  We present a simple randomized scheme for triangulating a set $P$ of $n$
points in the plane, and construct a kinetic data structure which maintains the
triangulation as the points of $P$ move continuously along piecewise algebraic
trajectories of constant description complexity. Our triangulation scheme
experiences an expected number of $O(n^2\beta_{s+2}(n)\log^2n)$ discrete
changes, and handles them in a manner that satisfies all the standard
requirements from a kinetic data structure: compactness, efficiency, locality
and responsiveness. Here $s$ is the maximum number of times where any specific
triple of points of $P$ can become collinear,
$\beta_{s+2}(q)=\lambda_{s+2}(q)/q$, and $\lambda_{s+2}(q)$ is the maximum
length of Davenport-Schinzel sequences of order $s+2$ on $n$ symbols. Thus,
compared to the previous solution of Agarwal et al.~\cite{AWY}, we achieve a
(slightly) improved bound on the number of discrete changes in the
triangulation. In addition, we believe that our scheme is simpler to implement
and analyze.
