Bézier curve

Cubic Bézier curve with four control points

Thebasis functionson the rangetin [0,1]for cubic Bézier curves:blue:y= (1 −t)³,green:y= 3(1 −t)²t,red:y= 3(1 −t)t²,andcyan:y=t³.

ABézier curve(/ˈbɛz.i.eɪ/BEH-zee-ay)is aparametric curveused incomputer graphicsand related fields.A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. The Bézier curve is named afterFrenchengineerPierre Bézier(1910–1999), who used it in the 1960s for designing curves for the bodywork ofRenaultcars.Other uses include the design of computerfontsand animation.Bézier curves can be combined to form aBézier spline,or generalized to higher dimensions to formBézier surfaces.TheBézier triangleis a special case of the latter.

Invector graphics,Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs,are combinations of linked Bézier curves. Paths are not bound by the limits ofrasterizedimages and are intuitive to modify.

Bézier curves are also used in the time domain, particularly inanimation,user interfacedesign and smoothing cursor trajectory in eye gaze controlled interfaces.For example, a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators orinterfacedesigners talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.

This also applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear.

Invention

The mathematical basis for Bézier curves—theBernstein polynomials—was established in 1912, but thepolynomialswere not applied to graphics until some 50 years later when mathematicianPaul de Casteljauin 1959 developedde Casteljau's algorithm,anumerically stablemethod for evaluating the curves, and became the first to apply them to computer-aided design at French automakerCitroën.De Casteljau's method was patented in France but not published until the 1980swhile the Bézier polynomials were widely publicised in the 1960s by theFrenchengineerPierre Bézier,who discovered them independently and used them to designautomobilebodies atRenault.

Specific cases

A Bézier curve is defined by asetofcontrol pointsP₀throughP_n,wherenis 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 asaffine combinations– that is, the coefficients sum to 1.

Linear Bézier curves

Given distinct pointsP₀andP₁,a linear Bézier curve is simply alinebetween those two points. The curve is given by

$\mathbf {B} (t)=\mathbf {P} _{0}+t(\mathbf {P} _{1}-\mathbf {P} _{0})=(1-t)\mathbf {P} _{0}+t\mathbf {P} _{1},\ 0\leq t\leq 1$

This is the simplest and is equivalent tolinear interpolation.The quantity$\mathbf {P} _{1}-\mathbf {P} _{0}$represents thedisplacement vectorfrom the start point to the end point.

Quadratic Bézier curves

Quadratic Béziers instring art:The end points (•) and control point (×) define the quadratic Bézier curve (⋯).

A quadratic Bézier curve is the path traced by thefunctionB(t), given pointsP₀,P₁,andP₂,

$\mathbf {B} (t)=(1-t)[(1-t)\mathbf {P} _{0}+t\mathbf {P} _{1}]+t[(1-t)\mathbf {P} _{1}+t\mathbf {P} _{2}],\ 0\leq t\leq 1$,

which can be interpreted as thelinear interpolantof corresponding points on the linear Bézier curves fromP₀toP₁and fromP₁toP₂respectively. Rearranging the preceding equation yields:

$\mathbf {B} (t)=(1-t)^{2}\mathbf {P} _{0}+2(1-t)t\mathbf {P} _{1}+t^{2}\mathbf {P} _{2},\ 0\leq t\leq 1.$

This can be written in a way that highlights the symmetry with respect toP₁:

$\mathbf {B} (t)=\mathbf {P} _{1}+(1-t)^{2}(\mathbf {P} _{0}-\mathbf {P} _{1})+t^{2}(\mathbf {P} _{2}-\mathbf {P} _{1}),\ 0\leq t\leq 1.$

Which immediately gives thederivativeof the Bézier curve with respect tot:

$\mathbf {B} '(t)=2(1-t)(\mathbf {P} _{1}-\mathbf {P} _{0})+2t(\mathbf {P} _{2}-\mathbf {P} _{1}),$

from which it can be concluded that thetangentsto the curve atP₀andP₂intersect atP₁.Astincreases from 0 to 1, the curve departs fromP₀in the direction ofP₁,then bends to arrive atP₂from the direction ofP₁.

