  In noncommutative algebraic geometry, noncommutative quadric hyerpersurfaces
are major objects of study. In this paper, we give a complete classification of
the homogeneous coordinate algebras $A$ of noncommutive conics
$\operatorname{Proj_{nc}} A$ embedded into Calabi-Yau quantum planes up to
isomorphism. We also give a complete classification of the point varieties of
$A$.
