  Nonsingular plane curves over a finite field $\mathbb{F}_q$ of degree $q+2$
passing through all the $\mathbb{F}_q$-points of the plane admita
representation by $3\times 3$ matrices over $\mathbb{F}_q$. We classify their
degenerations by means of the matrix representation, and also discuss the
similar problem for the affine-plane filling curves of degree $q+1$.