The second derivative of the Bézier curve with respect totis

$\mathbf {B} ''(t)=2(\mathbf {P} _{2}-2\mathbf {P} _{1}+\mathbf {P} _{0}).$

Cubic Bézier curves

Four pointsP₀,P₁,P₂andP₃in the plane or in higher-dimensional space define a cubic Bézier curve. The curve starts atP₀going towardP₁and arrives atP₃coming from the direction ofP₂.Usually, it will not pass throughP₁orP₂;these points are only there to provide directional information. The distance betweenP₁andP₂determines "how far" and "how fast" the curve moves towardsP₁before turning towardsP₂.

WritingB_Pi,Pj,Pk(t) for the quadratic Bézier curve defined by pointsP_i,P_j,andP_k,the cubic Bézier curve can be defined as an affine combination of two quadratic Bézier curves:

$\mathbf {B} (t)=(1-t)\mathbf {B} _{\mathbf {P} _{0},\mathbf {P} _{1},\mathbf {P} _{2}}(t)+t\mathbf {B} _{\mathbf {P} _{1},\mathbf {P} _{2},\mathbf {P} _{3}}(t),\ 0\leq t\leq 1.$

The explicit form of the curve is:

$\mathbf {B} (t)=(1-t)^{3}\mathbf {P} _{0}+3(1-t)^{2}t\mathbf {P} _{1}+3(1-t)t^{2}\mathbf {P} _{2}+t^{3}\mathbf {P} _{3},\ 0\leq t\leq 1.$

For some choices ofP₁andP₂the curve may intersect itself, or contain acusp.

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 tot= 1/3 andt= 2/3, the control points for the original Bézier curve can be recovered.

The derivative of the cubic Bézier curve with respect totis

$\mathbf {B} '(t)=3(1-t)^{2}(\mathbf {P} _{1}-\mathbf {P} _{0})+6(1-t)t(\mathbf {P} _{2}-\mathbf {P} _{1})+3t^{2}(\mathbf {P} _{3}-\mathbf {P} _{2})\,.$

The second derivative of the Bézier curve with respect totis

$\mathbf {B} ''(t)=6(1-t)(\mathbf {P} _{2}-2\mathbf {P} _{1}+\mathbf {P} _{0})+6t(\mathbf {P} _{3}-2\mathbf {P} _{2}+\mathbf {P} _{1})\,.$

General definition

Bézier curves can be defined for any degreen.

Recursive definition

A recursive definition for the Bézier curve of degreenexpresses it as a point-to-pointlinear combination(linear interpolation) of a pair of corresponding points in two Bézier curves of degreen− 1.

Let$\mathbf {B} _{\mathbf {P} _{0}\mathbf {P} _{1}\ldots \mathbf {P} _{k}}$denote the Bézier curve determined by any selection of pointsP₀,P₁,...,P_k.Then to start,

$\mathbf {B} _{\mathbf {P} _{0}}(t)=\mathbf {P} _{0}{\text{, and}}$

$\mathbf {B} (t)=\mathbf {B} _{\mathbf {P} _{0}\mathbf {P} _{1}\ldots \mathbf {P} _{n}}(t)=(1-t)\mathbf {B} _{\mathbf {P} _{0}\mathbf {P} _{1}\ldots \mathbf {P} _{n-1}}(t)+t\mathbf {B} _{\mathbf {P} _{1}\mathbf {P} _{2}\ldots \mathbf {P} _{n}}(t)$

This recursion is elucidated in theanimations below.

