Show that the cone ax^2 + by^2 + cz^2 - cxy - ayz - bzx = 0, is reciprocal cone of the cone (a^2 - bc)x^2 + (b^2 - ac)y^2 + (c^2 - ab)z^2 - 2(a^2 +bc)yz - 2(b^2 + ac)zx - 2(c^2 + ab)xy = 0
Our cone is
ax^2 + by^2 + cz^2 - ayz - bzx - cxy= 0 ------------------------------ (1)
Comparing (1) with a standard representation,
ax^2 + by*2 + cz^2 + 2fyz + 2gzx + 2hxy = 0, --------------------- (2)
we get
a = a, b = b, c = c, f = -a, g = -b , h = -c (ERROR: values should be f = -a/2, g = -b/2, g = -c/2, but
when Error is carried through, it gives the right answer!!
The reciprocal cone of (2) is given by
Ax^2 + By^2 + Cz^2 + 2Fyz + 2Gzx +2Hxy = 0, ------------------------- (2)
Where A, B, C, F, G, H are cofactors of a,b,c,f,g,h in
Δ = | a h g | i.e.
| h b f |
| g f c |
A = bc – f^2 = bc – (-a)^2 = bc – a^2
B = ac – g^2 = ac – (-b)^2 = ac – b^2
C = ab – h^2 = ab – (-c)^2 = ab – c^2
F = -( af – gh) = -(a*(-a) – (-b)*(-c)) = a^2 + bc
G = hf – gb = (-c)*(-a) – (-b)*b = b^2 + ac
H = -(hc – gf) = -((-c)*c – (-b)*(-a)) = c^2 + ab
Substituting for the above into (2), we get
(bc – a^2)x^2 + (ac – b^2)y^2 + (ab – c^2)z^2 + 2(a^2 + bc)yz + 2(b^2 + ac)zx + 2(c^2 + ab)xy = 0
Or,
(a^2 – bc)x^2 + (b^2 – ac)y^2 + (c^2 – ab)z^2 – 2(a^2 + bc)yz – 2(b^2 + ac)zx – 2(c^2 + ab)xy = 0