  We examine two natural operations to create numerical semigroups. We say that
a numerical semigroup $\mathcal{S}$ is $k$-normalescent if it is the projection
of the set of integer points in a $k$-dimensional polyhedral cone, and we say
that $\mathcal{S}$ is a $k$-quotient if it is the quotient of a numerical
semigroup with $k$ generators. We prove that all $k$-quotients are
$k$-normalescent, and although the converse is false in general, we prove that
the projection of the set of integer points in a cone with $k$ extreme rays
(possibly lying in a dimension smaller than $k$) is a $k$-quotient. The
discrete geometric perspective of studying cones is useful for studying
$k$-quotients: in particular, we use it to prove that the sum of a
$k_1$-quotient and a $k_2$-quotient is a $(k_1+k_2)$-quotient. In addition, we
prove several results about when a numerical semigroup is not $k$-normalescent.