Explicit definition

The formula can be expressed explicitly as follows (where t⁰and (1-t)⁰are extended continuously to be 1 throughout [0,1]):

${\begin{aligned}\mathbf {B} (t)&=\sum _{i=0}^{n}{n \choose i}(1-t)^{n-i}t^{i}\mathbf {P} _{i}\\&=(1-t)^{n}\mathbf {P} _{0}+{n \choose 1}(1-t)^{n-1}t\mathbf {P} _{1}+\cdots +{n \choose n-1}(1-t)t^{n-1}\mathbf {P} _{n-1}+t^{n}\mathbf {P} _{n},&&0\leqslant t\leqslant 1\end{aligned}}$

where$\scriptstyle {n \choose i}$are thebinomial coefficients.

For example, whenn= 5:

${\begin{aligned}\mathbf {B} (t)&=(1-t)^{5}\mathbf {P} _{0}+5t(1-t)^{4}\mathbf {P} _{1}+10t^{2}(1-t)^{3}\mathbf {P} _{2}+10t^{3}(1-t)^{2}\mathbf {P} _{3}+5t^{4}(1-t)\mathbf {P} _{4}+t^{5}\mathbf {P} _{5},&&0\leqslant t\leqslant 1.\end{aligned}}$

Terminology

Some terminology is associated with these parametric curves. We have

$\mathbf {B} (t)=\sum _{i=0}^{n}b_{i,n}(t)\mathbf {P} _{i},\ \ \ 0\leq t\leq 1$

where the polynomials

$b_{i,n}(t)={n \choose i}t^{i}(1-t)^{n-i},\ \ \ i=0,\ldots,n$

are known asBernstein basis polynomialsof degreen.

t⁰= 1, (1 −t)⁰= 1, and thebinomial coefficient,$\scriptstyle {n \choose i}$,is:

${n \choose i}={\frac {n!}{i!(n-i)!}}.$

The pointsP_iare calledcontrol pointsfor the Bézier curve. Thepolygonformed by connecting the Bézier points withlines,starting withP₀and finishing withP_n,is called theBézier polygon(orcontrol polygon). Theconvex hullof the Bézier polygon contains the Bézier curve.

Polynomial form

Sometimes it is desirable to express the Bézier curve as apolynomialinstead of a sum of less straightforwardBernstein polynomials.Application of thebinomial theoremto the definition of the curve followed by some rearrangement will yield

$\mathbf {B} (t)=\sum _{j=0}^{n}{t^{j}\mathbf {C} _{j}}$

where

$\mathbf {C} _{j}={\frac {n!}{(n-j)!}}\sum _{i=0}^{j}{\frac {(-1)^{i+j}\mathbf {P} _{i}}{i!(j-i)!}}=\prod _{m=0}^{j-1}(n-m)\sum _{i=0}^{j}{\frac {(-1)^{i+j}\mathbf {P} _{i}}{i!(j-i)!}}.$

This could be practical if$\mathbf {C} _{j}$can be computed prior to many evaluations of$\mathbf {B} (t)$;however one should use caution as high order curves may lacknumeric stability(de Casteljau's algorithmshould be used if this occurs). Note that theempty productis 1.

Properties

A cubic Bézier curve (yellow) can be made identical to a quadratic one (black) by

1. copying the end points, and

2. placing its 2 middle control points (yellow circles) 2/3 along line segments from the end points to the quadratic curve's middle control point (black rectangle).

