Higher-Dimensional Continuation

In computational and pure mathematics it is necessary to choose a representation or notation for the object being computed or manipulated. The parallel between notation and computational representation is quite close, although the computer is able to deal with expressions that are far too complicated to be dealt with by hand. A good notation makes manipulations easier and reduces errors, and a good computational representation makes operations easier to implement, thus reducing coding errors. The choice of representation is often not made explicitly, especially when the objects are simple.

The choice of representation also serves to distinguish between algorithms. Indeed, once the choice of a representation is made there are often fewer choices of how to proceed. We use this approach in Sect. 3.2 to analyze five algorithms for higher-dimensional continuation that are in the literature.

For one-dimensional continuation many issues are quite straightforward, since curves are easily represented as lists of points. However, representing surfaces and manifolds is more challenging. In Sect. 3.1.1 we discuss simplicial and cell complexes, which are general meshes that are commonly used in algebraic topology. Then we define manifolds and manifolds with boundary, which are the analogues of curves and arcs in one dimension.

Section 3.1.2 describes two fundamental geometric abstractions, the Voronoi and Delaunay tessellations, and a particular generalization of the Voronoi tessellation that represents the boundary of a union of spherical balls. In Sect. 3.1.3 we discuss how these topological and geometrical ideas have been used to represent manifolds and, finally, how to construct the neighborhood of a regular point.