  In this paper we construct planar polynomials of the type
$f_{A,B}(x)=x(x^{q^2}+Ax^{q}+Bx)\in \mathbb{F}_{q^3}[x]$, with $A,B \in
\mathbb{F}_{q}$. In particular we completely classify the pairs $(A,B)\in
\mathbb{F}_{q}^2$ such that $f_{A,B}(x)$ is planar using connections with
algebraic curves over finite fields.
