  Quasi-Monte Carlo (QMC) sampling has been developed for integration over
$[0,1]^s$ where it has superior accuracy to Monte Carlo (MC) for integrands of
bounded variation. Scrambled net quadrature gives allows replication based
error estimation for QMC with at least the same accuracy and for smooth enough
integrands even better accuracy than plain QMC. Integration over triangles,
spheres, disks and Cartesian products of such spaces is more difficult for QMC
because the induced integrand on a unit cube may fail to have the desired
regularity. In this paper, we present a construction of point sets for
numerical integration over Cartesian products of $s$ spaces of dimension $d$,
with triangles ($d=2$) being of special interest. The point sets are
transformations of randomized $(t,m,s)$-nets using recursive geometric
partitions. The resulting integral estimates are unbiased and their variance is
$o(1/n)$ for any integrand in $L^2$ of the product space. Under smoothness
assumptions on the integrand, our randomized QMC algorithm has variance
$O(n^{-1 - 2/d} (\log n)^{s-1})$, for integration over $s$-fold Cartesian
products of $d$-dimensional domains, compared to $O(n^{-1})$ for ordinary Monte
Carlo.
