Composite Bézier curve

[1] A continuous composite Bézier is also called a polybézier, by similarity to polyline, but whereas in polylines the points are connected by straight lines, in a polybézier the points are connected by Bézier curves.

A béziergon (also called bézigon) is a closed path composed of Bézier curves.

[6] Perhaps the most common use of composite Béziers is to describe the outline of each letter in a PostScript or PDF file.

A commonly desired property of splines is for them to join their individual curves together with a specified level of parametric or geometric continuity.

continuous within their own interval, there is always some amount of discontinuity where different curves meet.

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.

Given two cubic Bézier curves with control points

can be defined as follows: While the following continuity constraints are possible, they are rarely used with cubic Bézier splines, as other splines like the B-spline or the β-spline[7] will naturally handle higher constraints without loss of local control.

In case circular arc primitives are not supported in a particular environment, they may be approximated by Bézier curves.

of control points which result in the least approximation error for a given number of cubic segments.

Considering only the 90-degree unit-circular arc in the first quadrant, we define the endpoints

, respectively, as: From the definition of the cubic Bézier curve, we have: With the point

as the midpoint of the arc, we may write the following two equations: Solving these equations for the x-coordinate (and identically for the y-coordinate) yields: Note however that the resulting Bézier curve is entirely outside the circle, with a maximum deviation of the radius of about 0.00027.

By adding a small correction to intermediate points such as the magnitude of the radius deviation to 1 is reduced by a factor of about 3, to 0.000068 (at the expense of the derivability of the approximated circle curve at endpoints).

from an arbitrary number of cubic Bézier curves.

, placed at equal distances above and below the x-axis, spanning an arc of angle

: The control points may be written as:[11] TrueType fonts use composite Béziers composed of quadratic Bézier curves (2nd order curves).

To describe a typical type design as a computer font to any given accuracy, 3rd order Béziers require less data than 2nd order Béziers; and these in turn require less data than a series of straight lines.