# Control point (mathematics)

## Points used to define the shape of curves and surfaces / From Wikipedia, the free encyclopedia

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.^{[1]}

For Bézier curves, it has become customary to refer to the $d$-vectors $\mathbf {p} _{i}$ in a parametric representation ${\textstyle \sum _{i}\mathbf {p} _{i}\phi _{i}}$ 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.^{[2]} 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.