Editing Bézier curve (section)

==Specific cases==
A Bézier curve is defined by a [[set (mathematics)|set]] of ''[[Control point (mathematics)|control points]]'' '''P'''<sub>0</sub> through '''P'''<sub>''n''</sub>, where ''n'' is called the order of the curve (''n'' = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points generally do not lie on the curve. The sums in the following sections are to be understood as [[Affine space#Affine combinations and barycenter|affine combinations]] – that is, the coefficients sum to 1.

===Linear Bézier curves===
Given distinct points '''P'''<sub>0</sub> and '''P'''<sub>1</sub>, a linear Bézier curve is simply a [[straight line|line]] between those two points. The curve is given by

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

This is the simplest and is equivalent to [[linear interpolation]].<ref>{{cite book |url=https://books.google.com/books?id=Wq7QEAAAQBAJ&pg=PA33 |page=33 |title=
Stepping into Virtual Reality |author1=Mario A. Gutiérrez |author2=Frédéric Vexo |author3=Daniel Thalmann |publisher=Springer Nature |year=2023|isbn=9783031364877 }}</ref> The quantity <math>\mathbf{P}_1-\mathbf{P}_0</math> represents the [[displacement vector]] from the start point to the end point.

===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>

===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>