Kruissections

Kruissections are a concept in geometry and topology describing a way to decompose a space into a family of cross-sectional pieces obtained along a parameterized foliation. The term, inspired by the Dutch word kruis for cross, is used in some mathematical and computational contexts to discuss structured decompositions. There is no single universally accepted definition, and different authors may emphasize different aspects.

A common way to define a kruissection is as a collection of transversal cross-sections obtained from a foliation of a space X by one-dimensional leaves. If F is a smooth foliation of X and t runs over an index set T, then for a chosen discrete subset {t_i} of T, the kruissection is the family {C_i} where C_i is a cross-section corresponding to t_i. Each C_i is typically a submanifold of X, and the union of all C_i covers X; intersections among cross-sections occur along their boundaries or at lower-dimensional faces.

Starting with a space X that has a foliation by curves, one selects a parameter grid, picks cross-sections transverse to the leaves, ensures regularity (for example, piecewise-linear stages in computational use), and assembles them into a cover or a cell complex. The resulting division is used to study local structure and to build meshes.

Kruissections yield stratifications of X into cells that reflect the chosen foliation. They can produce grid-like decompositions on surfaces or more irregular partitions on complex 3-manifolds. The resulting structure depends on the foliation choice and on how the cross-sections are selected.

In computational geometry, kruissections assist in mesh generation, volumetric modeling, and visualization. In mathematics, they provide a framework for analyzing how global structure interacts with local cross-sections. See also: cross-section, foliation, mesh generation, cell complex.