<p>No, the $B$ matrix (basis coefficient matrix) does not change from one segment to the next. It is a property of the type of spline you're using, in this case cubic B-splines. If you used Bézier splines or Hermite splines instead, you'd have a different $B$ matrix.</p>

CC BY-SA 3.0
2016-10-27T17:42:26.637Z
2016-10-27T17:42:26.637Z
Nathan Reed
<p>No, $B$ is constant for given type of cubic spline, e.g. B-spline, Bezier, Hermite or Catmull-Rom cubic splines have different $B$ matrix. To make B-spline continuous, you need to copy 3 control points from the previous spline segment $a$ to the control points of spline segment $b$ and add a new point as the last control point of $b$ such that:</p>

<p>$Pb_{i-3}=Pa_{i-2}$, $Pb_{i-2}=Pa_{i-1}$, $Pb_{i-1}=Pa_i$ and $Pb_i$=new control point.</p>

CC BY-SA 3.0
2016-10-27T17:42:52.457Z
2016-10-27T17:42:52.457Z
JarkkoL
<p>If you use the same $\mathbf{B}$ matrix on each segment, then you are constructing a so-called "uniform" b-spline. This means that the parameter increment over each segment is a constant (typically $=1$). Uniform b-splines often have undesirable shapes if the control points are unevenly distributed. But, if the distances between control points don't vary too wildly, then you can use uniform b-splines and get some nice computational simplifications.</p>

CC BY-SA 3.0
2017-07-09T10:55:46.910Z
2017-07-09T10:55:46.910Z
bubba
<p>I would like to ask about uniform (periodic) cubic B-splines (approximation, no interpolation).</p>

<p>$$B=1/6\begin{bmatrix}-1&amp;3&amp;-3&amp;1\\3&amp;-6&amp;3&amp;0\\-3&amp;0&amp;3&amp;0\\1&amp;4&amp;1&amp;0\end{bmatrix}$$ </p>

<p>$B$ is the matrix of coefficients that allows calculating a single curve segment with the formula:</p>

<p>$$ Q_i(u)=[u^3 u^2 u^1 1]B \begin{bmatrix}P_{i-3}\\P_{i-2}\\P_{i-1}\\P_{i}\end{bmatrix} $$</p>

<p>When $i=3$ I calculate the first segment of the curve. My question is when $i=4$ and I want to calculate the second segment of the curve I must change the $4$ control points but does $B$ change or not? </p>

<interpolation><curve><mathematics>
Does the blending matrix change between calculating various curve segments in a uniform cubic B-splines approximation?
Nick
# Does the blending matrix change between calculating various curve segments in a uniform cubic B-splines approximation?

(Asked by: username_0 on 2016-10-27)

I would like to ask about uniform (periodic) cubic B-splines (approximation, no interpolation).

$$B=1/6\begin{bmatrix}-1&3&-3&1\\3&-6&3&0\\-3&0&3&0\\1&4&1&0\end{bmatrix}$$

$B$ is the matrix of coefficients that allows calculating a single curve segment with the formula:

$$ Q\_i(u)=[u^3 u^2 u^1 1]B \begin{bmatrix}P\_{i-3}\\P\_{i-2}\\P\_{i-1}\\P\_{i}\end{bmatrix} $$

When $i=3$ I calculate the first segment of the curve. My question is when $i=4$ and I want to calculate the second segment of the curve I must change the $4$ control points but does $B$ change or not?

**Tags:** interpolation, curve, mathematics


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## 3 answers

### Answer by username_1 on 2016-10-27. Score: 2 | Accepted answer

No, the $B$ matrix (basis coefficient matrix) does not change from one segment to the next. It is a property of the type of spline you're using, in this case cubic B-splines. If you used Bézier splines or Hermite splines instead, you'd have a different $B$ matrix.


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### Answer by username_2 on 2016-10-27. Score: 2

No, $B$ is constant for given type of cubic spline, e.g. B-spline, Bezier, Hermite or Catmull-Rom cubic splines have different $B$ matrix. To make B-spline continuous, you need to copy 3 control points from the previous spline segment $a$ to the control points of spline segment $b$ and add a new point as the last control point of $b$ such that:

$Pb\_{i-3}=Pa\_{i-2}$, $Pb\_{i-2}=Pa\_{i-1}$, $Pb\_{i-1}=Pa\_i$ and $Pb\_i$=new control point.


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### Answer by username_3 on 2017-07-09. Score: 0

If you use the same $\mathbf{B}$ matrix on each segment, then you are constructing a so-called "uniform" b-spline. This means that the parameter increment over each segment is a constant (typically $=1$). Uniform b-splines often have undesirable shapes if the control points are unevenly distributed. But, if the distances between control points don't vary too wildly, then you can use uniform b-splines and get some nice computational simplifications.


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