Editing Bézier curve (section)

===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}}