tabbtangent

Tabbtangent is a mathematical construct used to characterize the slope of a curve represented by tabulated data points. The term blends tabulated and tangent and is used in numerical analysis and data visualization to describe the instantaneous rate of change implied by a discrete table rather than by a closed-form function. It is typically defined for a sequence of pairs (x_i, y_i) with i = 1,...,n.

Computation and variants: The most common version is the central tabbtangent at interior points: T_i = (y_{i+1} - y_{i-1}) / (x_{i+1} - x_{i-1}). If x-values are equally spaced, this reduces to the standard difference quotient. For boundary points, forward or backward differences are used: T_1 = (y_2 - y_1) / (x_2 - x_1), T_n = (y_n - y_{n-1}) / (x_n - x_{n-1}). If spacing varies, a weighted central difference or linearization around the point can be applied.

Properties and limitations: Tabbtangents approximate the derivative of the underlying function at the corresponding x value, assuming the table samples a smooth function. They are sensitive to noise and outliers, especially with unequal spacing. The accuracy improves with denser sampling and less noise; higher-order schemes require additional points.

Applications: The construct is used in numerical differentiation, data smoothing, and interpolation. It provides a quick, interpretable surrogate for slope without requiring a global model. It can guide linear interpolation between points and inform slope-based rendering in computer graphics or trend analysis in data science.

See also: Tangent, Finite difference, Interpolation, Numerical differentiation.