====== S2
PASS S2 P1: rho^2 = Id
PASS S2 P1: rho fixes Pt
PASS S2 P1: rho^T A rho = A (preserves the absolute I)
PASS S2 P1: x, rho(x), Pt dependent (each line through pi(P) is invariant)
PASS S2 P1: polar line of pi(P) fixed pointwise (rho y = -y)
PASS S2 P2: <B rho x, rho x> = <Bx,x> on W_P, chart ellipse (a,b symbolic)
PASS S2 P2: whole pencil B - lam*A invariant
====== H2
PASS H2 P1: rho^2 = Id
PASS H2 P1: rho fixes Pt
PASS H2 P1: rho^T A rho = A (preserves the absolute I)
PASS H2 P1: x, rho(x), Pt dependent (each line through pi(P) is invariant)
PASS H2 P1: polar line of pi(P) fixed pointwise (rho y = -y)
PASS H2 P2: <B rho x, rho x> = <Bx,x> on W_P, chart ellipse (a,b symbolic)
PASS H2 P2: whole pencil B - lam*A invariant
S2 | parabola Y=X^2 (conic) | degree 1: dim of homogeneous invariants = 0  (0.0s)
S2 | parabola Y=X^2 (conic) | degree 2: dim of homogeneous invariants = 2  (0.0s)
       -x1**2 + x2*x3
       x1**2 + x2**2 + x3**2
S2 | parabola Y=X^2 (conic) | degree 3: dim of homogeneous invariants = 0  (0.1s)
S2 | parabola Y=X^2 (conic) | degree 4: dim of homogeneous invariants = 3  (0.2s)
       (x1**2 - x2*x3)**2
       -(x1**2 - x2*x3)*(x1**2 + x2**2 + x3**2)
       -x1**4 + 2*x1**2*x2**2 + 4*x1**2*x2*x3 + 2*x1**2*x3**2 + x2**4 + x3**4
S2 | parabola Y=X^2 (conic) | degree 5: dim of homogeneous invariants = 0  (0.4s)
S2 | parabola Y=X^2 (conic) | degree 6: dim of homogeneous invariants = 4  (0.9s)
       -(x1**2 - x2*x3)**3
       (x1**2 - x2*x3)**2*(x1**2 + x2**2 + x3**2)
       (x1**2 - x2*x3)*(x1**4 - 2*x1**2*x2**2 - 4*x1**2*x2*x3 - 2*x1**2*x3**2 - x2**4 - x3**4)
       -(x1**2 + x2**2 + x3**2)*(2*x1**4 - 2*x1**2*x2**2 - 6*x1**2*x2*x3 - 2*x1**2*x3**2 - x2**4 + x2**2*x3**2 - x3**4)
S2 | Y=X^2+X^3 (non-conic) | degree 1: dim of homogeneous invariants = 0  (0.0s)
S2 | Y=X^2+X^3 (non-conic) | degree 2: dim of homogeneous invariants = 1  (0.0s)
       x1**2 + x2**2 + x3**2
S2 | Y=X^2+X^3 (non-conic) | degree 3: dim of homogeneous invariants = 0  (0.3s)
S2 | Y=X^2+X^3 (non-conic) | degree 4: dim of homogeneous invariants = 1  (1.0s)
       (x1**2 + x2**2 + x3**2)**2
S2 | Y=X^2+X^3 (non-conic) | degree 5: dim of homogeneous invariants = 0  (3.4s)
S2 | Y=X^2+X^3 (non-conic) | degree 6: dim of homogeneous invariants = 1  (9.3s)
       (x1**2 + x2**2 + x3**2)**3
S2 | Y=X^2+X^4 (non-conic, symmetric) | degree 1: dim of homogeneous invariants = 0  (0.0s)
S2 | Y=X^2+X^4 (non-conic, symmetric) | degree 2: dim of homogeneous invariants = 1  (0.0s)
       x1**2 + x2**2 + x3**2
S2 | Y=X^2+X^4 (non-conic, symmetric) | degree 3: dim of homogeneous invariants = 0  (0.2s)
S2 | Y=X^2+X^4 (non-conic, symmetric) | degree 4: dim of homogeneous invariants = 1  (0.7s)
       (x1**2 + x2**2 + x3**2)**2
S2 | Y=X^2+X^4 (non-conic, symmetric) | degree 5: dim of homogeneous invariants = 0  (2.3s)
S2 | Y=X^2+X^4 (non-conic, symmetric) | degree 6: dim of homogeneous invariants = 1  (5.7s)
       (x1**2 + x2**2 + x3**2)**3
S2 | circle about the pole, chart radius 1/2 (conic, special) | degree 1: dim of homogeneous invariants = 1  (0.0s)
       x3
S2 | circle about the pole, chart radius 1/2 (conic, special) | degree 2: dim of homogeneous invariants = 2  (0.0s)
       x1**2 + x2**2
       x3**2
S2 | circle about the pole, chart radius 1/2 (conic, special) | degree 3: dim of homogeneous invariants = 2  (0.1s)
       x3*(x1**2 + x2**2)
       x3**3
S2 | circle about the pole, chart radius 1/2 (conic, special) | degree 4: dim of homogeneous invariants = 3  (0.4s)
       (x1**2 + x2**2)**2
       x3**2*(x1**2 + x2**2)
       x3**4
S2 | circle about the pole, chart radius 1/2 (conic, special) | degree 5: dim of homogeneous invariants = 3  (1.0s)
       x3*(x1**2 + x2**2)**2
       x3**3*(x1**2 + x2**2)
       x3**5
S2 | circle about the pole, chart radius 1/2 (conic, special) | degree 6: dim of homogeneous invariants = 4  (2.5s)
