
== M1: a^2=8, b^2=3, lambda=72/25, focus=((-1)*sqrt(5), 0)
  ok   P0 on ellipse
  ok   P1 on ellipse
  ok   P2 on ellipse
  ok   side 0 tangent to caustic
  ok   side 1 tangent to caustic
  ok   side 2 tangent to caustic
  ok   reflection law at P0
  ok   reflection law at P1
  ok   reflection law at P2
  ok   vertices pairwise distinct (least period N)
  ok   polar vertex = pole of side line (Lemma 3.1(b))
  ok   inverse of focal pedal = polar polygon (reading (ii) = A_pol)
  ok   polar side 0 touches circle (O_p, a/b^2)
  ok   polar side 1 touches circle (O_p, a/b^2)
  ok   polar side 2 touches circle (O_p, a/b^2)
  ok   dual vertex 0 on circle (O_p, b^2/a)
  ok   dual vertex 1 on circle (O_p, b^2/a)
  ok   dual vertex 2 on circle (O_p, b^2/a)
  ok   point-wise dual = midpoint polygon of dual
    A          = (54/25)*sqrt(6)    (~ 5.29089784441)
    Adag       = (1/16)*sqrt(6) + (1/64)*sqrt(15)    (~ 0.213608473708)
    Apol       = (32/3)*sqrt(6) + (8/3)*sqrt(15)    (~ 36.4558461796)
    Adual      = (27/80)*sqrt(6) + (27/320)*sqrt(15)    (~ 1.15348575803)
    Apt        = (27/320)*sqrt(6) + (27/1280)*sqrt(15)    (~ 0.288371439506)
    Aped       = (96/625)*sqrt(6) + (-24/625)*sqrt(15)    (~ 0.227519063997)
    Aant       = (36)*sqrt(6) + (-9)*sqrt(15)    (~ 53.3247806243)
    Aip        = (3/800)*sqrt(6)    (~ 0.00918558653544)

== M2: a^2=8, b^2=3, lambda=72/25, focus=((-1)*sqrt(5), 0)
  ok   P0 on ellipse
  ok   P1 on ellipse
  ok   P2 on ellipse
  ok   side 0 tangent to caustic
  ok   side 1 tangent to caustic
  ok   side 2 tangent to caustic
  ok   reflection law at P0
  ok   reflection law at P1
  ok   reflection law at P2
  ok   vertices pairwise distinct (least period N)
  ok   polar vertex = pole of side line (Lemma 3.1(b))
  ok   inverse of focal pedal = polar polygon (reading (ii) = A_pol)
  ok   polar side 0 touches circle (O_p, a/b^2)
  ok   polar side 1 touches circle (O_p, a/b^2)
  ok   polar side 2 touches circle (O_p, a/b^2)
  ok   dual vertex 0 on circle (O_p, b^2/a)
  ok   dual vertex 1 on circle (O_p, b^2/a)
  ok   dual vertex 2 on circle (O_p, b^2/a)
  ok   point-wise dual = midpoint polygon of dual
    A          = 144/25    (~ 5.76)
    Adag       = 9/64    (~ 0.140625)
    Apol       = 24    (~ 24)
    Adual      = 243/320    (~ 0.759375)
    Apt        = 243/1280    (~ 0.18984375)
    Aped       = 216/625    (~ 0.3456)
    Aant       = 96    (~ 96)
    Aip        = 27/3200    (~ 0.0084375)

-- Table 3 (N = 3) against the note
  ok   Table 3 M1 A
  ok   Table 3 M1 Adag
  ok   Table 3 M1 Apol
  ok   Table 3 M1 Adual
  ok   Table 3 M1 Aped
  ok   Table 3 M1 Aant
  ok   Table 3 M1 Aip
  ok   M1: A_pt = A_dual/4
  ok   M1: A_pol/A_dag = 512/3
  ok   M1: A_pol/A_dual = 2560/81
  ok   M1: A_ped*A_dual = 6561/25000 (reading (iii))
  ok   Table 3 M2 A
  ok   Table 3 M2 Adag
  ok   Table 3 M2 Apol
  ok   Table 3 M2 Adual
  ok   Table 3 M2 Aped
  ok   Table 3 M2 Aant
  ok   Table 3 M2 Aip
  ok   M2: A_pt = A_dual/4
  ok   M2: A_pol/A_dag = 512/3
  ok   M2: A_pol/A_dual = 2560/81
  ok   M2: A_ped*A_dual = 6561/25000 (reading (iii))
    Aip*Adual(M1) = 243/32000 + (243/256000)*sqrt(10)    (~ 0.0105954432477)
    Aip*Adual(M2) = 6561/1024000    (~ 0.0064072265625)
  ok   Thm 3(A) reading (i) M1 value
  ok   Thm 3(A) reading (i) M2 value
  ok   difference = 243/204800 + 243 sqrt10/256000
  ok   difference > 0
    Apol*Adual(M1) = 999/40 + (27/5)*sqrt(10)    (~ 42.0512993649)
    Apol*Adual(M2) = 729/40    (~ 18.225)
  ok   Thm 3(A) reading (ii) M1 value
  ok   Thm 3(A) reading (ii) M2 value
  ok   reading (ii) differs
  ok   point-wise dual, reading (i): products differ
  k908,b at M1: (i) 1  (ii) 2.30734152893878  (iii) 0.433399211801962
  ok   k908,b holds under reading (i) at M1
  k908,b at M2: (i) 1  (ii) 1  (iii) 1
  ok   k908,b holds under reading (i) at M2

