I'm pretty comfortable with Bezier curves (not as much with b-splines, nurbs, hermite, catmull rom, etc), such that i know how to generate a bezier curve of any degree using the bernstein polynomials (aka binomial expansion), or can alternately evaluate them via the De Casteljeau algorithm.

However, I sometimes see references to RATIONAL splines as being something different than regular splines.

Can anyone explain the difference? Preferably in the context of rational Bezier curves vs "regular".

Thank you!


Rational Bezier curves are Bezier curves with control points that are associated with a weighting value. The integral Bezier curves is represented as


while a rational Bezier curve is represented as

$C(t)={\sum_{i=0}^n{B_{i,n}(t)w_iP_i}\over \sum_{i=0}^n{B_{i,n}(t)w_i}}$

where $w_i$ are non-negative weighting values associated with each control points $P_i$.

Rational Bezier curves are most often used to represent conic sections analytically, which cannot be done by integral Bezier curves.

You still can use all the algorithms associated with integral Bezier curves for rational Bezier curves but you will need to treat the rational Bezier curves as the projective image of a integral Bezier curve in a higher dimension space (or homogenous space).


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