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.[1]

For Bézier curves, it has become customary to refer to the ${\displaystyle d}$-vectors p${\displaystyle _{i}}$ in a parametric representation ${\displaystyle \sum _{i}}$ p${\displaystyle _{i}\phi _{i}}$ of a curve or surface in ${\displaystyle d}$-space as control points, while the scalar-valued functions ${\displaystyle \phi _{i}}$, 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 ${\displaystyle \phi _{i}}$ 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.

References

1. Salomon, David (2007), Curves and Surfaces for Computer Graphics, Springer, p. 11, ISBN 9780387284521.
2. Guha, Sumanta (2010), Computer Graphics Through OpenGL: From Theory to Experiments, CRC Press, p. 663, ISBN 9781439846209.