• The curve begins at$\mathbf {P} _{0}$and ends at$\mathbf {P} _{n}$;this is the so-calledendpoint interpolationproperty.
• The curve is a lineif and only ifall the control points arecollinear.
• The start and end of the curve istangentto the first and last section of the Bézier polygon, respectively.
• A curve can be split at any point into two subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.
• Some curves that seem simple, such as thecircle,cannot be described exactly by a Bézier orpiecewiseBézier curve; though a four-piece cubic Bézier curve can approximate a circle (seecomposite Bézier curve), with a maximum radial error of less than one part in a thousand, when each inner control point (or offline point) is the distance$\textstyle {\frac {4\left({\sqrt {2}}-1\right)}{3}}$horizontally or vertically from an outer control point on a unit circle. More generally, ann-piece cubic Bézier curve can approximate a circle, when each inner control point is the distance$\textstyle {\frac {4}{3}}\tan(t/4)$from an outer control point on a unit circle, where${\textstyle t=2\pi /n}$(i.e.$t=360^{\circ }/n$), and$n>2$.
• Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degreenBézier curve is also a degreemcurve for anym>n.In detail, a degreencurve with control points$\mathbf {P} _{0},\,\dots,\,\mathbf {P} _{n}$is equivalent (including the parametrization) to the degreen+ 1 curve with control points$\mathbf {P} '_{0},\,\dots,\,\mathbf {P} '_{n+1}$,where$\mathbf {P} '_{k}={\tfrac {k}{n+1}}\mathbf {P} _{k-1}+\left(1-{\tfrac {k}{n+1}}\right)\mathbf {P} _{k}$,$\forall k=0,\,1,\,\dots,\,n,\,n+1$and define$\mathbf {P} _{n+1}:=\mathbf {P} _{0}$,$\mathbf {P} _{-1}:=\mathbf {P} _{n}$.
• Bézier curves have thevariation diminishing property.What this means in intuitive terms is that a Bézier curve does not "undulate" more than the polygon of its control points, and may actually "undulate" less than that.
• There is nolocal controlin degreenBézier curves—meaning that any change to a control point requires recalculation of and thus affects the aspect of the entire curve, "although the further that one is from the control point that was changed, the smaller is the change in the curve".
• A Bézier curve of order higher than two may intersect itself or have acuspfor certain choices of the control points.

Second-order curve is a parabolic segment

Equivalence of a quadratic Bézier curve and a parabolic segment

A quadratic Bézier curve is also a segment of aparabola.As a parabola is aconic section,some sources refer to quadratic Béziers as "conic arcs".With reference to the figure on the right, the important features of the parabola can be derived as follows:

1. Tangents to the parabola at the endpoints of the curve (A and B) intersect at its control point (C).
2. If D is the midpoint of AB, the tangent to the curve which isperpendicularto CD (dashed cyan line) defines its vertex (V). Its axis of symmetry (dash-dot cyan) passes through V and is perpendicular to the tangent.
3. E is either point on the curve with a tangent at 45° to CD (dashed green). If G is the intersection of this tangent and the axis, the line passing through G and perpendicular to CD is the directrix (solid green).
4. The focus (F) is at the intersection of the axis and a line passing through E and perpendicular to CD (dotted yellow). The latus rectum is the line segment within the curve (solid yellow).

Derivative

The derivative for a curve of ordernis

$\mathbf {B} '(t)=n\sum _{i=0}^{n-1}b_{i,n-1}(t)(\mathbf {P} _{i+1}-\mathbf {P} _{i}).$

Constructing Bézier curves

Linear curves

Lettdenote the fraction of progress (from 0 to 1) the pointB(t) has made along its traversal fromP₀toP₁.For example, whent=0.25,B(t) is one quarter of the way from pointP₀toP₁.Astvaries from 0 to 1,B(t) draws a line fromP₀toP₁.

Animation of a linear Bézier curve,tin [0,1]

Quadratic curves

For quadratic Bézier curves one can construct intermediate pointsQ₀andQ₁such that astvaries from 0 to 1:

• PointQ₀(t) varies fromP₀toP₁and describes a linear Bézier curve.
• PointQ₁(t) varies fromP₁toP₂and describes a linear Bézier curve.
• PointB(t) is interpolated linearly betweenQ₀(t) toQ₁(t) and describes a quadratic Bézier curve.