== R1: a^2=25, b^2=16, lambda=400/41, focus=(-3, 0)
  ok   P0 on ellipse
  ok   P1 on ellipse
  ok   P2 on ellipse
  ok   P3 on ellipse
  ok   side 0 tangent to caustic
  ok   side 1 tangent to caustic
  ok   side 2 tangent to caustic
  ok   side 3 tangent to caustic
  ok   reflection law at P0
  ok   reflection law at P1
  ok   reflection law at P2
  ok   reflection law at P3
  ok   vertices pairwise distinct (least period N)
  ok   polar vertex = pole of side line (Lemma 3.1(b))
  ok   inverse of focal pedal = polar polygon (reading (ii) = A_pol)
  ok   polar side 0 touches circle (O_p, a/b^2)
  ok   polar side 1 touches circle (O_p, a/b^2)
  ok   polar side 2 touches circle (O_p, a/b^2)
  ok   polar side 3 touches circle (O_p, a/b^2)
  ok   dual vertex 0 on circle (O_p, b^2/a)
  ok   dual vertex 1 on circle (O_p, b^2/a)
  ok   dual vertex 2 on circle (O_p, b^2/a)
  ok   dual vertex 3 on circle (O_p, b^2/a)
  ok   point-wise dual = midpoint polygon of dual
    A          = 40    (~ 40)
    Adag       = 1/10    (~ 0.1)
    Apol       = 125/256    (~ 0.48828125)
    Adual      = 2048/125    (~ 16.384)
    Apt        = 1024/125    (~ 8.192)
    Aped       = 32000/1681    (~ 19.0362879239)
    Aant       = 80    (~ 80)
    Aip        = 4096/210125    (~ 0.019493158834)

== R2: a^2=25, b^2=16, lambda=400/41, focus=(-3, 0)
  ok   P0 on ellipse
  ok   P1 on ellipse
  ok   P2 on ellipse
  ok   P3 on ellipse
  ok   side 0 tangent to caustic
  ok   side 1 tangent to caustic
  ok   side 2 tangent to caustic
  ok   side 3 tangent to caustic
  ok   reflection law at P0
  ok   reflection law at P1
  ok   reflection law at P2
  ok   reflection law at P3
  ok   vertices pairwise distinct (least period N)
  ok   polar vertex = pole of side line (Lemma 3.1(b))
  ok   inverse of focal pedal = polar polygon (reading (ii) = A_pol)
  ok   polar side 0 touches circle (O_p, a/b^2)
  ok   polar side 1 touches circle (O_p, a/b^2)
  ok   polar side 2 touches circle (O_p, a/b^2)
  ok   polar side 3 touches circle (O_p, a/b^2)
  ok   dual vertex 0 on circle (O_p, b^2/a)
  ok   dual vertex 1 on circle (O_p, b^2/a)
  ok   dual vertex 2 on circle (O_p, b^2/a)
  ok   dual vertex 3 on circle (O_p, b^2/a)
  ok   point-wise dual = midpoint polygon of dual
    A          = 1600/41    (~ 39.0243902439)
    Adag       = 41/400    (~ 0.1025)
    Apol       = 1025/2048    (~ 0.50048828125)
    Adual      = 10496/625    (~ 16.7936)
    Apt        = 5248/625    (~ 8.3968)
    Aped       = 800/41    (~ 19.512195122)
    Aant       = 3200/41    (~ 78.0487804878)
    Aip        = 328/15625    (~ 0.020992)

