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 used to determine the shape of a

spline curve In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree poly ...

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 t ...

or higher-dimensional object. For

Bézier curve A Bézier curve ( ) is a 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 approximate a real-world shape t ...

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

parametric representation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric o ...

\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 repres ...

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 of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, ...

, 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 or 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 the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expresse ...

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

References

Splines (mathematics) {{geometry-stub