Definition: Rational Bézier curve, central projection A rational Bézier curve in \(\mathbb{R}^d\;(d=2,\,3)\) is the central projection of a polynomial Bézier curve in \(\mathbb{R}^{d+1}\).
Definition: Rational Bézier curve, analytical view Given control points \(\mathbf{b}_0,\ldots,\mathbf{b}_n \in \mathbb{R}^d, d \geq 2\), and weights \(w_0\ldots,w_n\geq 0,\;w_i\in\mathbb{R}, \; \text{and}\;w_0 = w_n = 1\), then \[ R(u):=\frac{\sum_{i=0}^n\,B_i^n(u)\cdot w_i\cdot\mathbf{b}_i} {\sum_{i=0}^n\,B_i^n(u)\cdot w_i},\quad u\in[0,1] \] is called rational Bézier curve over the interval \([0,1]\).
The points \(\mathbf{b}^h_n=(w_i\cdot\mathbf{b}_i,\, w_i)\in\mathbb{R}^{d+1}\) are the control points of the Bézier curve in \(\mathbb{R}^{d+1}\).
de Casteljau in homogeneous coordinatesEvaluate de Casteljau in the next higher dimension and project the result onto the hyperplane \(w=1\) \[ \begin{pmatrix} w_i^j\ \mathbf{b}_i^j \\ w_i^j\ \end{pmatrix}= \frac{1-u}{t-s}\, \begin{pmatrix} w^{j-1}_i\ \mathbf{b}_i^{j-1} \\ w^{j-1}_i\ \end{pmatrix}+\frac{u-s}{t-s}\,\begin{pmatrix} w_{i+1}^{j-1}\ \mathbf{b}_{i+1}^{j-1} \\ w_{i+1}^{j-1}\ \end{pmatrix} \]
Select and drag control points in order to interact with the graphic representation. Use the sliders to change the values of the weights, and observe how the curve behaves.