N = 4  lam = 4*u**2*(u**2 + 1)**2/(u**4 + 6*u**2 + 1)  D = 16*(u**8*w**2 + 4*u**6*w**2 + 4*u**4*w**4 - 2*u**4*w**2 + 4*u**4 + 4*u**2*w**2 + w**2)/((w**2 + 1)**2*(u**4 + 6*u**2 + 1)**2)
check P2 on unit circle: True
check chord P1P2 tangent (relation): True
check chord P2P3 tangent: 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 0.4s
  antipedal/pedal areas w.r.t. f1 computed, elapsed 0.9s
  antipedal/pedal areas w.r.t. f2 computed, elapsed 1.3s
  antipedal/pedal areas w.r.t. O computed, elapsed 1.9s

=== identities along the whole family (w-free means constant) ===
  A'*A  (k106)          = 32*u**2*(u**2 + 1)**2   [w-free: constant along the family]
  A*_f1 / A             = 2   [w-free: constant along the family]
  A*_f2 / A             = 2   [w-free: constant along the family]
  A*_f1 - A*_f2         = 0   [w-free: constant along the family]
  k401 f1: A'*A*_f1     = 64*u**2*(u**2 + 1)**2   [w-free: constant along the family]
  k401 f2: A'*A*_f2     = 64*u**2*(u**2 + 1)**2   [w-free: constant along the family]
  k401 O : A'*A*_O      : NOT constant (degree 8 in w). value(w=0) = 64*u**2*(u**2 + 1)**2 ; value(w=1) = 64*u**2*(u**2 + 1)**2
  k402 f1: A'/A*_f1     : NOT constant (degree 8 in w). value(w=0) = 1 ; value(w=1) = 1
  k402 O : A'/A*_O      : NOT constant (degree 8 in w). value(w=0) = 1 ; value(w=1) = 1
  k403 f1: A_f1*A*_f1   = 256*u**4*(u**2 + 1)**4/(u**4 + 6*u**2 + 1)**2   [w-free: constant along the family]
  k403 O : A_O*A*_O     : NOT constant (degree 8 in w). value(w=0) = 256*u**4*(u**2 + 1)**4/(u**4 + 6*u**2 + 1)**2 ; value(w=1) = 256*u**4*(u**2 + 1)**4/(u**4 + 6*u**2 + 1)**2
  k404 f1: A*_f1/A_f1   : NOT constant (degree 8 in w). value(w=0) = (u**4 + 6*u**2 + 1)**2/(4*u**2*(u**2 + 1)**2) ; value(w=1) = (u**4 + 6*u**2 + 1)**2/(4*u**2*(u**2 + 1)**2)
  k204 f1: A/A_f1       : NOT constant (degree 8 in w). value(w=0) = (u**4 + 6*u**2 + 1)**2/(8*u**2*(u**2 + 1)**2) ; value(w=1) = (u**4 + 6*u**2 + 1)**2/(8*u**2*(u**2 + 1)**2)
  k204 O : A/A_O        : NOT constant (degree 8 in w). value(w=0) = (u**4 + 6*u**2 + 1)**2/(8*u**2*(u**2 + 1)**2) ; value(w=1) = (u**4 + 6*u**2 + 1)**2/(8*u**2*(u**2 + 1)**2)
  k303a f1: A'*A'_f1    : NOT constant (degree 8 in w). value(w=0) = 32*u**2*(u**2 + 1)**2 ; value(w=1) = 32*u**2*(u**2 + 1)**2
  k303a O : A'*A'_O     : NOT constant (degree 8 in w). value(w=0) = 32*u**2*(u**2 + 1)**2 ; value(w=1) = 32*u**2*(u**2 + 1)**2
  A'/A'' (k113)         = (u**4 + 6*u**2 + 1)**2/(4*u**2*(u**2 + 1)**2)   [w-free: constant along the family]
  A*_O / A              : NOT constant (degree 8 in w). value(w=0) = 2 ; value(w=1) = 2
  A'*_O / A'            : NOT constant (degree 8 in w). value(w=0) = (u**4 + 6*u**2 + 1)**2/(8*u**2*(u**2 + 1)**2) ; value(w=1) = (u**4 + 6*u**2 + 1)**2/(8*u**2*(u**2 + 1)**2)
  A'*_f1 / A'           = 2   [w-free: constant along the family]

total elapsed 2.3s
cas4 rc=0
