074218, -0.056381, -0.330997, 0.152555] ; certified lower bound 0.056381
   crossing at fraction [0.68451199931727202, 0.68451199931727202] of segment 1 and [0.95013143732202376, 0.95013143732202376] of segment 4
   CROSS segments (1,5): signs (o1,o2,o3,o4) = -++- ; o = [[-0.036585521363144899, -0.036585521363144899], [0.58112920806938673, 0.58112920806938673], [0.40527716882276332, 0.40527716882276332], [-0.21243756060976823, -0.21243756060976823]]
   distances of endpoints from the other line: [-0.056381, 0.895569, 0.269625, -0.141332] ; certified lower bound 0.056381
   crossing at fraction [0.65609115261841722, 0.65609115261841722] of segment 1 and [0.059227212206441102, 0.059227212206441102] of segment 5
cut at v2: crossing pairs [(0, 3), (0, 4)] ; certified-disjoint pairs 13 ; undecided [] ; adjacent ok True
   end point w_n = ([-0.19767294960268669, -0.19767294960268669], [-0.013775113519959518, -0.013775113519959518])
   CROSS segments (0,3): signs (o1,o2,o3,o4) = +--+ ; o = [[0.69705345675206676, 0.69705345675206676], [-0.036585521363144899, -0.036585521363144899], [-0.50218468368672387, -0.50218468368672387], [0.23145429442848772, 0.23145429442848772]]
   distances of endpoints from the other line: [1.074218, -0.056381, -0.330997, 0.152555] ; certified lower bound 0.056381
   crossing at fraction [0.68451199931727202, 0.68451199931727202] of segment 0 and [0.95013143732202376, 0.95013143732202376] of segment 3
   CROSS segments (0,4): signs (o1,o2,o3,o4) = -++- ; o = [[-0.036585521363144899, -0.036585521363144899], [0.58112920806938673, 0.58112920806938673], [0.40527716882276332, 0.40527716882276332], [-0.21243756060976823, -0.21243756060976823]]
   distances of endpoints from the other line: [-0.056381, 0.895569, 0.269625, -0.141332] ; certified lower bound 0.056381
   crossing at fraction [0.65609115261841722, 0.65609115261841722] of segment 0 and [0.059227212206441102, 0.059227212206441102] of segment 4
cut at v3: crossing pairs [] ; certified-disjoint pairs 15 ; undecided [] ; adjacent ok True
   end point w_n = ([-0.058726430449017292, -0.058726430449017292], [0.42433982314902075, 0.42433982314902075])
cut at v4: crossing pairs [(1, 6), (2, 6)] ; certified-disjoint pairs 13 ; undecided [] ; adjacent ok True
   end point w_n = ([-0.32415694310213095, -0.32415694310213095], [-0.050361250595544228, -0.050361250595544228])
   CROSS segments (1,6): signs (o1,o2,o3,o4) = -++- ; o = [[-0.41254805592456512, -0.41254805592456512], [0.59311558318580726, 0.59311558318580726], [0.75781189344374067, 0.75781189344374067], [-0.24785174566663179, -0.24785174566663179]]
   distances of endpoints from the other line: [-0.271916, 0.390931, 0.552544, -0.180716] ; certified lower bound 0.180716
   crossing at fraction [0.75354409165485414, 0.75354409165485414] of segment 1 and [0.41022469131877171, 0.41022469131877171] of segment 6
   CROSS segments (2,6): signs (o1,o2,o3,o4) = +--+ ; o = [[0.43134796740521802, 0.43134796740521802], [-0.83164652359704094, -0.83164652359704094], [-0.24785174566663179, -0.24785174566663179], [1.0151427453356272, 1.0151427453356272]]
   distances of endpoints from the other line: [0.286969, -0.553282, -0.180716, 0.740172] ; certified lower bound 0.180716
   crossing at fraction [0.19624135135375542, 0.19624135135375542] of segment 2 and [0.34152798803019208, 0.34152798803019208] of segment 6
cut at v5: crossing pairs [(0, 5), (1, 5)] ; certified-disjoint pairs 13 ; undecided [] ; adjacent ok True
   end point w_n = ([0.17917250585764735, 0.17917250585764735], [0.35393019160169709, 0.35393019160169709])
   CROSS segments (0,5): signs (o1,o2,o3,o4) = -++- ; o = [[-0.41254805592456512, -0.41254805592456512], [0.59311558318580726, 0.59311558318580726], [0.75781189344374067, 0.75781189344374067], [-0.24785174566663179, -0.24785174566663179]]
   distances of endpoints from the other line: [-0.271916, 0.390931, 0.552544, -0.180716] ; certified lower bound 0.180716
   crossing at fraction [0.75354409165485414, 0.75354409165485414] of segment 0 and [0.41022469131877171, 0.41022469131877171] of segment 5
   CROSS segments (1,5): signs (o1,o2,o3,o4) = +--+ ; o = [[0.43134796740521802, 0.43134796740521802], [-0.83164652359704094, -0.83164652359704094], [-0.24785174566663179, -0.24785174566663179], [1.0151427453356272, 1.0151427453356272]]
   distances of endpoints from the other line: [0.286969, -0.553282, -0.180716, 0.740172] ; certified lower bound 0.180716
   crossing at fraction [0.19624135135375542, 0.19624135135375542] of segment 1 and [0.34152798803019208, 0.34152798803019208] of segment 5
cut at v6: crossing pairs [(0, 4), (0, 6), (3, 6)] ; certified-disjoint pairs 12 ; undecided [] ; adjacent ok True
   end point w_n = ([0.066142495102978469, 0.066142495102978469], [0.37057549904467241, 0.37057549904467241])
   CROSS segments (0,4): signs (o1,o2,o3,o4) = +--+ ; o = [[0.43134796740521802, 0.43134796740521802], [-0.83164652359704094, -0.83164652359704094], [-0.24785174566663179, -0.24785174566663179], [1.0151427453356272, 1.0151427453356272]]
   distances of endpoints from the other line: [0.286969, -0.553282, -0.180716, 0.740172] ; certified lower bound 0.180716
   crossing at fraction [0.19624135135375542, 0.19624135135375542] of segment 0 and [0.34152798803019208, 0.34152798803019208] of segment 4
   CROSS segments (0,6): signs (o1,o2,o3,o4) = -++- ; o = [[-0.83609076178538089, -0.83609076178538089], [0.55701747723750128, 0.55701747723750128], [0.50643658263570668, 0.50643658263570668], [-0.8866716563871756, -0.8866716563871756]]
   distances of endpoints from the other line: [-0.556239, 0.370575, 0.333799, -0.584417] ; certified lower bound 0.333799
   crossing at fraction [0.36352996016369715, 0.36352996016369715] of segment 0 and [0.60016209678855248, 0.60016209678855248] of segment 6
   CROSS segments (3,6): signs (o1,o2,o3,o4) = +--+ ; o = [[0.69705345675206676, 0.69705345675206676], [-0.036585521363144899, -0.036585521363144899], [-0.50218468368672387, -0.50218468368672387], [0.23145429442848772, 0.23145429442848772]]
   distances of endpoints from the other line: [1.074218, -0.056381, -0.330997, 0.152555] ; certified lower bound 0.056381
   crossing at fraction [0.68451199931727202, 0.68451199931727202] of segment 3 and [0.95013143732202376, 0.95013143732202376] of segment 6
