1.3.2 Arithmetic operations of polynomials in Bernstein form

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1.3.2 Arithmetic operations of polynomials in Bernstein form

Arithmetic operations between polynomials are often required for shape interrogation (see for example Chaps. 4, 5, etc.). Farouki and Rajan [106] provide formulae for such arithmetic operations of polynomials in Bernstein form. Let the two polynomials

and

of degree

and

with Bernstein coefficients

and

be as follows:

$\displaystyle f(t) = \sum_{i=0}^mf_i^mB_{i,m}(t),\;\;\;\;\;\;g(t) = \sum_{i=0}^ng_i^nB_{i,n}(t),\;\;\;\;\;\;0\leq t \leq 1\;.$

(1.25)

Addition and subtraction
If the degrees of the two polynomials are the same, i.e.

, we simply add or subtract the coefficients

$\displaystyle f(t) + g(t) = \sum_{i=0}^m(f_i^m \pm g_i^m) B_{i,m}(t)\;.$

(1.26)

, we need to first degree elevate

times using (1.23) and then add or subtract the coefficients

$\displaystyle f(t) + g(t) = \sum_{i=0}^m\left(f_i^m \pm \sum_{j=max(0, i-m+n)}^... ...nd{pmatrix}}{\begin{pmatrix}m \cr i \cr\end{pmatrix}} g_j^n\right)B_{i,m}(t)\;.$

(1.27)

Multiplication
Multiplication of two polynomials of degree

and

yields a degree

polynomial

$\displaystyle f(t)g(t) = \sum_{i=0}^{m+n}\left(\sum_{j=max(0, i-n)}^{min(m,i)} ... ...\begin{pmatrix}m+n \cr i \cr\end{pmatrix}} f_j^mg_{i-j}^n\right)B_{i,m+n}(t)\;.$

(1.28)

Next: 1.3.3 Numerical condition of Up: 1.3 Bézier curves and Previous: 1.3.1 Bernstein polynomials Contents Index

December 2009