Editing Bézier curve (section)

==Applications==
===Computer graphics===
[[Image:Bézier curve in Adobe Illustrator CS2.png|left|thumb|Bézier path in [[Adobe Illustrator]] ]]
Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the [[convex hull]] of its [[Control point (mathematics)|control points]], the points can be graphically displayed and used to manipulate the curve intuitively. [[Affine transformation]]s such as [[translation (geometry)|translation]] and [[rotation]] can be applied on the curve by applying the respective transform on the control points of the curve.

[[Quadratic function|Quadratic]] and [[Cubic function|cubic]] Bézier curves are most common. Higher degree curves are more [[computationally expensive]] to evaluate. When more complex shapes are needed, low order Bézier curves are patched together, producing a [[composite Bézier curve]]. A composite Bézier curve is commonly referred to as a "path" in [[vector graphics]] languages (like [[PostScript]]), vector graphics standards (like [[Scalable vector graphics|SVG]]) and vector graphics programs (like [[Artline (program)|Artline]]<!-- 1990 -->, [[Timeworks Publisher]]<!-- early 1990s -->, [[Adobe Illustrator]], [[CorelDraw]], [[Inkscape]], and [[Allegro (software library)|Allegro]]). In order to join Bézier curves into a composite Bézier curve without kinks, a property called [[Smoothness#Geometric continuity|''G1 continuity'']] suffices 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.

[[File:Bezier curves composition ray-traced in 3D.png|thumb|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.<ref>Alexander Reshetov and David Luebke, Phantom Ray-Hair Intersector. In Proceedings of the ACM on Computer Graphics and Interactive Techniques (August 1, 2018). [https://research.nvidia.com/publication/2018-08_Phantom-Ray-Hair-Intersector]</ref>]]
The simplest method for scan converting ([[Rasterisation|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.<ref name="Xuexiang"/>

Analytical methods where a Bézier is intersected with each scan line involve finding [[zero of a function|roots]] of cubic polynomials (for cubic Béziers) and dealing with multiple roots, so they are not often used in practice.<ref name="Xuexiang">{{cite web |title=Complex Quadratic Bézier Curve on Unit Circle |author1=Xuexiang Li |name-list-style=amp |author2=Junxiao Xue |publisher=School of Software, Zhengzhou University |location=Zhengzhou, China |url=https://www.atlantis-press.com/php/download_paper.php?id=12403}}</ref>

The rasterisation algorithm used in [[Metafont]] is based on discretising the curve, so that it is approximated by a sequence of "[[rook (chess)|rook]] moves" 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 <math> y = \pm x </math>), then the curve is decomposed into smaller segments such that the ''direction'' of a curve segment stays within one sector; since the curve velocity is a second degree polynomial, finding the <math>t</math> values where it is parallel to one of these lines can be done by solving [[quadratic equation]]s. 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.<ref>Parts 19–22 of {{cite book |last1=Knuth |first1=Donald E. |author-link=Donald Knuth |title=Metafont: The Program |date=1986 |publisher=Addison-Wesley |isbn=0-201-13438-1}}</ref>

There is also a modified curve form of [[Bresenham's line algorithm|Bresenham's line drawing algorithm]] by 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 of [[integer]] additions and subtractions.<ref name=Zingl>{{cite report|last=Zingl|first=Alois|title=A Rasterizing Algorithm for Drawing Curves|year=2012|url=http://members.chello.at/~easyfilter/Bresenham.pdf}}<br>HTML abstract and demo: {{cite web |date=2016|last=Zingl|first=Alois|title=Bresenham |url=http://members.chello.at/easyfilter/bresenham.html |website=members.chello.at}}</ref>

===Animation===
In animation applications, such as [[Adobe Flash]] and [[Synfig]], 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.<ref>{{cite web |url=https://www.adobe.com/devnet/flash/learning_guide/animation/part07.html |title=Using motion paths in animations |website=Adobe |access-date=2019-04-11}}</ref><ref>{{cite web |url=https://wiki.synfig.org/Doc:Following_a_Spline |title=Following a Spline |website=Synfig Wiki |access-date=2019-04-11}}</ref>

