A novel mathematical framework for modeling folds in structural geology is presented. All the main fold classes from the classical literature: parallel folds, similar folds, and other fold types with convergent and divergent dip isogons are modeled in 3D space by linear and non-linear first-order partial differential equations. The equations are derived from a static Hamilton-Jacobi equation in the context of isotropic and anisotropic front propagation. The proposed Hamilton-Jacobi framework represents folded geological volumes in an Eulerian context as a time of arrival field relative to a reference layer. Metric properties such as distances, gradients (dip and strike), curvature, and their spatial variations can then be easily derived and represented as three-dimensional continua covering the whole geological volume. The model also serves as a basis for distributing properties in folded geological volumes.