  This paper introduces a novel, lightweight method to solve the visibility
problem for 2D grids. The proposed method evaluates the existence of
lines-of-sight from a source point to all other grid cells in a single pass
with no preprocessing and independently of the number and shape of obstacles.
It has a compute and memory complexity of $\mathcal{O}(n)$, where $n =
n_{x}\times{} n_{y}$ is the size of the grid, and requires at most ten
arithmetic operations per grid cell. In the proposed approach, we use a linear
first-order hyperbolic partial differential equation to transport the
visibility quantity in all directions. In order to accomplish that, we use an
entropy-satisfying upwind scheme that converges to the true visibility polygon
as the step size goes to zero. This dynamic-programming approach allows the
evaluation of visibility for an entire grid orders of magnitude faster than
typical ray-casting algorithms. We provide a practical application of our
proposed algorithm by posing the visibility quantity as a heuristic and
implementing a deterministic, local-minima-free path planner, setting apart the
proposed planner from traditional methods. Lastly, we provide necessary
algorithms and an open-source implementation of the proposed methods.
