Editing Bézier curve (section)

===Quadratic Bézier curves===
[[File:Quadratic Beziers in string art.svg|thumb|upright|Quadratic Béziers in [[string art]]: The end points ('''•''') and control point ('''×''') define the quadratic Bézier curve ('''⋯''').]]
A quadratic Bézier curve is the path traced by the [[function (mathematics)|function]] '''B'''(''t''), given points '''P'''<sub>0</sub>, '''P'''<sub>1</sub>, and '''P'''<sub>2</sub>,

: <math>\mathbf{B}(t) = (1 - t)[(1 - t) \mathbf P_0 + t \mathbf P_1] + t [(1 - t) \mathbf P_1 + t \mathbf P_2],\ 0 \le t \le 1</math>,

which can be interpreted as the [[Linear interpolation|linear interpolant]] of corresponding points on the linear Bézier curves from '''P'''<sub>0</sub> to '''P'''<sub>1</sub> and from '''P'''<sub>1</sub> to '''P'''<sub>2</sub> respectively. Rearranging the preceding equation yields:

: <math>\mathbf{B}(t) = (1 - t)^{2}\mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^{2}\mathbf{P}_2,\ 0 \le t \le 1.</math>

This can be written in a way that highlights the symmetry with respect to '''P'''<sub>1</sub>:

: <math>\mathbf{B}(t) = \mathbf{P}_1+(1 - t)^{2}(\mathbf{P}_0 - \mathbf{P}_1) + t^{2}(\mathbf{P}_2-\mathbf{P}_1),\ 0 \le t \le 1.</math>

Which immediately gives the [[derivative]] of the Bézier curve with respect to ''t'':

: <math>\mathbf{B}'(t) = 2 (1 - t) (\mathbf{P}_1 - \mathbf{P}_0) + 2 t (\mathbf{P}_2 - \mathbf{P}_1),</math>

from which it can be concluded that the [[tangent]]s to the curve at '''P'''<sub>0</sub> and '''P'''<sub>2</sub> intersect at '''P'''<sub>1</sub>. As ''t'' increases from 0 to 1, the curve departs from '''P'''<sub>0</sub> in the direction of '''P'''<sub>1</sub>, then bends to arrive at '''P'''<sub>2</sub> from the direction of '''P'''<sub>1</sub>.

The second derivative of the Bézier curve with respect to ''t'' is

: <math>\mathbf{B}''(t) = 2(\mathbf{P}_2 - 2 \mathbf{P}_1 + \mathbf{P}_0).</math>