  Let $k$ be an algebraically closed field of characteristic $0$ and $A$ a
graded $k$-algebra finitely generated in degree $1$. In this paper, for
$3$-dimensional quadratic AS-regular algebras except for Type EC, we give a
complete list of twisted superpotentials and a complete list of superpotentials
such that derivation-quotient algebras are $3$-dimensional quadratic Calabi-Yau
AS-regular algebras. For an algebra $A$ of Type EC, we give a criterion when
$A$ is AS-regular. As an application, for an algebra $A$ of any type, we show
that there exists a Calabi-Yau AS-regular algebra $S$ such that $A$ and $S$ are
graded Morita equivalent. This result tells us that, for a $3$-dimensional
quadratic AS-regular algebra $A$, to study the noncommutative projective scheme
for $A$ defined by Artin-Zhang is reduced to study the noncommutative
projective scheme for $S$ for the Calabi-Yau AS-regular algebra $S$.
