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Degree Reduction of Bezier Curves

Degree Reduction of Bezier Curves

Continuing with my reading efforts in Curves and Surfaces for GACD by Gerald Farin, I will talk today about a topic that comes up in chapter 6: elevating and reducing the degree of a Bézier curve.

Elevating the dimensionality

First, we start out by proving that the degree of a Bézier curve can always be elevated without the curve being changed.

A curve $$\vec{x}(t)$$ with the control polygon $$B = {\vec{b}_0, \ldots, \vec{b}_n}$$ can be represented as a Bézier curve using the Bernstein Polynomials $$B_i^n$$ as basis:

\begin{equation*}\vec{x}(t) = \sum_{i=0}^{n} \vec{b}_i B_i^n(t)\end{equation*}

We will now prove that we can increase the degree from $$n$$ to $$n+1$$ and still keep the same curve, given we choose then new control polygon accordingly.

We start out by rewriting $$\vec{x}(t)$$ as

\begin{eqnarray*}\vec{x}(t) &=& (1-t) \vec{x}(t) + t \vec{x}(t) \\ &=& \sum_{i=0}^n (1-t) \vec{b}_i B_i^n (t) + \sum_{i=0}^n t \vec{b}_i B_i^n (t)\end{eqnarray*}

And now we show that the terms can be rewritten as a sum from 0 to n+1. Let's start with the first term:

\begin{eqnarray*}(1-t) B_i^n (t) &=& (1-t) \frac{n!}{(n-i)! i!} t^i (1-t)^{n-i} \\ &=& \frac{n + 1 - i}{n + 1} \frac{(n+1)!}{(n+1 -i)! i!} t^i (1-t)^{(n+1)-i} \\ &=& \frac{n+1 - i}{n+1} B_i^{n+1}(t)\end{eqnarray*}

Easy. Now the second one

\begin{eqnarray*}t B_i^n(t) &=& t \frac{n!}{(n-i)! i!} t^i (1-t)^{n-i} \\ &=& \frac{i+1}{n+1} \frac{(n+1)!}{(n+1)!(i+1)!} t^{i+1} (1-t)^{n-i} \\ &=& \frac{i+1}{n+1} B_{i+1}^{n+1} (t)\end{eqnarray*}

Now, we plug this into the summation formula. We can readjust the summation variables cleverly by noting that we only add 0 terms and we get the formula:

\begin{eqnarray*}\vec{x}(t) &=& \sum_{i=0}^{n+1} \frac{n+1-i}{n+1} \vec{b}_i B_i^{n+1} (t) + \sum_{i=0}^{n+1} \frac{i}{n+1} \vec{b}_{i-1} B_i^{n+1} (t) \\ &=& \sum_{i=0}^{n+1} \Big( \big(1 - \frac{i}{n+1}\big) b_i + \frac{i}{n+1} b_{i-1}) \Big) B_i^{n+1} (t)\end{eqnarray*}

Now all there is to do is to compare the coefficients for old control polygon points and new polygon points and we have our solution: the new points are linear interpolations of the old ones. We can find the new points $$\vec{B}^{(1)}$$ from the old $$\vec{B}$$ by a simple Matrix multiplication:

\begin{equation*}\mat{M} \vec{B} = \vec{B^{(1)}}\end{equation*}

with the Matrix $$\mat{M}$$ being of shape $$n+1 \times n$$ and looking something like this:

\begin{equation*}\mat{M} = \begin{pmatrix} 1 & & & \\ \frac{1}{n+1} & 1- \frac{1}{n+1} & & \\ & \frac{2}{n+1} & 1- \frac{2}{n+1} & \\ \vdots \\ & & \frac{n}{n+1} & 1- \frac{n}{n+1} & \\ & & & 1 \end{pmatrix}\end{equation*}

Degree Reduction

The hard work is already done now: It only remains to say that we indeed find the best approximation to our Bézier curve (in the least squares sense) by finding the least-squares solution to the inverse of the system of equations given in the Matrix equation. Therefore, we find the best degree reduced control polygon by

\begin{equation*}\vec{B} = (\mat{M}^T \mat{M})^{-1} \mat{M}^T \vec{B}^{(1)}\end{equation*}

And here is an example from inside Blender. The highlighted objects are the reduced control polygon (n=4) and its Bézier curve. The non highlighted are the originals with one degree higher (5).

Reduced control Polygon and original + splines

As always, you can find the source code for this in the corresponding GitHub project.

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