Construction of a quadratic Bézier curve		Animation of a quadratic Bézier curve,tin [0,1]

Higher-order curves

For higher-order curves one needs correspondingly more intermediate points. For cubic curves one can construct intermediate pointsQ₀,Q₁,andQ₂that describe linear Bézier curves, and pointsR₀andR₁that describe quadratic Bézier curves:

Construction of a cubic Bézier curve		Animation of a cubic Bézier curve,tin [0,1]

For fourth-order curves one can construct intermediate pointsQ₀,Q₁,Q₂andQ₃that describe linear Bézier curves, pointsR₀,R₁andR₂that describe quadratic Bézier curves, and pointsS₀andS₁that describe cubic Bézier curves:

Construction of a quartic Bézier curve		Animation of a quartic Bézier curve,tin [0,1]

For fifth-order curves, one can construct similar intermediate points.

Animation of a fifth-order Bézier curve,tin [0,1] in red. The Bézier curves for each of the lower orders are also shown.

These representations rest on the process used inDe Casteljau's algorithmto calculate Bézier curves.

Offsets (or stroking) of Bézier curves

The curve at a fixed offset from a given Bézier curve, called anoffset or parallel curvein mathematics (lying "parallel" to the original curve, like the offset between rails in arailroad track), cannot be exactly formed by a Bézier curve (except in some trivial cases). In general, the two-sided offset curve of a cubic Bézier is a 10th-orderalgebraic curveand more generally for a Bézier of degreenthe two-sided offset curve is an algebraic curve of degree 4n− 2.However, there areheuristicmethods that usually give an adequate approximation for practical purposes.

In the field ofvector graphics,painting two symmetrically distanced offset curves is calledstroking(the Bézier curve or in general a path of several Bézier segments).The conversion from offset curves to filled Bézier contours is of practical importance in convertingfontsdefined inMetafont,which require stroking of Bézier curves, to the more widely usedPostScript type 1 fonts,which only require (for efficiency purposes) the mathematically simpler operation of filling a contour defined by (non-self-intersecting) Bézier curves.

Degree elevation

A Bézier curve of degreencan be converted into a Bézier curve of degreen+ 1with the same shape.This is useful if software supports Bézier curves only of specific degree. For example, systems that can only work with cubic Bézier curves can implicitly work with quadratic curves by using their equivalent cubic representation.

To do degree elevation, we use the equality$\mathbf {B} (t)=(1-t)\mathbf {B} (t)+t\mathbf {B} (t).$Each component$\mathbf {b} _{i,n}(t)\mathbf {P} _{i}$is multiplied by (1 −t) andt,thus increasing a degree by one, without changing the value. Here is the example of increasing degree from 2 to 3.

${\begin{aligned}(1-t)^{2}\mathbf {P} _{0}+2(1-t)t\mathbf {P} _{1}+t^{2}\mathbf {P} _{2}&=(1-t)^{3}\mathbf {P} _{0}+2(1-t)^{2}t\mathbf {P} _{1}+(1-t)t^{2}\mathbf {P} _{2}+(1-t)^{2}t\mathbf {P} _{0}+2(1-t)t^{2}\mathbf {P} _{1}+t^{3}\mathbf {P} _{2}\\&=(1-t)^{3}\mathbf {P} _{0}+(1-t)^{2}t\left(\mathbf {P} _{0}+2\mathbf {P} _{1}\right)+(1-t)t^{2}\left(2\mathbf {P} _{1}+\mathbf {P} _{2}\right)+t^{3}\mathbf {P} _{2}\\&=(1-t)^{3}\mathbf {P} _{0}+3(1-t)^{2}t{\tfrac {1}{3}}\left(\mathbf {P} _{0}+2\mathbf {P} _{1}\right)+3(1-t)t^{2}{\tfrac {1}{3}}\left(2\mathbf {P} _{1}+\mathbf {P} _{2}\right)+t^{3}\mathbf {P} _{2}\end{aligned}}$

In other words, the original start and end points are unchanged. The new control points are$\mathbf {P'} _{1}={\tfrac {1}{3}}\left(\mathbf {P} _{0}+2\mathbf {P} _{1}\right)$and$\mathbf {P'} _{2}={\tfrac {1}{3}}\left(2\mathbf {P} _{1}+\mathbf {P} _{2}\right)$.

