Alpha-plane
= Alpha-plane
{title2=$A_\ell=\{W:\ell\subset W\}\cong\mathbb{CP}^2$}
An alpha-plane in the compact <twistor correspondence> is the set of two-planes containing a fixed line $\ell$ in <twistor space>. Its affine portion has tangent vectors $u^A\pi^{A'}$ for fixed primed spinor $\pi$, so any two tangent vectors are orthogonal and null. Two distinct compact alpha-planes meet at the unique two-plane spanned by their defining lines; that point can lie at conformal infinity.