================ S2 ================
PASS S2: <AV,P> = 0 after elimination
PASS S2: rho_P is an involution (M^2 = I), any P with <AP,P> != 0
PASS S2: rho_P fixes P
PASS S2: rho_P = -1 on the polar plane of P (M V = -V for <AV,P> = 0)
PASS S2: rho_P is an isometry of <A.,.> (M^T A M = A)
PASS S2: rho_P(aP+bV) = aP-bV, i.e. sigma_P: [a:b] -> [a:-b]
PASS S2: <A(aP+bV),(aP+bV)> = a^2<AP,P> + b^2<AV,V> (no cross term)
PASS S2: [a:b]->[a:-b] swaps the two absolute points b = [-I*a, I*a] and fixes [1:0],[0:1]
PASS S2: <Ax(t),x(t)> = c^2<AP,P> + 2cs<AP,V> + s^2<AV,V>
PASS S2: with <AP,P> = eps, <AV,V> = 1 this equals eps for all t
PASS S2: rho_P(x(t)) = x(-t)
PASS S2: <B rho x, rho x> - <Bx,x> = -4ab<BP,V> on W_P (generic symmetric B)
PASS S2: circle frame: <APc,Pc> = eps, <Ae2,e2> = 1, <APc,e2> = 0
PASS S2: circle x3=const: (rho_P x)_3 = x_3 on its tangent geodesic
PASS S2: F = F_e + F_o, F_e even, F_o odd
PASS S2: G_e - F_e and G_o - F_o are multiples of (eps<Ax,x> - 1): they agree on Sigma
PASS S2: G_e homogeneous of degree 4, G_o of degree 5
PASS S2: R_e = G_e/(eps<Ax,x>)^m, R_o = G_o^2/(eps<Ax,x>)^(2m+1) are 0-homogeneous
PASS S2: F(x(t)) is even in t along the tangent geodesic of the circle
PASS S2: even/odd parts of the circle integral
PASS S2: homogenised parts agree with F_e, F_o on Sigma
PASS S2: descended integrals R_e, R_o are sigma_P-invariant on the projective tangent line
================ H2 ================
PASS H2: <AV,P> = 0 after elimination
PASS H2: rho_P is an involution (M^2 = I), any P with <AP,P> != 0
PASS H2: rho_P fixes P
PASS H2: rho_P = -1 on the polar plane of P (M V = -V for <AV,P> = 0)
PASS H2: rho_P is an isometry of <A.,.> (M^T A M = A)
PASS H2: rho_P(aP+bV) = aP-bV, i.e. sigma_P: [a:b] -> [a:-b]
PASS H2: <A(aP+bV),(aP+bV)> = a^2<AP,P> + b^2<AV,V> (no cross term)
PASS H2: [a:b]->[a:-b] swaps the two absolute points b = [-a, a] and fixes [1:0],[0:1]
PASS H2: <Ax(t),x(t)> = c^2<AP,P> + 2cs<AP,V> + s^2<AV,V>
PASS H2: with <AP,P> = eps, <AV,V> = 1 this equals eps for all t
PASS H2: rho_P(x(t)) = x(-t)
PASS H2: <B rho x, rho x> - <Bx,x> = -4ab<BP,V> on W_P (generic symmetric B)
PASS H2: circle frame: <APc,Pc> = eps, <Ae2,e2> = 1, <APc,e2> = 0
PASS H2: circle x3=const: (rho_P x)_3 = x_3 on its tangent geodesic
PASS H2: F = F_e + F_o, F_e even, F_o odd
PASS H2: G_e - F_e and G_o - F_o are multiples of (eps<Ax,x> - 1): they agree on Sigma
PASS H2: G_e homogeneous of degree 4, G_o of degree 5
PASS H2: R_e = G_e/(eps<Ax,x>)^m, R_o = G_o^2/(eps<Ax,x>)^(2m+1) are 0-homogeneous
PASS H2: F(x(t)) is even in t along the tangent geodesic of the circle
PASS H2: even/odd parts of the circle integral
PASS H2: homogenised parts agree with F_e, F_o on Sigma
PASS H2: descended integrals R_e, R_o are sigma_P-invariant on the projective tangent line
================
ALL CHECKS PASSED
