Editing Bézier curve (section)

==Rational Bézier curves==
[[Image:Rational Bezier curve-conic sections.svg|thumb|Segments of conic sections represented exactly by rational Bézier curves]]
The rational Bézier curve adds adjustable weights to provide closer approximations to arbitrary shapes. The numerator is a weighted Bernstein-form Bézier curve and the denominator is a weighted sum of [[Bernstein polynomial]]s. Rational Bézier curves can, among other uses, be used to represent segments of [[conic section]]s exactly, including circular arcs.<ref>{{cite web |url=http://www.cl.cam.ac.uk/teaching/2000/AGraphHCI/SMEG/node5.html |title=Some Mathematical Elements of Graphics: Rational B-splines |author=Neil Dodgson |date=2000-09-25 |access-date=2009-02-23}}</ref>

Given ''n''&nbsp;+ 1 control points '''P'''<sub>0</sub>, ..., '''P'''<sub>''n''</sub>, the rational Bézier curve can be described by
:<math>\mathbf{B}(t) = \frac{
\sum_{i=0}^n b_{i,n}(t) \mathbf{P}_{i}w_i
}
{
\sum_{i=0}^n b_{i,n}(t) w_i
},</math>
or simply
:<math>\mathbf{B}(t) = \frac{
\sum_{i=0}^n {n \choose i} t^i (1-t)^{n-i}\mathbf{P}_{i}w_i
}
{
\sum_{i=0}^n {n \choose i} t^i (1-t)^{n-i}w_i
}.</math>
The expression can be extended by using number systems besides [[real number|reals]] for the weights. In the [[complex plane]] the points {1}, {-1}, and {1} with weights {<math>i</math>}, {1}, and {<math>-i</math>} generate a full circle with radius one. For curves with points and weights on a circle, the weights can be scaled without changing the curve's shape.<ref>{{ cite journal |author= J. Sánchez-Reyes |journal=Computer Aided Geometric Design |volume=26 |issue=8 |date=November 2009|pages=865–876 | title=Complex rational Bézier curves |doi=10.1016/j.cagd.2009.06.003 }}</ref>  Scaling the central weight of the above curve by 1.35508 gives a more uniform parameterization.