Editing B-spline (section)

== Cubic B-Splines ==
A cubic B-spline curve <math>\mathbf{C}(t)</math> with a normalized parameter <math>t \in [0,1]</math> is defined by four nodes (i.e. ''[[Control point (mathematics)|control points]]'') <math>\textbf{b}_0</math>, <math>\textbf{b}_1</math>, <math>\textbf{b}_2</math>, and <math>\textbf{b}_3</math>. It forms a polynomial of degree 3 that can be written as
: <math>\mathbf{C}(t) = \frac{1}{6} \; \begin{bmatrix}t^3 & t^2 & t & 1\end{bmatrix} 
\begin{bmatrix}
-1 & 3 & -3 & 1\\
3 & -6 & 3 & 0\\
-3 & 0 & 3 & 0\\
1 & 4 & 1 & 0
\end{bmatrix} 
\begin{bmatrix}\mathbf{b}_0\\\mathbf{b}_1\\\mathbf{b}_2\\\mathbf{b}_3 \end{bmatrix} </math>.
This corresponds to B-spline polynomials
: <math>\begin{align}
B_0(t) &= \frac{1}{6} ( -t^3 + 3t^2 - 3t + 1 )\\
B_1(t) &= \frac{1}{6} ( 3t^3 - 6t^2 + 4 )\\
B_2(t) &= \frac{1}{6} ( -3t^3 + 3t^2 + 3t + 1 )\\
B_3(t) &= \frac{1}{6} t^3
\end{align}
 </math>
and the curve can be evaluated as <math>\mathbf{C}(t) = \sum_{i=0}^3 B_i(t)\,\mathbf{b}_i</math>. Expanding this, we can write the full polynomial form as below
: <math>\mathbf{C}(t) = \frac{1}{6}
\biggl( (-\mathbf{b}_0 + 3\mathbf{b}_1 - 3\mathbf{b}_2 + \mathbf{b}_3 ) t^3
+ ( 3\mathbf{b}_0 - 6\mathbf{b}_1 + 3\mathbf{b}_2 ) t^2
+ ( -3\mathbf{b}_0 + 3\mathbf{b}_2 ) t 
+ ( \mathbf{b}_0 + 4\mathbf{b}_1 + \mathbf{b}_2 ) \biggr)
 </math>.
Since this is a cubic polynomial, we can also write it as a cubic [[Bézier curve]] with control points  <math>\textbf{P}_0</math>, <math>\textbf{P}_1</math>, <math>\textbf{P}_2</math>, and <math>\textbf{P}_3</math>, such that
: <math>\begin{align}
\mathbf{P}_0 &= \frac{1}{6} ( \mathbf{b}_0 + 4\mathbf{b}_1 + \mathbf{b}_2 ), \\
\mathbf{P}_1 &= \frac{1}{3} ( 2\mathbf{b}_1 + \mathbf{b}_2 ), \\
\mathbf{P}_2 &= \frac{1}{3} ( \mathbf{b}_1 + 2\mathbf{b}_2 ), \\
\mathbf{P}_3 &= \frac{1}{6} ( \mathbf{b}_1 + 4\mathbf{b}_2 + \mathbf{b}_3 ).
\end{align} </math>
A piecewise cubic B-spline is formed by a set of nodes and each four consecutive nodes define a cubic piece of the curve with the formulation above.