In solid-state NMR, spectra of powdered samples are dominated by the anisotropy of the involved interactions. All internal interactions can be described by second-rank tensors, which account for the observed broadening of the lines. A well-known representation of these tensorial effects is the so-called "representation ellipsoid". It allows a direct pictorial representation of any first-order interaction. However, an ellipsoid can be defined only for strictly positive principal components of a given tenser. More complicated surfaces, such as ovaloids, were introduced recently for the direct representation of tensorial properties and for every set of principal components. In this paper, we show that the "representation ellipsoid" can be extended to generalized quadrics (including cylinders, hyperboloids, and degenerate surfaces), avoiding the use of ovaloids. Moreover, such quadrics can be used for a very simple representation of macroscopic reorientation techniques of samples such as magic-angle spinning, variable-angle spinning, and switching-angle spinning, as well as for the description of rapid anisotropic molecular motions. No explicit reference to Legendre polynomials was made. This article is the first step for a Cartesian representation of higher-order NMR interactions and higher-order macroscopic trajectories such as dynamic-angle spinning, double-rotation, or multiple-quantum magic-angle spinning.