       (x1**2 + x2**2)**3
       x3**2*(x1**2 + x2**2)**2
       x3**4*(x1**2 + x2**2)
       x3**6
H2 | parabola Y=X^2 (conic) | degree 1: dim of homogeneous invariants = 0  (0.0s)
H2 | parabola Y=X^2 (conic) | degree 2: dim of homogeneous invariants = 2  (0.0s)
       -x1**2 + x2*x3
       -x1**2 - x2**2 + x3**2
H2 | parabola Y=X^2 (conic) | degree 3: dim of homogeneous invariants = 0  (0.0s)
H2 | parabola Y=X^2 (conic) | degree 4: dim of homogeneous invariants = 3  (0.2s)
       (x1**2 - x2*x3)**2
       (x1**2 - x2*x3)*(x1**2 + x2**2 - x3**2)
       3*x1**4 + 2*x1**2*x2**2 - 4*x1**2*x2*x3 - 2*x1**2*x3**2 + x2**4 + x3**4
H2 | parabola Y=X^2 (conic) | degree 5: dim of homogeneous invariants = 0  (0.4s)
H2 | parabola Y=X^2 (conic) | degree 6: dim of homogeneous invariants = 4  (0.8s)
       -(x1**2 - x2*x3)**3
       -(x1**2 - x2*x3)**2*(x1**2 + x2**2 - x3**2)
       -(x1**2 - x2*x3)*(3*x1**4 + 2*x1**2*x2**2 - 4*x1**2*x2*x3 - 2*x1**2*x3**2 + x2**4 + x3**4)
       -(x1**2 + x2**2 - x3**2)*(4*x1**4 + 2*x1**2*x2**2 - 6*x1**2*x2*x3 - 2*x1**2*x3**2 + x2**4 + x2**2*x3**2 + x3**4)
H2 | Y=X^2+X^3 (non-conic) | degree 1: dim of homogeneous invariants = 0  (0.0s)
H2 | Y=X^2+X^3 (non-conic) | degree 2: dim of homogeneous invariants = 1  (0.0s)
       -x1**2 - x2**2 + x3**2
H2 | Y=X^2+X^3 (non-conic) | degree 3: dim of homogeneous invariants = 0  (0.2s)
H2 | Y=X^2+X^3 (non-conic) | degree 4: dim of homogeneous invariants = 1  (1.0s)
       (x1**2 + x2**2 - x3**2)**2
H2 | Y=X^2+X^3 (non-conic) | degree 5: dim of homogeneous invariants = 0  (3.4s)
H2 | Y=X^2+X^3 (non-conic) | degree 6: dim of homogeneous invariants = 1  (9.4s)
       -(x1**2 + x2**2 - x3**2)**3
H2 | Y=X^2+X^4 (non-conic, symmetric) | degree 1: dim of homogeneous invariants = 0  (0.0s)
H2 | Y=X^2+X^4 (non-conic, symmetric) | degree 2: dim of homogeneous invariants = 1  (0.0s)
       -x1**2 - x2**2 + x3**2
H2 | Y=X^2+X^4 (non-conic, symmetric) | degree 3: dim of homogeneous invariants = 0  (0.2s)
H2 | Y=X^2+X^4 (non-conic, symmetric) | degree 4: dim of homogeneous invariants = 1  (0.7s)
       (x1**2 + x2**2 - x3**2)**2
H2 | Y=X^2+X^4 (non-conic, symmetric) | degree 5: dim of homogeneous invariants = 0  (2.2s)
H2 | Y=X^2+X^4 (non-conic, symmetric) | degree 6: dim of homogeneous invariants = 1  (5.5s)
       -(x1**2 + x2**2 - x3**2)**3
H2 | circle about the pole, chart radius 1/2 (conic, special) | degree 1: dim of homogeneous invariants = 1  (0.0s)
       x3
H2 | circle about the pole, chart radius 1/2 (conic, special) | degree 2: dim of homogeneous invariants = 2  (0.0s)
       x1**2 + x2**2
       x3**2
H2 | circle about the pole, chart radius 1/2 (conic, special) | degree 3: dim of homogeneous invariants = 2  (0.1s)
       x3*(x1**2 + x2**2)
       x3**3
H2 | circle about the pole, chart radius 1/2 (conic, special) | degree 4: dim of homogeneous invariants = 3  (0.4s)
       (x1**2 + x2**2)**2
       x3**2*(x1**2 + x2**2)
       x3**4
H2 | circle about the pole, chart radius 1/2 (conic, special) | degree 5: dim of homogeneous invariants = 3  (1.0s)
       x3*(x1**2 + x2**2)**2
       x3**3*(x1**2 + x2**2)
       x3**5
H2 | circle about the pole, chart radius 1/2 (conic, special) | degree 6: dim of homogeneous invariants = 4  (2.5s)
       (x1**2 + x2**2)**3
       x3**2*(x1**2 + x2**2)**2
       x3**4*(x1**2 + x2**2)
       x3**6
PASS S2 P4: parametrisation lies on <Ax,x> = eps
PASS S2 P4: homogenised even part = F_even on Sigma~
PASS S2 P4: homogenised odd part = F_odd on Sigma~
PASS S2 P4: Ge/s^m and Go^2/s^(2m-1) are 0-homogeneous
PASS H2 P4: parametrisation lies on <Ax,x> = eps
PASS H2 P4: homogenised even part = F_even on Sigma~
PASS H2 P4: homogenised odd part = F_odd on Sigma~
PASS H2 P4: Ge/s^m and Go^2/s^(2m-1) are 0-homogeneous
PASS P5: R(q) x R(v) = -R(q x v) for the reflection R in n-perp
PASS P5: -R(X) = rho_n(X) = 2<X,n>n/<n,n> - X
=====
ALL PASS
