Editing Bézier curve (section)

==Degree elevation==
A Bézier curve of degree ''n'' can be converted into a Bézier curve of degree ''n''&nbsp;+ 1 ''with the same shape''. This is useful if software supports Bézier curves only of specific degree. For example, systems that can only work with cubic Bézier curves can implicitly work with quadratic curves by using their equivalent cubic representation.

To do degree elevation, we use the equality <math>\mathbf{B}(t) = (1-t)\mathbf{B}(t) + t\mathbf{B}(t).</math> Each component <math>\mathbf{b}_{i,n}(t)\mathbf{P}_i</math> is multiplied by (1&nbsp;&minus;&nbsp;''t'') and&nbsp;''t'', thus increasing a degree by one, without changing the value. Here is the example of increasing degree from 2 to&nbsp;3.

:<math>\begin{align}
(1 - t)^2 \mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^2 \mathbf{P}_2 &= (1 - t)^3 \mathbf{P}_0 + 2(1 - t)^2 t\mathbf{P}_1 + (1 - t)t^2 \mathbf{P}_2 + (1 - t)^{2}t\mathbf{P}_0 + 2(1 - t)t^2 \mathbf{P}_1 + t^3 \mathbf{P}_2 \\
  &= (1 - t)^3 \mathbf{P}_0 + (1 - t)^2 t \left( \mathbf{P}_0 + 2\mathbf{P}_1\right) + (1 - t) t^2 \left(2\mathbf{P}_1 + \mathbf{P}_2\right) + t^{3}\mathbf{P}_2 \\
  &= (1 - t)^3 \mathbf{P}_0 + 3(1 - t)^2 t \tfrac{1}{3} \left( \mathbf{P}_0 + 2\mathbf{P}_1 \right) + 3(1 - t) t^2 \tfrac{1}{3} \left( 2\mathbf{P}_1 + \mathbf{P}_2 \right) + t^{3}\mathbf{P}_2
\end{align}</math>

In other words, the original start and end points are unchanged. The new control points are <math>\mathbf{P'}_1 = \tfrac{1}{3} \left( \mathbf{P}_0 + 2\mathbf{P}_1 \right)</math> and <math>\mathbf{P'}_2 = \tfrac{1}{3} \left( 2\mathbf{P}_1 + \mathbf{P}_2 \right)</math>.

For arbitrary ''n'' we use equalities<ref name="farin1997"/>

:<math>\begin{cases} {n + 1 \choose i}(1 - t)\mathbf{b}_{i, n} = {n \choose i} \mathbf{b}_{i, n + 1} \\ {n + 1 \choose i + 1} t\mathbf{b}_{i, n} = {n \choose i} \mathbf{b}_{i + 1, n + 1} \end{cases} \quad \implies \quad \begin{cases} (1 - t)\mathbf{b}_{i, n} = \frac{n + 1 - i}{n + 1} \mathbf{b}_{i, n + 1} \\ t\mathbf{b}_{i, n} = \frac{i + 1}{n + 1} \mathbf{b}_{i + 1, n + 1} \end{cases} </math>

Therefore:

:<math>\begin{align}
\mathbf{B}(t) &= (1 - t)\sum_{i=0}^n \mathbf{b}_{i, n}(t)\mathbf{P}_i + t\sum_{i=0}^n \mathbf{b}_{i, n}(t)\mathbf{P}_i \\
                &= \sum_{i=0}^n \frac{n + 1 - i}{n + 1}\mathbf{b}_{i, n + 1}(t)\mathbf{P}_i + \sum_{i=0}^n \frac{i + 1}{n + 1}\mathbf{b}_{i + 1, n + 1}(t)\mathbf{P}_i \\
                &= \sum_{i=0}^{n + 1} \left(\frac{i}{n + 1}\mathbf{P}_{i - 1} + \frac{n + 1 - i}{n + 1}\mathbf{P}_i\right) \mathbf{b}_{i, n + 1}(t) \\
                &= \sum_{i=0}^{n + 1} \mathbf{b}_{i, n + 1}(t)\mathbf{P'}_i
\end{align}</math>

introducing arbitrary <math>\mathbf{P}_{-1}</math> and <math>\mathbf{P}_{n + 1}</math>.

Therefore, new control points are<ref name="farin1997">{{cite book |title=Curves and surfaces for computer-aided geometric design |first=Gerald |last=Farin |publisher=[[Elsevier]] Science & Technology Books |year=1997 |isbn=978-0-12-249054-5 |edition=4}}</ref>

: <math>\mathbf{P'}_i = \frac{i}{n + 1}\mathbf{P}_{i - 1} + \frac{n + 1 - i}{n + 1}\mathbf{P}_i,\quad i=0, \ldots, n + 1.</math>

===Repeated degree elevation===
The concept of degree elevation can be repeated on a control polygon '''R''' to get a sequence of control polygons '''R''', '''R'''<sub>1</sub>, '''R'''<sub>2</sub>, and so on. After ''r'' degree elevations, the polygon '''R'''<sub>''r''</sub> has the vertices '''P'''<sub>0,''r''</sub>, '''P'''<sub>1,''r''</sub>, '''P'''<sub>2,''r''</sub>, ..., '''P'''<sub>''n''+''r'',''r''</sub> given by <ref name="farin1997"/>

:<math>\mathbf{P}_{i,r} = \sum_{j=0}^n \mathbf{P}_j \tbinom nj \frac{\tbinom{r}{i-j}}{\tbinom{n+r}{i}}.</math>

It can also be shown that for the underlying Bézier curve ''B'',

:<math>\mathbf{\lim_{r \to \infty}R_r} = \mathbf{B}.</math>

===Degree reduction===
Degree reduction can only be done exactly when the curve in question is originally elevated from a lower degree.<ref name="ffbezier">{{cite web |title=Bézier Splines |url=https://fontforge.org/docs/techref/bezier.html |website=FontForge 20230101 documentation}}</ref> A number of approximation algorithms have been proposed and used in practice.<ref>{{cite journal |last1=Eck |first1=Matthias |title=Degree reduction of Bézier curves |journal=Computer Aided Geometric Design |date=August 1993 |volume=10 |issue=3–4 |pages=237–251 |doi=10.1016/0167-8396(93)90039-6}}</ref><ref>{{cite conference |last1=Rababah |first1=Abedallah |last2=Ibrahim |first2=Salisu |chapter=Geometric Degree Reduction of Bézier Curves |title=Mathematics and Computing |conference=ICMC 2018, Varanasi, India |date=2018 |pages=87–95 |doi=10.1007/978-981-13-2095-8_8 |isbn=978-981-13-2094-1 }}</ref>