  We consider blowups at a general point of weighted projective planes and,
more generally, of toric surfaces with Picard number one. We give a unifying
construction of negative curves on these blowups such that all previously known
families appear as boundary cases of this. The classification consists of two
classes of said curves, each depending on two parameters. Every curve in these
two classes is algebraically related to other curves in both classes; this
allows us to find their defining equations inductively. For each curve in our
classification, we consider a family of blowups in which the curve defines an
extremal class in the effective cone. We give a complete classification of
these blowups into Mori Dream Spaces and non-Mori Dream Spaces. Our approach
greatly simplifies previous proofs, avoiding positive characteristic methods
and higher cohomology.
