  Most quasi-Monte Carlo research focuses on sampling from the unit cube. Many
problems, especially in computer graphics, are defined via quadrature over the
unit triangle. Quasi-Monte Carlo methods for the triangle have been developed
by Pillands and Cools (2005) and by Brandolini et al. (2013). This paper
presents two QMC constructions in the triangle with a vanishing discrepancy.
The first is a version of the van der Corput sequence customized to the unit
triangle. It is an extensible digital construction that attains a discrepancy
below 12/sqrt(N). The second construction rotates an integer lattice through an
angle whose tangent is a quadratic irrational number. It attains a discrepancy
of O(log(N)/N) which is the best possible rate. Previous work strongly
indicated that such a discrepancy was possible, but no constructions were
available. Scrambling the digits of the first construction improves its
accuracy for integration of smooth functions. Both constructions also yield
convergent estimates for integrands that are Riemann integrable on the triangle
without requiring bounded variation.