In 3D animation, Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation.<ref>{{cite web |title=Advanced Graphics Lecture Notes|publisher=University of Cambridge Computer Laboratory|first=Neil A.|last=Dodgson |url=https://www.cl.cam.ac.uk/teaching/1213/AdvGraph/smeg13.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.cl.cam.ac.uk/teaching/1213/AdvGraph/smeg13.pdf |archive-date=2022-10-09 |url-status=live|website=cl.cam.ac.uk|date=1999}}</ref> Bézier curves are now very frequently used to control the animation easing in [[CSS]], [[JavaScript]], [[JavaFx]] and [[Flutter (software)|Flutter SDK]].<ref name=":0" />

===Fonts===
[[TrueType]] fonts use composite Bézier curves composed of '''quadratic''' Bézier curves. Other languages and imaging tools (such as [[PostScript]], [[Asymptote (vector graphics language)|Asymptote]], [[Metafont]], and [[Scalable vector graphics|SVG]]) use composite Béziers composed of '''cubic''' Bézier curves for drawing curved shapes. [[OpenType]] fonts can use either kind of curve, depending on which font technology underlies the OpenType wrapper.<ref>{{cite web |title=The difference between CFF and TTF |url=https://www.linotype.com/8120/the-difference-between-cff-and-ttf.html |archive-url=https://web.archive.org/web/20170703232935/https://www.linotype.com/8120/the-difference-between-cff-and-ttf.html |archive-date=2017-07-03 |website=Know How |publisher=Linotype |access-date=3 July 2018 |ref=CFF_TTF |quote=The OpenType format was formulated in 1996. By 2003, it began to replace two competing formats: the Type1 fonts, developed by Adobe and based on [P]ost[S]cript, and the TrueType fonts, specified by Microsoft and Apple. (...) TTF stands for TrueTypeFont and indicates that the font data is the same as in the TrueType fonts. CFF stands for the Type1 font format. Strictly speaking, it refers to the Compact Font Format, which is used in the compression processes for the Type2 fonts. (...) the cubic Bézier format of the Type1 fonts is more space-saving compared to the quadratic format of the TrueType fonts. Some kilobytes can be saved in large, elaborate fonts which may represent an advantage on the Web. On the other hand, the more detailed hinting information of the TrueType fonts is useful for very extensive optimization for screen use.}}</ref>

Font engines, like [[FreeType]], draw the font's curves (and lines) on a pixellated surface using a process known as [[font rasterization]].<ref name=freetype>{{cite web |title=FreeType Glyph Conventions / VI. FreeType outlines |website=The Free Type Project |date=13 February 2018 |url=http://www.freetype.org/freetype2/docs/glyphs/glyphs-6.html}}<br>{{cite web |title=FreeType Glyph Conventions – Version 2.1 / VI. FreeType outlines |date=6 March 2011 |url=http://www.freetype.org/freetype2/docs/glyphs/glyphs-6.html |archive-url=https://web.archive.org/web/20110929201958/http://www.freetype.org/freetype2/docs/glyphs/glyphs-6.html |url-status=dead |archive-date=2011-09-29}}</ref> 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 in [[Microsoft Excel]] also uses this algorithm.<ref>{{cite web |url=http://www.xlrotor.com/resources/files.shtml |title=smooth_curve_bezier_example_file.xls |website=Rotating Machinery Analysis, Inc. |access-date=2011-02-05 |url-status=dead |archive-url=https://web.archive.org/web/20110718131613/http://www.xlrotor.com/resources/files.shtml |archive-date=2011-07-18 }}</ref>

Because arcs of circles and [[ellipse]]s cannot 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) into [[non-uniform rational B-spline|NURBS]], 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.{{citation needed|date=October 2015}}

===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 the [[end effector]]s.<ref name=Malik2021>{{cite journal|last1=Malik|first1=Aryslan|last2=Henderson|first2=Troy|last3=Prazenica|first3=Richard|title=Trajectory Generation for a Multibody Robotic System using the Product of Exponentials Formulation|journal=AIAA Scitech 2021 Forum|date=January 2021|pages=2016|doi=10.2514/6.2021-2016|isbn=978-1-62410-609-5 |s2cid=234251587 |url=https://arc.aiaa.org/doi/abs/10.2514/6.2021-2016}}</ref> 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.<ref name=Malik2021 />