Formgrenze

Formgrenze denotes the boundary in a parameter space that separates regions where a mathematical form meets certain qualitative properties from regions where it does not. The term is commonly used in German-language texts in geometry, algebraic geometry, computational design, and numerical analysis.

Etymology: from Form (shape, form) and Grenze (boundary). It refers not to a fixed geometric boundary in physical space but to a locus defined by algebraic conditions, often where degeneracy occurs.

In algebraic geometry, the formgrenze often coincides with the zero locus of the discriminant or a determinant associated with the form. For a quadratic form defined by a symmetric matrix A, the formgrenze is the set of parameter values for which det(A)=0, i.e., where the form becomes singular. For higher-degree forms, the discriminant vanishes on the formgrenze, marking degenerate or reducible cases.

Examples: In a family of conic sections, the formgrenze separates ellipses, parabolas, and hyperbolas from degenerate conics. In optimization, crossing the formgrenze can change the number of local extrema or feasibility.

Computations and study: The formgrenze is typically studied via discriminants, resultants, or Gröbner basis computations. Singularity theory and stratification describe how the boundary decomposes into strata with different properties.

See also: discriminant, determinant, degeneracy, boundary, algebraic geometry, moduli spaces.