Control point (mathematics)

A control point is a member of a set of points for a curve (or, more generally, for surfaces and even higher-dimensional objects) where said set can be used in a function to define the curve.

In CAGD, based on Bézier's representation of a polynomial curve called a Bézier curve, it has become customary to refer to the $d$-vectors p$_{i}\$ in a parametric representation $\sum _{i}$ p$_{i}\phi _{i}\$ of a curve or surface in $d$-space as control points, while the scalar-valued functions $\phi _{i}$, defined over the relevant parameter domain, are the corresponding weight or blending functions. Some would reasonably insist, in order to give intuitive geometric meaning to the word `control', that the blending functions form a nonnegative partition of unity, i.e., the $\phi _{i}$ are nonnegative and sum to 1. This is the case for Bézier's representation of a polynomial curve as well as for the B-spline representation of a spline curve or tensor-product spline surface.