**Property A7.** *Arithmetic Operations*

*Arithmetic operations of Bernstein polynomials require that both curves be defined on the same time interval* [*t*0, *tf* ]*. In case they are not, the de Casteljau algorithm can be used to split them at an intersecting time interval.*

*The sum and difference of two polynomials of the same order can be performed by simply adding and subtracting their Bernstein coefficients, respectively. For Bernstein polynomials of different order, the Degree Elevation (Property A6) may be used to increase the order of the lower order Bernstein polynomial.*

*Given two Bernstein polynomials, Fm*(*t*) *and Gn*(*t*)*, with degrees m and n, respectively, and having Bernstein coefficients X*0,*m*, ... , *Xm*,*<sup>m</sup> and Y*0,*n*, ... ,*Yn*,*n, the product Cm*+*n*(*t*) = *Fm*(*t*)*Gn*(*t*) *is a Bernstein polynomial of degree* (*m* + *n*) *with coefficients Pk*,*m*<sup>+</sup>*n, k* ∈ {0, ... , *m* + *n*} *given by*

$$P\_{k,m+n} = \sum\_{j=\max(0,k-n)}^{\min(m,k)} \frac{\binom{m}{j} \binom{n}{k-j}}{\binom{m+n}{k}} X\_{j,m} Y\_{k-j,n}.\tag{A9}$$

*The ratio between two Bernstein polynomials, Fn*(*t*) *and Gn*(*t*)*, with coefficients X*0,*n*, ... , *Xn*,*<sup>n</sup> and Y*0,*n*, ... ,*Yn*,*n, i.e., Rn*(*t*) = *Fn*(*t*)/*Gn*(*t*)*, can be expressed as a rational Bernstein polynomial as defined in Equation* (6)*, with coefficients and weights*

$$P\_{i, \mathfrak{n}} = \frac{X\_{i, \mathfrak{n}}}{Y\_{i, \mathfrak{n}}}, \quad w\_{i, \mathfrak{n}} = Y\_{i, \mathfrak{n}}.$$

*respectively.*

**Property A8.** *The de Casteljau Algorithm Extended to Rational Bernstein Polynomials The de Casteljau algorithm can be extended to rational Bernstein polynomials (see [54]). Letting*

$$w\_{i,n}^r(t) = \sum\_{j=0}^r w\_{i+j,n} B\_{i,r}(t),\tag{A10}$$

*we can determine a point on an nth order rational Bernstein polynomial using the recursive equation*

$$\begin{split} P\_{i,n}^{r}(t) &= \left(\frac{t\_f - t}{t\_f - t\_0}\right) \frac{w\_{i,n}^{r-1}(t)}{w\_{i,n}^{r}(t)} P\_{i,n}^{r-1}(t) \\ &+ \left(\frac{t - t\_0}{t\_f - t\_0}\right) \frac{w\_{i+1,n}^{r-1}(t)}{w\_{i,n}^{r}(t)} P\_{i+1,n}^{r-1}(t). \end{split} \tag{A11}$$

*Moreover, the recursive relationship can be used to split the rational Bernstein polynomial into two nth order rational Bernstein polynomials with weights*

$$w^0\_{0,\mathfrak{u}\prime} w^1\_{0,\mathfrak{u}\prime} \dots \,\_{\prime} w^{\mathfrak{n}}\_{0,\mathfrak{n}} \quad \text{and} \quad w^{\mathfrak{n}}\_{0,\mathfrak{n}\prime} w^{\mathfrak{n}-1}\_{1,\mathfrak{n}\prime} \dots \,\_{\prime} w^0\_{\mathfrak{n},\mathfrak{n}\prime} \dots$$

*and coefficients*

*P*0 0,*n*, *<sup>P</sup>*<sup>1</sup> 0,*n*,..., *<sup>P</sup><sup>n</sup>* 0,*<sup>n</sup> and P<sup>n</sup>* 0,*n*, *<sup>P</sup>n*−<sup>1</sup> 1,*<sup>n</sup>* ,..., *<sup>P</sup>*<sup>0</sup> *<sup>n</sup>*,*<sup>n</sup>* .

**Property A9.** *Degree Elevation for Rational Bernstein Polynomials*

*Degree elevation can be extended to rational Bernstein polynomials (see [54]). The nth order rational Bernstein polynomial given by Equation* (6) *can be rewritten as a rational Bernstein polynomial of order n* + 1 *with weights*

$$w\_{i,n+1} = \frac{i}{n+1}w\_{i-1,n} + \left(1 - \frac{i}{n+1}\right)w\_{i,n}.$$

*where w*0,*n*+<sup>1</sup> = *w*0,*<sup>n</sup> and wn*+1,*n*+<sup>1</sup> = *wn*,*n, and coefficients*

$$P\_{i,n+1} = \frac{\frac{i}{n+1} w\_{i,n} P\_{i,n} + \left(1 - \frac{i}{n+1}\right) w\_{i+1,n} P\_{i+1,n}}{\frac{i}{n+1} w\_{i,n} + \left(1 - \frac{i}{n+1}\right) w\_{i+1,n}}.$$

#### **References**

