Editing B-spline (section)

==Computer-aided design and computer graphics ==
In [[computer-aided design]] and [[computer graphics]] applications, a spline curve is sometimes represented as <math>C(t)</math>, a parametric curve of some real parameter <math>t</math>. In this case the curve <math>C(t)</math> can be treated as two or three separate coordinate functions <math>(x(t), y(t))</math>, or <math>(x(t), y(t), z(t))</math>. The coordinate functions <math> x(t)</math>, <math>y(t)</math> and <math>z(t)</math> are each spline functions, with a common set of knot values <math>t_1, t_2, \ldots, t_n</math>.

Because a B-splines form basis functions, each of the coordinate functions can be expressed as a linear sum of B-splines, so we have

:<math>
\begin{align}
X(t) &= \sum_i x_i B_{i,n}(t), \\
Y(t) &= \sum_i y_i B_{i,n}(t), \\
Z(t) &= \sum_i z_i B_{i,n}(t).
\end{align}
</math>

The weights <math>x_i</math>, <math>y_i</math> and <math>z_i</math> can be combined to form points <math>P_i=(x_i,y_i,z_i)</math> in 3-d space. These points <math>P_i</math> are commonly known as control points.

Working in reverse, a sequence of control points, knot values, and order of the B-spline define a parametric curve. This representation of a curve by control points has several useful properties:
# The control points <math>P_i</math> define a curve. If the control points are all transformed together in some way, such as being translated, rotated, scaled, or moved by any affine transformation, then the corresponding curve is transformed in the same way.
# Because the B-splines are non-zero for just a finite number of knot intervals, if a single control point is moved, the corresponding change to the parametric curve is just over the parameter range of a small number knot intervals.
# Because <math>\sum_i B_{i,n}(x) = 1</math>, and at all times each <math>B_{i,n}(x) \geq 0</math>, then the curve remains inside the bounding box of the control points. Also, in some sense, the curve broadly follows the control points.

A less desirable feature is that the parametric curve does not interpolate the control points. Usually the curve does not pass through the control points.