  For an element in $BS(1,n) = \langle t,a | tat^{-1} = a^n \rangle$ written in
the normal form $t^{-u}a^vt^w$ with $u,w \geq 0$ and $v \in \mathbb{Z}$, we
exhibit a geodesic word representing the element and give a formula for its
word length with respect to the generating set $\{t,a\}$. Using this word
length formula, we prove that there are sets of elements of positive density of
positive, negative and zero conjugation curvature, as defined by Bar Natan,
Duchin and Kropholler.
