computer-aided geometric design Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...

a control point is a member of a set of

points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...

used to determine the shape of a

spline curve In mathematics, a spline is a Function (mathematics), function defined piecewise by polynomials. In interpolation, interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, eve ...

or, more generally, a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

or higher-dimensional object. For

Bézier curve A Bézier curve ( , ) is a parametric equation, parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approxima ...

s, it has become customary to refer to the -vectors in a

parametric representation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters. In the case of a single parameter, parametric equations are commonly used to ...

\sum_i \mathbf p_i \phi_i

of a curve or surface in -space as control points, while the scalar-valued functions , defined over the relevant parameter domain, are the corresponding ''weight'' or ''

blending function In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represe ...

s''. Some would reasonably insist, in order to give intuitive geometric meaning to the word "control", that the blending functions form a

partition of unity In mathematics, a partition of unity on a topological space is a Set (mathematics), set of continuous function (topology), continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood (mathem ...

, i.e., that the are nonnegative and sum to one. This property implies that the curve lies within the

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

of its control points.. This is the case for Bézier's representation of a polynomial curve as well as for the

B-spline In numerical analysis, a B-spline (short for basis spline) is a type of Spline (mathematics), spline function designed to have minimal Support (mathematics), support (overlap) for a given Degree of a polynomial, degree, smoothness, and set of bre ...

representation of a spline curve or tensor-product spline surface.

References

Splines (mathematics) {{geometry-stub