For arbitrarynwe use equalities

${\begin{cases}{n+1 \choose i}(1-t)\mathbf {b} _{i,n}={n \choose i}\mathbf {b} _{i,n+1}\\{n+1 \choose i+1}t\mathbf {b} _{i,n}={n \choose i}\mathbf {b} _{i+1,n+1}\end{cases}}\quad \implies \quad {\begin{cases}(1-t)\mathbf {b} _{i,n}={\frac {n+1-i}{n+1}}\mathbf {b} _{i,n+1}\\t\mathbf {b} _{i,n}={\frac {i+1}{n+1}}\mathbf {b} _{i+1,n+1}\end{cases}}$

Therefore:

${\begin{aligned}\mathbf {B} (t)&=(1-t)\sum _{i=0}^{n}\mathbf {b} _{i,n}(t)\mathbf {P} _{i}+t\sum _{i=0}^{n}\mathbf {b} _{i,n}(t)\mathbf {P} _{i}\\&=\sum _{i=0}^{n}{\frac {n+1-i}{n+1}}\mathbf {b} _{i,n+1}(t)\mathbf {P} _{i}+\sum _{i=0}^{n}{\frac {i+1}{n+1}}\mathbf {b} _{i+1,n+1}(t)\mathbf {P} _{i}\\&=\sum _{i=0}^{n+1}\left({\frac {i}{n+1}}\mathbf {P} _{i-1}+{\frac {n+1-i}{n+1}}\mathbf {P} _{i}\right)\mathbf {b} _{i,n+1}(t)\\&=\sum _{i=0}^{n+1}\mathbf {b} _{i,n+1}(t)\mathbf {P'} _{i}\end{aligned}}$

introducing arbitrary$\mathbf {P} _{-1}$and$\mathbf {P} _{n+1}$.

Therefore, new control points are

$\mathbf {P'} _{i}={\frac {i}{n+1}}\mathbf {P} _{i-1}+{\frac {n+1-i}{n+1}}\mathbf {P} _{i},\quad i=0,\ldots,n+1.$

Repeated degree elevation

The concept of degree elevation can be repeated on a control polygonRto get a sequence of control polygonsR,R₁,R₂,and so on. Afterrdegree elevations, the polygonR_rhas the verticesP_0,r,P_1,r,P_2,r,...,P_n+r,rgiven by

$\mathbf {P} _{i,r}=\sum _{j=0}^{n}\mathbf {P} _{j}{\tbinom {n}{j}}{\frac {\tbinom {r}{i-j}}{\tbinom {n+r}{i}}}.$

It can also be shown that for the underlying Bézier curveB,

$\mathbf {\lim _{r\to \infty }R_{r}} =\mathbf {B}.$

Degree reduction

Degree reduction can only be done exactly when the curve in question is originally elevated from a lower degree.A number of approximation algorithms have been proposed and used in practice.

Rational Bézier curves

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 ofBernstein polynomials.Rational Bézier curves can, among other uses, be used to represent segments ofconic sectionsexactly, including circular arcs.

Givenn+ 1 control pointsP₀,...,P_n,the rational Bézier curve can be described by

$\mathbf {B} (t)={\frac {\sum _{i=0}^{n}b_{i,n}(t)\mathbf {P} _{i}w_{i}}{\sum _{i=0}^{n}b_{i,n}(t)w_{i}}},$

or simply

$\mathbf {B} (t)={\frac {\sum _{i=0}^{n}{n \choose i}t^{i}(1-t)^{n-i}\mathbf {P} _{i}w_{i}}{\sum _{i=0}^{n}{n \choose i}t^{i}(1-t)^{n-i}w_{i}}}.$

The expression can be extended by using number systems besidesrealsfor the weights. In thecomplex planethe points {1}, {-1}, and {1} with weights {$i$}, {1}, and {$-i$} 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.Scaling the central weight of the above curve by 1.35508 gives a more uniform parameterization.

