  The Closest Point method, initially developed by Ruuth and Merriman, allows
for the numerical solution of surface partial differential equations without
the need for a parameterization of the surface itself. Surface quantities are
embedded into the surrounding domain by assigning each value at a given spatial
location to the corresponding value at the closest point on the surface. This
embedding allows for surface derivatives to be replaced by their Cartesian
counterparts (e.g. $\nabla_s = \nabla$). This equivalence is only valid on the
surface, and thus, interpolation is used to enforce what is known as the side
condition away from the surface. To improve upon the method, this work derives
an operator embedding that incorporates curvature information, making it valid
in a neighborhood of the surface. With this, direct enforcement of the side
condition is no longer needed. Comparisons in $\mathbb{R}^2$ and $\mathbb{R}^3$
show that the resulting Curvature-Augmented Closest Point method has better
accuracy and requires less memory, through increased matrix sparsity, than the
Closest Point method, while maintaining similar matrix condition numbers. To
demonstrate the utility of the method in a physical application, simulations of
inextensible, bi-lipid vesicles evolving toward equilibrium shapes are also
included.