-- Table 3 (N = 4) against the note
  ok   Table 3 R1 A
  ok   Table 3 R1 Adag
  ok   Table 3 R1 Apol
  ok   Table 3 R1 Adual
  ok   Table 3 R1 Aped
  ok   Table 3 R1 Aant
  ok   Table 3 R1 Aip
  ok   R1: A_pt = A_dual/2
  ok   R1: A_pol/A_dag = 2a^4/b^4 = 625/128
  ok   R1: A_pol/A_dual = 2a^6/b^10
  ok   R1: A_ped/A_dual = 2a^6/(b^2(a^2+b^2)^2)
  ok   R1: A*A_dag = 4
  ok   R1: A_dag*A_ant = 8
  ok   R1: A_ant/A = 2
  ok   Table 3 R2 A
  ok   Table 3 R2 Adag
  ok   Table 3 R2 Apol
  ok   Table 3 R2 Adual
  ok   Table 3 R2 Aped
  ok   Table 3 R2 Aant
  ok   Table 3 R2 Aip
  ok   R2: A_pt = A_dual/2
  ok   R2: A_pol/A_dag = 2a^4/b^4 = 625/128
  ok   R2: A_pol/A_dual = 2a^6/b^10
  ok   R2: A_ped/A_dual = 2a^6/(b^2(a^2+b^2)^2)
  ok   R2: A*A_dag = 4
  ok   R2: A_dag*A_ant = 8
  ok   R2: A_ant/A = 2
    Aip/Adual(R1) = 2/1681    (~ 0.00118976799524)
    Aip/Adual(R2) = 1/800    (~ 0.00125)
  ok   Thm 4(A): 2/1681 at R1
  ok   Thm 4(A): 1/800 at R2
  FerVanish formula at R1: (hk)^2 = 400, predicted 0.0194931588340274, A_ip = 0.0194931588340274
  ok   FerVanish N=4 formula agrees at R1 (to 1e-14)
  FerVanish formula at R2: (hk)^2 = 1681/4, predicted 0.020992, A_ip = 0.020992
  ok   FerVanish N=4 formula agrees at R2 (to 1e-14)

== H: a^2=1, b^2=1/4, lambda=1/9, focus=((-1/2)*sqrt(3), 0)
  ok   P0 on ellipse
  ok   P1 on ellipse
  ok   P2 on ellipse
  ok   P3 on ellipse
  ok   P4 on ellipse
  ok   P5 on ellipse
  ok   side 0 tangent to caustic
  ok   side 1 tangent to caustic
  ok   side 2 tangent to caustic
  ok   side 3 tangent to caustic
  ok   side 4 tangent to caustic
  ok   side 5 tangent to caustic
  ok   reflection law at P0
  ok   reflection law at P1
  ok   reflection law at P2
  ok   reflection law at P3
  ok   reflection law at P4
  ok   reflection law at P5
  ok   vertices pairwise distinct (least period N)
  ok   polar vertex = pole of side line (Lemma 3.1(b))
  ok   inverse of focal pedal = polar polygon (reading (ii) = A_pol)
  ok   polar side 0 touches circle (O_p, a/b^2)
  ok   polar side 1 touches circle (O_p, a/b^2)
  ok   polar side 2 touches circle (O_p, a/b^2)
  ok   polar side 3 touches circle (O_p, a/b^2)
  ok   polar side 4 touches circle (O_p, a/b^2)
  ok   polar side 5 touches circle (O_p, a/b^2)
  ok   dual vertex 0 on circle (O_p, b^2/a)
  ok   dual vertex 1 on circle (O_p, b^2/a)
  ok   dual vertex 2 on circle (O_p, b^2/a)
  ok   dual vertex 3 on circle (O_p, b^2/a)
  ok   dual vertex 4 on circle (O_p, b^2/a)
  ok   dual vertex 5 on circle (O_p, b^2/a)
  ok   point-wise dual = midpoint polygon of dual
    A          = (5/9)*sqrt(5)    (~ 1.2422599875)
    Adag       = (15/2)*sqrt(5)    (~ 16.7705098312)
    Apol       = (32)*sqrt(5)    (~ 71.55417528)
    Adual      = (5/96)*sqrt(5)    (~ 0.116461873828)
    Apt        = (25/768)*sqrt(5)    (~ 0.0727886711426)
    Aped       = (50/81)*sqrt(5)    (~ 1.380288875)
    Aant       = 0    (~ 0)
    Aip        = (25/36)*sqrt(5)    (~ 1.55282498437)
  ok   H: six distinct antipedal vertices
  ok   H: A_ant = 0 exactly
  ok   H: A = 5 sqrt5/9
  ok   H: A_dag = 15 sqrt5/2
  ok   H: A_pol/A_dag = 64/15
  ok   H: A_ip/A_dual = 40/3
  ok   H: A_ped/A_dual = 320/27
  R(1, 1/4, 1/9) = 0
  ok   H: R = 0

ALL EXACT CHECKS PASSED
