What is the difference between a rational spline and a "regular" spline?

1,397

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

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

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).

Share:
1,397
Alan Wolfe
Author by

Alan Wolfe

I'm a professional game programmer specializing in audio, skeletal animation, online engineering and non traditional graphics. I'm always researching something new and digesting it into simple code implementations with intuitive explanations. Check out my blog for the latest: http://blog.demofox.org I also have a 1st degree black belt in kempo and am currently studying krav maga. I have also some exposure to bjj, Thai boxing, catch wrestling, chi Na, and others.

Updated on June 12, 2020

Comments

  • Alan Wolfe
    Alan Wolfe over 3 years

    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!