Control point (mathematics)

Points used to define the shape of curves and surfaces

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 ⁠ $d$ {\displaystyle d}⁠-vectors ⁠ $\mathbf {p} _{i}$ {\displaystyle \mathbf {p} _{i}}⁠ in a parametric representation ${\textstyle \sum _{i}\mathbf {p} _{i}\phi _{i}}$ {\textstyle \sum _{i}\mathbf {p} _{i}\phi _{i}} of a curve or surface in ⁠ $d$ {\displaystyle d}⁠-space as control points, while the scalar-valued functions ⁠ $\phi _{i}$ {\displaystyle \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 ⁠ $\phi _{i}$ {\displaystyle \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.