tabbtangenten

Tabbtangenten is a term found in certain discussions of numerical analysis and geometric modeling to denote tangent lines derived from tabulated data. In this usage, a function is given only at discrete abscissas x_i with values f_i, and tabbtangenten refers to the tangent line at x_i computed from the neighboring tabulated information, typically by finite differences or local slope estimation. The idea is to use these tangent lines to enable efficient local interpolation and derivative estimation without requiring an explicit analytic form of the function.

Origin and usage: The term appears mainly in niche German-language texts and some computer graphics references that emphasize data-driven interpolation. It is not part of the standard nomenclature in most numerical analysis handbooks, where related concepts are described as piecewise linear or Hermite interpolation based on tabulated data. The precise definition of tabbtangenten can vary: some sources define a tangent at x_i with slope s_i = (f_{i+1}-f_{i-1})/(x_{i+1}-x_{i-1}) using central differences, others use one-sided differences, or fit a local polynomial.

Applications: Tabbtangenten are used to construct simple interpolants, to approximate derivatives from discrete data, and to guide shading or rendering in graphics pipelines where function values are sampled on a grid.

Limitations: The accuracy depends on data quality, spacing, and the chosen slope scheme. Without smoothing, interpolants can exhibit overshoots or non-smooth derivatives.

See also: interpolation, tangent line, Hermite interpolation, finite difference.