Control point (mathematics)

In computer-aided geometric design a control point is a member of a set of points used to determine the shape of a spline curve or, more generally, a surface or higher-dimensional object.

For Bézier curves, 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 partition of unity, i.e., that the $\phi _{i}$ are nonnegative and sum to one. This property implies that the curve lies within the convex hull of its control points. 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.