  The classical $S_n$ equations of Carlson and Lee have been a mainstay in
multi-dimensional radiation transport calculations. In this paper, an
alternative to the $S_n$ equations, the "Lagrange Discrete Ordinate" (LDO)
equations are derived. These equations are based on an interpolatory framework
for functions on the unit sphere in three dimensions. While the LDO equations
retain the formal structure of the classical $S_n$ equations, they have a
number of important differences. The LDO equations naturally allow the angular
flux to be evaluated in directions other than those found in the quadrature
set. To calculate the scattering source in the LDO equations, no spherical
harmonic moments are needed--only values of the angular flux. Moreover, the LDO
scattering source preserves the eigenstructure of the continuous scattering
operator. The formal similarity of the LDO equations with the $S_n$ equations
should allow easy modification of mature 3D $S_n$ codes such as PARTISN or
PENTRAN to solve the LDO equations. Numerical results are shown that
demonstrate the spectral convergence (in angle) of the LDO equations for smooth
solutions and the ability to mitigate ray effects by increasing the angular
resolution of the LDO equations.