Applications

Computer graphics

Bézier path inAdobe Illustrator

Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in theconvex hullof itscontrol points,the points can be graphically displayed and used to manipulate the curve intuitively.Affine transformationssuch astranslationandrotationcan be applied on the curve by applying the respective transform on the control points of the curve.

QuadraticandcubicBézier curves are most common. Higher degree curves are morecomputationally expensiveto evaluate. When more complex shapes are needed, low order Bézier curves are patched together, producing acomposite Bézier curve.A composite Bézier curve is commonly referred to as a "path" invector graphicslanguages (likePostScript), vector graphics standards (likeSVG) and vector graphics programs (likeArtline,Timeworks Publisher,Adobe Illustrator,CorelDraw,Inkscape,andAllegro). In order to join Bézier curves into a composite Bézier curve without kinks, a property calledG1 continuitysuffices to force the control point at which two constituent Bézier curves meet to lie on the line defined by the two control points on either side.

Abstract composition of cubic Bézier curves ray-traced in 3D. Ray intersection with swept volumes along curves is calculated with Phantom Ray-Hair Intersector algorithm.

The simplest method for scan converting (rasterizing) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments. However, this does not guarantee that the rasterized output looks sufficiently smooth, because the points may be spaced too far apart. Conversely it may generate too many points in areas where the curve is close to linear. A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line to within a small tolerance. If not, the curve is subdivided parametrically into two segments, 0 ≤t≤ 0.5 and 0.5 ≤t≤ 1, and the same procedure is applied recursively to each half. There are also forward differencing methods, but great care must be taken to analyse error propagation.

Analytical methods where a Bézier is intersected with each scan line involve findingrootsof cubic polynomials (for cubic Béziers) and dealing with multiple roots, so they are not often used in practice.

The rasterisation algorithm used inMetafontis based on discretising the curve, so that it is approximated by a sequence of "rookmoves "that are purely vertical or purely horizontal, along the pixel boundaries. To that end, the plane is first split into eight 45° sectors (by the coordinate axes and the two lines$y=\pm x$), then the curve is decomposed into smaller segments such that thedirectionof a curve segment stays within one sector; since the curve velocity is a second degree polynomial, finding the$t$values where it is parallel to one of these lines can be done by solvingquadratic equations.Within each segment, either horizontal or vertical movement dominates, and the total number of steps in either direction can be read off from the endpoint coordinates; in for example the 0–45° sector horizontal movement to the right dominates, so it only remains to decide between which steps to the right the curve should make a step up.

There is also a modified curve form ofBresenham's line drawing algorithmby Zingl that performs this rasterization by subdividing the curve into rational pieces and calculating the error at each pixel location such that it either travels at a 45° angle or straight depending on compounding error as it iterates through the curve. This reduces the next step calculation to a series ofintegeradditions and subtractions.

Animation

In animation applications, such asAdobe FlashandSynfig,Bézier curves are used to outline, for example, movement. Users outline the wanted path in Bézier curves, and the application creates the needed frames for the object to move along the path.

In 3D animation, Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation.Bézier curves are now very frequently used to control the animation easing inCSS,JavaScript,JavaFxandFlutter SDK.

Fonts

TrueTypefonts use composite Bézier curves composed ofquadraticBézier curves. Other languages and imaging tools (such asPostScript,Asymptote,Metafont,andSVG) use composite Béziers composed ofcubicBézier curves for drawing curved shapes.OpenTypefonts can use either kind of curve, depending on which font technology underlies the OpenType wrapper.

Font engines, likeFreeType,draw the font's curves (and lines) on a pixellated surface using a process known asfont rasterization.Typically font engines and vector graphics engines render Bézier curves by splitting them recursively up to the point where the curve is flat enough to be drawn as a series of linear or circular segments. The exact splitting algorithm is implementation dependent, only the flatness criteria must be respected to reach the necessary precision and to avoid non-monotonic local changes of curvature. The "smooth curve" feature of charts inMicrosoft Excelalso uses this algorithm.

