Control point (mathematics)

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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 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.[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[edit]

  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 .