Editing Bézier curve (section)

===Cubic Bézier curves===
Four points '''P'''<sub>0</sub>, '''P'''<sub>1</sub>, '''P'''<sub>2</sub> and '''P'''<sub>3</sub> in the plane or in higher-dimensional space define a cubic Bézier curve.
The curve starts at '''P'''<sub>0</sub> going toward '''P'''<sub>1</sub> and arrives at '''P'''<sub>3</sub> coming from the direction of '''P'''<sub>2</sub>. Usually, it will not pass through '''P'''<sub>1</sub> or '''P'''<sub>2</sub>; these points are only there to provide directional information. The distance between '''P'''<sub>1</sub> and '''P'''<sub>2</sub> determines "how far" and "how fast" the curve moves towards '''P'''<sub>1</sub> before turning towards '''P'''<sub>2</sub>.

Writing '''B'''<sub>'''P'''<sub>''i''</sub>,'''P'''<sub>''j''</sub>,'''P'''<sub>''k''</sub></sub>(''t'') for the quadratic Bézier curve defined by points '''P'''<sub>''i''</sub>, '''P'''<sub>''j''</sub>, and '''P'''<sub>''k''</sub>, the cubic Bézier curve can be defined as an affine combination of two quadratic Bézier curves:

:<math>\mathbf{B}(t) = (1-t)\mathbf{B}_{\mathbf P_0,\mathbf P_1,\mathbf P_2}(t) + t \mathbf{B}_{\mathbf P_1,\mathbf P_2,\mathbf P_3}(t),\ 0 \le t \le 1.</math>

The explicit form of the curve is:

:<math>\mathbf{B}(t) = (1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3,\ 0 \le t \le 1.</math>

For some choices of '''P'''<sub>1</sub> and '''P'''<sub>2</sub> the curve may intersect itself, or contain a [[Cusp (singularity)|cusp]].

Any series of 4 distinct points can be converted to a cubic Bézier curve that goes through all 4 points in order.
Given the starting and ending point of some cubic Bézier curve, and the points along the curve corresponding to ''t''&nbsp;= 1/3 and ''t''&nbsp;=&nbsp;2/3, the control points for the original Bézier curve can be recovered.<ref>{{cite web |author=John Burkardt |url-status=dead |archive-url=https://web.archive.org/web/20131225210855/http://people.sc.fsu.edu/~jburkardt/html/bezier_interpolation.html |url=http://people.sc.fsu.edu/~jburkardt/html/bezier_interpolation.html |title=Forcing Bezier Interpolation |archive-date=2013-12-25}}</ref>

The derivative of the cubic Bézier curve with respect to ''t'' is
: <math>\mathbf{B}'(t) = 3(1-t)^2(\mathbf{P}_1 - \mathbf{P}_0) + 6(1-t)t(\mathbf{P}_2 - \mathbf{P}_1) + 3t^2(\mathbf{P}_3 - \mathbf{P}_2) \,.</math>

The second derivative of the Bézier curve with respect to ''t'' is
: <math>\mathbf{B}''(t) = 6(1-t)(\mathbf{P}_2 - 2 \mathbf{P}_1 + \mathbf{P}_0) + 6t(\mathbf{P}_3 - 2 \mathbf{P}_2 + \mathbf{P}_1) \,.</math>