throughpoints

Throughpoints are a set of points that a geometric object, such as a line or curve, is required to pass through. The term is used across mathematics, computer graphics, and data analysis to describe the data points that define an interpolating object. In this context, the objective is to construct a function or path that exactly or approximately goes through each throughpoint.

In basic geometry, two throughpoints determine a straight line. When more than two throughpoints are given, the object can be chosen (or constrained) to interpolate the points, using models such as polynomials or splines.

Polynomial interpolation yields a single polynomial that passes through all throughpoints, with methods such as Lagrange or Newton forms. Splines, particularly cubic splines, build a piecewise-defined curve that passes through the throughpoints while enforcing smoothness of values and derivatives across segments. In computer graphics, throughpoints are distinct from control points, which influence the shape of a curve but are not themselves required to lie on it.

Applications include data visualization, CAD, animation, and geographic information systems, where accuracy at the specified points is important.

A key consideration is the trade-off between exact interpolation and numerical stability or smoothness: high-degree polynomial interpolation can exhibit oscillations (the Runge phenomenon), so practitioners often use splines or local interpolation to balance fidelity with stability.