================ N = 3, a^2 = 8, b^2 = 3, caustic a''^2 = 128/25, b''^2 = 3/25 (lambda = 72/25) ================
M1: certified billiard 3-periodic (on ellipse, sides tangent to caustic, reflection law): True
--- M1: O_p = (-4*sqrt(5)/3, 0), R* = 2*sqrt(2)/3
    A       = 54*sqrt(6)/25   ~ 5.290897844411664692106134
    Adag    = (sqrt(15) + 4*sqrt(6))/64   ~ 0.2136084737084395199682563
    Apol    = (8*sqrt(15) + 32*sqrt(6))/3   ~ 36.45584617957367807458240
    AdualR  = (27*sqrt(15) + 108*sqrt(6))/320   ~ 1.153485758025573407828584
    AdualP  = (27*sqrt(15) + 108*sqrt(6))/1280   ~ 0.2883714395063933519571460
    Aped    = (-24*sqrt(15) + 96*sqrt(6))/625   ~ 0.2275190639971313474922190
    Aant    = -9*sqrt(15) + 36*sqrt(6)   ~ 53.32478062432765956848884
    ApedX   = 3*sqrt(6)/800   ~ 0.009185586535436917868239815
    k813 Apol/Adag                     = 512/3   ~ 170.666666666666666666666666667
    k814R Apol/AdualR                  = 2560/81   ~ 31.6049382716049382716049382716
    k814P Apol/AdualP                  = 10240/81   ~ 126.419753086419753086419753086
    k815(i) ApedX*AdualR               = 243*sqrt(10)/256000 + 243/32000   ~ 0.0105954432477379538190458247316
    k815(i)P ApedX*AdualP              = 243*sqrt(10)/1024000 + 243/128000   ~ 0.00264886081193448845476145618291
    k815(iii) Aped*AdualR              = 6561/25000   ~ 0.262440000000000000000000000000
    k816(i) ApedX/AdualR               = (16 - 2*sqrt(10))/1215   ~ 0.00796332895445534266337630692275
    k816(i)P ApedX/AdualP              = (64 - 8*sqrt(10))/1215   ~ 0.0318533158178213706535052276910
    k816(iii) Aped/AdualR              = (18944 - 4096*sqrt(10))/30375   ~ 0.197244796837870559872675952000
    k804 A*Adag                        = 81*sqrt(10)/800 + 81/100   ~ 1.13018061309204840736488797137
    Aant/A                             = 50/3 - 25*sqrt(10)/12   ~ 10.0785882079825430583356384491
    k818 Adag/Aant                     = (8*sqrt(10) + 37)/15552   ~ 0.00400580126551871364814757898376
M2: certified billiard 3-periodic (on ellipse, sides tangent to caustic, reflection law): True
--- M2: O_p = (-4*sqrt(5)/3, 0), R* = 2*sqrt(2)/3
    A       = 144/25   ~ 5.760000000000000000000000
    Adag    = 9/64   ~ 0.1406250000000000000000000
    Apol    = 24   ~ 24.00000000000000000000000
    AdualR  = 243/320   ~ 0.7593750000000000000000000
    AdualP  = 243/1280   ~ 0.1898437500000000000000000
    Aped    = 216/625   ~ 0.3456000000000000000000000
    Aant    = 96   ~ 96.00000000000000000000000
    ApedX   = 27/3200   ~ 0.008437500000000000000000000
    k813 Apol/Adag                     = 512/3   ~ 170.666666666666666666666666667
    k814R Apol/AdualR                  = 2560/81   ~ 31.6049382716049382716049382716
    k814P Apol/AdualP                  = 10240/81   ~ 126.419753086419753086419753086
    k815(i) ApedX*AdualR               = 6561/1024000   ~ 0.00640722656250000000000000000000
    k815(i)P ApedX*AdualP              = 6561/4096000   ~ 0.00160180664062500000000000000000
    k815(iii) Aped*AdualR              = 6561/25000   ~ 0.262440000000000000000000000000
    k816(i) ApedX/AdualR               = 1/90   ~ 0.0111111111111111111111111111111
    k816(i)P ApedX/AdualP              = 2/45   ~ 0.0444444444444444444444444444444
    k816(iii) Aped/AdualR              = 512/1125   ~ 0.455111111111111111111111111111
    k804 A*Adag                        = 81/100   ~ 0.810000000000000000000000000000
    Aant/A                             = 50/3   ~ 16.6666666666666666666666666667
    k818 Adag/Aant                     = 3/2048   ~ 0.00146484375000000000000000000000
================ N = 4, a = 5, b = 4, caustic lambda = 400/41 ================
R1 rhombus: certified billiard 4-periodic: True
--- R1 rhombus: O_p = (-51/16, 0), R* = 5/16
    A       = 40   ~ 40.00000000000000000000000
    Adag    = 1/10   ~ 0.1000000000000000000000000
    Apol    = 125/256   ~ 0.4882812500000000000000000
    AdualR  = 2048/125   ~ 16.38400000000000000000000
    AdualP  = 1024/125   ~ 8.192000000000000000000000
    Aped    = 32000/1681   ~ 19.03628792385484830458061
    Aant    = 80   ~ 80.00000000000000000000000
    ApedX   = 4096/210125   ~ 0.01949315883402736466389054
    k813 Apol/Adag                     = 625/128   ~ 4.88281250000000000000000000000
    k814R Apol/AdualR                  = 15625/524288   ~ 0.0298023223876953125000000000000
