In
geometric modelling and in
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, a composite Bézier curve or Bézier spline is a
spline made out of
Bézier curves that is at least
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
. In other words, a composite Bézier curve is a series of Bézier curves joined end to end where the last point of one curve coincides with the starting point of the next curve. Depending on the application, additional smoothness requirements (such as
or
continuity) may be added.
A continuous composite Bézier is also called a polybezier, by similarity to
polyline
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...
, but whereas in polylines the points are connected by straight lines, in a polybezier the points are connected by Bézier curves. A beziergon (also called bezigon) is a closed path composed of
Bézier curves. It is similar to a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
in that it connects a set of
vertices by lines, but whereas in polygons the vertices are connected by straight lines, in a beziergon the vertices are connected by Bézier curves. Some authors even call a
composite Bézier curve a "Bézier spline"; the latter term is however used by other authors as a synonym for the (non-composite) Bézier curve, and they add "composite" in front of "Bézier spline" to denote the composite case.
Perhaps the most common use of composite Béziers is to describe the outline of each letter in a
PostScript or
PDF file. Such outlines are composed of one beziergon for
open letters
An open letter is a letter that is intended to be read by a wide audience, or a letter intended for an individual, but that is nonetheless widely distributed intentionally.
Open letters usually take the form of a letter addressed to an indiv ...
, or multiple beziergons for closed letters. Modern
vector graphics
Vector graphics is a form of computer graphics in which visual images are created directly from geometric shapes defined on a Cartesian plane, such as points, lines, curves and polygons. The associated mechanisms may include vector display ...
and
computer font
A computer font is implemented as a digital data file containing a set of graphically related glyphs. A computer font is designed and created using a font editor. A computer font specifically designed for the computer screen, and not for print ...
systems like
PostScript,
Asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
,
Metafont,
OpenType
OpenType is a format for scalable computer fonts. It was built on its predecessor TrueType, retaining TrueType's basic structure and adding many intricate data structures for prescribing typographic behavior. OpenType is a registered trademark ...
, and
SVG use composite Bézier curves composed of cubic Bézier curves (3rd order curves) for drawing curved shapes.
Smooth joining
A commonly desired property of splines is for them to join their individual curves together with a specified level of parametric or geometric
continuity. While individual curves in the spline are fully
continuous within their own interval, there is always some amount of discontinuity where different curves meet.
The Bézier spline is fairly unique in that it's one of the few splines that doesn't guarantee any higher degree of continuity than
. It is, however, possible to arrange control points to guarantee various levels of continuity across joins, though this can come at a loss of local control if the constraint is too strict for the given degree of the Bézier spline.
Smoothly joining cubic Béziers
Given two cubic Bézier curves with control points