N = 6  lam = 4*u**2*(u**2 + 1)**2/(u + 1)**4  D = 16*(u**8*w**2 + 4*u**7*w**2 + 4*u**6*w**2 + 4*u**5*w**4 + 4*u**5*w**2 + 4*u**5 + 4*u**4*w**4 - 2*u**4*w**2 + 4*u**4 + 4*u**3*w**4 + 4*u**3*w**2 + 4*u**3 + 4*u**2*w**2 + 4*u*w**2 + w**2)/((u + 1)**8*(w**2 + 1)**2)
check P2 on unit circle: True
check chord P1P2 tangent (relation): True
check P3 on unit circle: True
check chord P2P3 tangent: True
check chord P3P4 tangent (closure, Cayley condition for lam): True
check P3 != P2: True
vertices in Q(u,w)[s]: computed; reflection law at every vertex: True
A'' tangency of all sides to the caustic: checked (assert).  elapsed 3.2s
  antipedal/pedal areas w.r.t. f1 computed, elapsed 8.7s
  antipedal/pedal areas w.r.t. f2 computed, elapsed 14.3s
  antipedal/pedal areas w.r.t. O computed, elapsed 21.4s

=== identities along the whole family (w-free means constant) ===
  A'*A  (k106)          = 32*u**2*(u**2 + 1)**2*(u**2 + u + 1)*(u**2 + 4*u + 1)/(u + 1)**4   [w-free: constant along the family]
  A*_f1 / A             = -(u + 1)**4*(u**2 - 4*u + 1)/(8*u**2*(u**2 + u + 1))   [w-free: constant along the family]
  A*_f2 / A             = -(u + 1)**4*(u**2 - 4*u + 1)/(8*u**2*(u**2 + u + 1))   [w-free: constant along the family]
  A*_f1 - A*_f2         = 0   [w-free: constant along the family]
  k401 f1: A'*A*_f1     = -4*(u**2 + 1)**2*(u**2 - 4*u + 1)*(u**2 + 4*u + 1)   [w-free: constant along the family]
  k401 f2: A'*A*_f2     = -4*(u**2 + 1)**2*(u**2 - 4*u + 1)*(u**2 + 4*u + 1)   [w-free: constant along the family]
  k401 O : A'*A*_O      : NOT constant (degree 12 in w). value(w=0) = (u**2 + 1)**2*(u**2 + 4*u + 1)*(u**6 - 4*u**5 + 11*u**4 + 8*u**3 + 11*u**2 - 4*u + 1)/(u*(u**2 + u + 1)) ; value(w=1) = 64*u**2*(u**2 + u + 1)*(u**6 + 2*u**5 - u**4 + 8*u**3 - u**2 + 2*u + 1)/((u**2 + 1)*(u**2 + 4*u + 1))
  k402 f1: A'/A*_f1     : NOT constant (degree 12 in w). value(w=0) = -u*(u**2 + 4*u + 1)/((u**2 - 4*u + 1)*(u**2 + u + 1)) ; value(w=1) = -16*u**2*(u**2 + u + 1)**2/((u**2 + 1)*(u**2 - 4*u + 1)*(u**2 + 4*u + 1)**2)
  k402 O : A'/A*_O      : NOT constant (degree 12 in w). value(w=0) = 4*u**2*(u**2 + 4*u + 1)/(u**6 - 4*u**5 + 11*u**4 + 8*u**3 + 11*u**2 - 4*u + 1) ; value(w=1) = (u**2 + 1)**2*(u**2 + u + 1)/(u**6 + 2*u**5 - u**4 + 8*u**3 - u**2 + 2*u + 1)
  k403 f1: A_f1*A*_f1   : NOT constant (degree 12 in w). value(w=0) = -64*u*(u**2 + 1)**3*(u**2 - 4*u + 1)*(u**2 + u + 1)**3/(u + 1)**8 ; value(w=1) = -4*(u**2 + 1)**4*(u**2 - 4*u + 1)*(u**2 + 4*u + 1)**3/(u + 1)**8
  k403 O : A_O*A*_O     : NOT constant (degree 12 in w). value(w=0) = 8*(u**2 + 1)**2*(u**2 + u + 1)*(u**4 + 2*u**3 + 6*u**2 + 2*u + 1)*(u**6 - 4*u**5 + 11*u**4 + 8*u**3 + 11*u**2 - 4*u + 1)/(u + 1)**8 ; value(w=1) = 32*u**2*(u**2 + 4*u + 1)*(u**4 + 2*u**3 + 6*u**2 + 2*u + 1)*(u**6 + 2*u**5 - u**4 + 8*u**3 - u**2 + 2*u + 1)/(u + 1)**8
  k404 f1: A*_f1/A_f1   = -(u + 1)**8*(u**2 - 4*u + 1)/(16*u**2*(u**2 + 1)*(u**2 + u + 1)**2)   [w-free: constant along the family]
  k204 f1: A/A_f1       = (u + 1)**4/(2*(u**2 + 1)*(u**2 + u + 1))   [w-free: constant along the family]
  k204 O : A/A_O        = (u + 1)**4/(u**4 + 2*u**3 + 6*u**2 + 2*u + 1)   [w-free: constant along the family]
  k303a f1: A'*A'_f1    = 8*u*(u**2 + 1)**2*(u**2 + u + 1)*(u**2 + 4*u + 1)/(u + 1)**2   [w-free: constant along the family]
  k303a O : A'*A'_O     = 4*u*(u**2 + 1)*(u**2 + u + 1)*(u**2 + 4*u + 1)   [w-free: constant along the family]
  A'/A'' (k113)         = (u + 1)**8/(4*u*(u**2 + 1)*(u**2 + u + 1)*(u**2 + 4*u + 1))   [w-free: constant along the family]
  A*_O / A              : NOT constant (degree 12 in w). value(w=0) = (u + 1)**4*(u**6 - 4*u**5 + 11*u**4 + 8*u**3 + 11*u**2 - 4*u + 1)/(32*u**3*(u**2 + u + 1)**2) ; value(w=1) = 2*(u + 1)**4*(u**6 + 2*u**5 - u**4 + 8*u**3 - u**2 + 2*u + 1)/((u**2 + 1)**3*(u**2 + 4*u + 1)**2)
  A'*_O / A'            : NOT constant (degree 12 in w). value(w=0) = 2*(u + 1)**4*(u**4 - u**3 + 3*u**2 - u + 1)/((u**2 + 1)**2*(u**2 + u + 1)*(u**2 + 4*u + 1)) ; value(w=1) = (u + 1)**4*(u**4 - 4*u**3 + 18*u**2 - 4*u + 1)/(8*u**2*(u**2 + u + 1)*(u**2 + 4*u + 1))
  A'*_f1 / A'           = -(u + 1)**2*(u**4 - 4*u**3 - 6*u**2 - 4*u + 1)/(2*u*(u**2 + u + 1)*(u**2 + 4*u + 1))   [w-free: constant along the family]

total elapsed 23.6s
cas6 rc=0