    k814P Apol/AdualP                  = 15625/262144   ~ 0.0596046447753906250000000000000
    k815(i) ApedX*AdualR               = 8388608/26265625   ~ 0.319375914336704342653182629387
    k815(i)P ApedX*AdualP              = 4194304/26265625   ~ 0.159687957168352171326591314694
    k815(iii) Aped*AdualR              = 524288/1681   ~ 311.890541344437834622248661511
    k816(i) ApedX/AdualR               = 2/1681   ~ 0.00118976799524092801903628792385
    k816(i)P ApedX/AdualP              = 4/1681   ~ 0.00237953599048185603807257584771
    k816(iii) Aped/AdualR              = 15625/13448   ~ 1.16188280785246876859012492564
    k804 A*Adag                        = 4   ~ 4.00000000000000000000000000000
    Aant/A                             = 2   ~ 2.00000000000000000000000000000
    k818 Adag/Aant                     = 1/800   ~ 0.00125000000000000000000000000000
R2 rectangle: certified billiard 4-periodic: True
--- R2 rectangle: O_p = (-51/16, 0), R* = 5/16
    A       = 1600/41   ~ 39.02439024390243902439024
    Adag    = 41/400   ~ 0.1025000000000000000000000
    Apol    = 1025/2048   ~ 0.5004882812500000000000000
    AdualR  = 10496/625   ~ 16.79360000000000000000000
    AdualP  = 5248/625   ~ 8.396800000000000000000000
    Aped    = 800/41   ~ 19.51219512195121951219512
    Aant    = 3200/41   ~ 78.04878048780487804878049
    ApedX   = 328/15625   ~ 0.02099200000000000000000000
    k813 Apol/Adag                     = 625/128   ~ 4.88281250000000000000000000000
    k814R Apol/AdualR                  = 15625/524288   ~ 0.0298023223876953125000000000000
    k814P Apol/AdualP                  = 15625/262144   ~ 0.0596046447753906250000000000000
    k815(i) ApedX*AdualR               = 3442688/9765625   ~ 0.352531251200000000000000000000
    k815(i)P ApedX*AdualP              = 1721344/9765625   ~ 0.176265625600000000000000000000
    k815(iii) Aped*AdualR              = 8192/25   ~ 327.680000000000000000000000000
    k816(i) ApedX/AdualR               = 1/800   ~ 0.00125000000000000000000000000000
    k816(i)P ApedX/AdualP              = 1/400   ~ 0.00250000000000000000000000000000
    k816(iii) Aped/AdualR              = 15625/13448   ~ 1.16188280785246876859012492564
    k804 A*Adag                        = 4   ~ 4.00000000000000000000000000000
    Aant/A                             = 2   ~ 2.00000000000000000000000000000
    k818 Adag/Aant                     = 1681/1280000   ~ 0.00131328125000000000000000000000
    claimed closed forms for N=4: 2a^4/b^4 = 625/128 , 2a^6/b^10 = 15625/524288
================ N = 6, a = 1, b = 1/2, caustic lambda = 1/9 ================
H: certified billiard 6-periodic: True
H: six distinct vertices: True
--- H: O_p = (-5*sqrt(3)/2, 0), R* = 4
    A       = 5*sqrt(5)/9   ~ 1.242259987499883164671763
    Adag    = 15*sqrt(5)/2   ~ 16.77050983124842272306880
    Apol    = 32*sqrt(5)   ~ 71.55417527999327028509356
    AdualR  = 5*sqrt(5)/96   ~ 0.1164618738281140466879778
    AdualP  = 25*sqrt(5)/768   ~ 0.07278867114257127917998612
    Aped    = 50*sqrt(5)/81   ~ 1.380288874999870182968626
    Aant    = 0   ~ 0
    ApedX   = 25*sqrt(5)/36   ~ 1.552824984374853955839704
    k813 Apol/Adag                     = 64/15   ~ 4.26666666666666666666666666667
    k814R Apol/AdualR                  = 3072/5   ~ 614.400000000000000000000000000
    k814P Apol/AdualP                  = 24576/25   ~ 983.040000000000000000000000000
    k815(i) ApedX*AdualR               = 625/3456   ~ 0.180844907407407407407407407407
    k815(i)P ApedX*AdualP              = 3125/27648   ~ 0.113028067129629629629629629630
    k815(iii) Aped*AdualR              = 625/3888   ~ 0.160751028806584362139917695473
    k816(i) ApedX/AdualR               = 40/3   ~ 13.3333333333333333333333333333
    k816(i)P ApedX/AdualP              = 64/3   ~ 21.3333333333333333333333333333
    k816(iii) Aped/AdualR              = 320/27   ~ 11.8518518518518518518518518519
    k804 A*Adag                        = 125/6   ~ 20.8333333333333333333333333333
    Aant/A                             = 0   ~ 0
    k818 Adag/Aant: antipedal area is exactly 0 -> undefined
    antipedal polygon of H w.r.t. f1: [(1, (-2*sqrt(15) - sqrt(5))/10), (sqrt(3)/2, -sqrt(5)/5), (-1, (-sqrt(5) + 2*sqrt(15))/10), (-1, (-2*sqrt(15) + sqrt(5))/10), (sqrt(3)/2, sqrt(5)/5), (1, (sqrt(5) + 2*sqrt(15))/10)]
