  We introduce a new concept for generating optimal quadrature rules for
splines. Given a target spline space where we aim to generate an optimal
quadrature rule, we build an associated source space with known optimal
quadrature and transfer the rule from the source space to the target one,
preserving the number of quadrature points and therefore optimality. The
quadrature nodes and weights are, considered as a higher-dimensional point, a
zero of a particular system of polynomial equations. As the space is
continuously deformed by modifying the source knot vector, the quadrature rule
gets updated using polynomial homotopy continuation. For example, starting with
$C^1$ cubic splines with uniform knot sequences, we demonstrate the methodology
by deriving the optimal rules for uniform $C^2$ cubic spline spaces where the
rule was only conjectured heretofore. We validate our algorithm by showing that
the resulting quadrature rule is independent of the path chosen between the
target and the source knot vectors as well as the source rule chosen.
