[alogo] Bezier General

Curve of degree n, determined by n+1 control points P0, ..., Pn, using the Bernstein polynomials Bn,i(x).
The curve is defined by the equation c(t) = Bn,0(x)*P0 + ...+Bn,i(x)*Pi + ...+ Bn,n(x)*Pn.
It is contained in the convex hull of the set of control points. Here the curve is constructed using a small script (EUC_Scripts\EUC_Curves\GeneralBezier.txt). n is the degree of the Bezier curve coinciding with the degree of the Bernstein polynomials used. The parameter t varies in the interval [0,1]. The curve below has degree 5 and 6 control points that can be freely moved.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]

Most useful for applications are the "piecewise" Bezier curves, build from Bezier "Arcs" i.e. Bezier curves of degree 3. Usually the arcs are joined so that their tangents vary continuously at the joins. This kind of piecewise Bezier curves can be produced through a build in tool of EucliDraw. Below is such an example build up from 6 arcs. Each arc has 4 control points. At each join two control points are identified. Thus there are 6*3 = 18 control points in total.

[0_0] [0_1] [0_2]

See Also



Duncan Marsh. Applied Geometry for Computer Graphics and CAD Berlin, Springer, 1999, p. 135

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