  As the use of spectral/$hp$ element methods, and high-order finite element
methods in general, continues to spread, community efforts to create efficient,
optimized algorithms associated with fundamental high-order operations have
grown. Core tasks such as solution expansion evaluation at quadrature points,
stiffness and mass matrix generation, and matrix assembly have received
tremendousattention. With the expansion of the types of problems to which
high-order methods are applied, and correspondingly the growth in types of
numerical tasks accomplished through high-order methods, the number and types
of these core operations broaden. This work focuses on solution expansion
evaluation at arbitrary points within an element. This operation is core to
many postprocessing applications such as evaluation of streamlines and
pathlines, as well as to field projection techniques such as mortaring. We
expand barycentric interpolation techniques developed on an interval to 2D
(triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and
hexahedra) spectral/$hp$ element methods. We provide efficient algorithms for
their implementations, and demonstrate their effectiveness using the
spectral/$hp$ element library Nektar++.
