[alogo] Rational Bezier Curve

Curve of degree n, determined by n+1 control points P0, ..., Pn, using the Bernstein polynomials Bn,i(x) and n+1 additional (positive) numbers w0, ..., wn, called weights. The curve is defined by the equation:
c(t) = (1/BSum(x))*(Bn,0(x)*w0*P0 + ...+Bn,i(x)*wi*Pi + ...+ Bn,n(x)*wn*Pn),
where the function BSum(x) = Bn,0(x)*w0 + ...+Bn,i(x)*wi + ...+ Bn,n(x)*wn.
c(t), for positive weights wi, 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\RationalBezier.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.
Rational Bezier curves of degree 2 can represent exactly arcs of conic sections. This is illustrated in the file ConicThroughBezierRational.html .

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]
[2_0] [2_1] [2_2]

See Also



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

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