An intuitive explanation of third-order surface behavior

We present a novel parameterization-independent exposition of the third-order geometric behavior of a surface point. Unlike existing algebraic expositions, our work produces an intuitive explanation of third-order shape, analogous to the principal curvatures and directions that describe second-order shape. We extract four parameters that provide a quick and concise understanding of the third-order surface behavior at any given point. Our shape parameters are useful for easily characterizing different third-order surface shapes without having to use tensor algebra. Our approach generalizes to higher orders, allowing us to extract similarly intuitive parameters that fully describe fourth- and higher-order surface behavior.