Jacobi sets of multiple Morse functions have applications in Computer Graphics, Structural Biology (e.g., understanding protein interaction), Oceanography, and Celestial Mechanics. (For some discussion of this see biogeometry.cs.duke.edu/meetings/ITR/03jun/presentations/ungor.03jun.pdf and/or http://www.cs.duke.edu/~edels/TriTop/). As an example, imagine the gravitational potential of the planets in our solar system defining a smooth map on the three-dimensional space. The planets themselves form local maxima. An interesting saddle point lies between the Earth and the Moon where the gravitational pull cancels. The trajectory of this point can be modeled as a portion of the Jacobi curve of two smooth functions on space-time: the gravitational potential and time. More generally, the Jacobi set of two or more generic smooth functions on a manifold contains all points at which the matrix of partial derivatives has rank deficiency at least one. These notions will be described and illustrated.

Inspired by the smooth concept, we study an analogous notion for piecewise linear maps over a triangulated manifold and we describe a combinatorial algorithm based on a definition of critical points. As a particularly easy special case, this algorithm computes the contour (or silhouette) of a triangulated surface embedded in space.