Because arcs of circles andellipsescannot be exactly represented by Bézier curves, they are first approximated by Bézier curves, which are in turn approximated by arcs of circles. This is inefficient as there exists also approximations of all Bézier curves using arcs of circles or ellipses, which can be rendered incrementally with arbitrary precision. Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves (or surfaces) intoNURBS,that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition. This approach also preserves the curve definition under all linear or perspective 2D and 3D transforms and projections.

Robotics

Because the control polygon allows to tell whether or not the path collides with any obstacles, Bézier curves are used in producing trajectories of theend effectors.Furthermore, joint space trajectories can be accurately differentiated using Bézier curves. Consequently, the derivatives of joint space trajectories are used in the calculation of the dynamics and control effort (torque profiles) of the robotic manipulator.

See also

• Bézier surface
• B-spline
• GEM/4andGEM/5
• Hermite curve
• NURBS
• String art– Bézier curves are also formed by many common forms of string art, where strings are looped across a frame of nails.
• Variation diminishing property of Bézier curves

References

Sources

• Rida T. Farouki (August 2012)."The Bernstein polynomial basis: A centennial retrospective"(PDF).Computer Aided Geometric Design.29(6): 379–419.doi:10.1016/j.cagd.2012.03.001.
• Paul Bourke (2009-07-19).Bézier Surfaces (in 3D).Archived fromthe originalon 2009-07-19.
• Donald Knuth(1986).Metafont: the Program.Addison-Wesley. pp. 123–131.Excellent discussion of implementation details; available for free as part of the TeX distribution.
• Thomas Sederberg.Bézier curves(PDF).Archived fromthe original(PDF)on 2006-02-21.Retrieved2005-09-14.
• J. D. Foley; et al. (1992).Computer Graphics: Principles and Practice in C(2nd ed.). Addison Wesley.
• Rajiv Chandel (2014-03-20)."Implementing Bezier Curves in games".

Further reading

• A Primer on Bézier Curves– an open source online book explaining Bézier curves and associated graphics algorithms, with interactive graphics
• Cubic Bezier Curves – Under the Hood (video)– video showing how computers render a cubic Bézier curve, by Peter Nowell
• From Bézier to BernsteinFeature Column fromAmerican Mathematical Society
• "Bézier curve",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Prautzsch, Hartmut; Boehm, Wolfgang; Paluszny, Marco (2002).Bézier and B-Spline Techniques.Springer Science & Business Media.ISBN978-3-540-43761-1.
• Gallier, Jean(1999)."Chapter 5. Polynomial Curves as Bézier Curves".Curves and Surfaces in Geometric Modeling: Theory and Algorithms.Morgan Kaufmann.This book is out of print and freely available from the author.
• Farin, Gerald E. (2002).Curves and Surfaces for CAGD: A Practical Guide(5th ed.). Morgan Kaufmann.ISBN978-1-55860-737-8.
• Weisstein, Eric W."Bézier Curve".MathWorld.
• Hoffmann, Gernot."Bézier Curves"(PDF).Archived fromthe original(PDF)on 2006-12-02.(60 pages)
• Ahn, Young Joon (2004)."Approximation of circular arcs and offset curves by Bézier curves of high degree".Journal of Computational and Applied Mathematics.167(2): 405–416.Bibcode:2004JCoAM.167..405A.doi:10.1016/j.cam.2003.10.008.
• Davies, Jason."Animated Bézier curves".
• Hovey, Chad (20 May 2022)."Bézier Geometry".GitHub.
• Hovey, Chad (2022). Formulation and Python Implementation of Bézier and B-Spline Geometry.SAND2022-7702C.(153 pages)

External links

Computer code

• TinySpline: Open source C-library for NURBS, B-splines and Bézier curves with bindings for various languages
• C++ library to generate Bézier functions at compile time
• Simple Bézier curve implementation via recursive method in Python