1. N = 4 rhombus, symbolic a > b > 0 (c^2 = a^2 - b^2)
   tangency + reflection law + on E: True
   focus f1: A = 2*a*b, A^dag = 2/(a*b), A'^dag = 1/(a*b), A* = 4*a*b, kappa = 1/2, kappa* = 2
   focus f2: A = 2*a*b, A^dag = 2/(a*b), A'^dag = 1/(a*b), A* = 4*a*b, kappa = 1/2, kappa* = 2
   outer polygon: [(-a, b), (-a, -b), (a, -b), (a, b)]
2. N = 6, a = 2, b = 1, lambda = 4/9 (exact)
   tangency + reflection law + on E: True
   focus f1: A = 20*sqrt(5)/9, A* = 0 (distinct antipedal vertices: 6), Abar = 200*sqrt(5)/81, kappa_p = 10/9
   focus f2: A = 20*sqrt(5)/9, A* = 0 (distinct antipedal vertices: 6), Abar = 200*sqrt(5)/81, kappa_p = 10/9
   closed form kappa* (eq. R) at lambda = 4/9: 0
3a/3b. Kriebel examples: a^2 = 21, b^2 = 16, lambda = 336/25
   H: billiard/tangency exact: True; signed area A = 128*sqrt(21)/25 (sign gives the orientation)
      outer-vertex inversion areas B+ = (-sqrt(5) + 7*sqrt(21))/1920, B- = (sqrt(5) + 7*sqrt(21))/1920, product = 1/3600
      original-vertex inversion product D+ D- = 1/324
      outer focal pedal areas B+ = (7*sqrt(5) + 49*sqrt(21))/10, B- = (-7*sqrt(5) + 49*sqrt(21))/10, product = 12544/25
      B+/D- = 2016/5, B-/D+ = 2016/5
      original focal pedal product A+ A- = 7225344/390625, A+ = (84*sqrt(5) + 588*sqrt(21))/625, B+/A+ = 125/24
   V: billiard/tangency exact: True; signed area A = 588/25 (sign gives the orientation)
      outer-vertex inversion areas B+ = 1/60, B- = 1/60, product = 1/3600
      original-vertex inversion product D+ D- = 1/324
      outer focal pedal areas B+ = 112/5, B- = 112/5, product = 12544/25
      B+/D- = 2016/5, B-/D+ = 2016/5
      original focal pedal product A+ A- = 7225344/390625, A+ = 2688/625, B+/A+ = 125/24
3b (even control). x^2/4 + y^2 = 1, lambda = 4/5
   D: exact billiard: True, B+ = 4, B- = 4, product = 16
   C: exact billiard: True, B+ = 5, B- = 5, product = 25
3d. x^2/25 + y^2/16 = 1, F = (3,0), rhombus (5,0),(0,4),(-5,0),(0,-4)
   A^dag_F = 1/10, A'^dag_F = 1/20
