[Math] the difference between a rational spline and a “regular” spline

bezier-curvespline

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!

Best Answer

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).

Related Solutions

[Math] the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline

Roughly speaking ...

A spline is a curve that's formed from a collection of simple segments strung end-to-end so that their junctions are fairly smooth. There are exotic splines that use trigonometric and hyperbolic functions, but most splines consist of polynomial segments, so those are the only ones considered in the discussion below.

If there is only one (polynomial) segment, the spline is often called a Bézier curve.

If each segment is expressed in Bézier form (using Bernstein basis functions), then you might say that the spline is a "Bézier spline", though this term is not standard, AFAIK.

If each polynomial segment has degree 3, the spline is called a cubic spline.

If each segment is described by its ending positions and derivatives, it is said to be in "Hermite" form.

The b-spline approach gives a way of ensuring continuity between segments. In fact, you can show that every spline can be represented in b-spline form. So, in that sense, every spline is a b-spline.

For more than you would ever want to know about the subject, you can search for "spline" or "b-spline" in this bibliography.

[Math] Finding the mid-point of a B-spline curve.

My understanding is that a B-spline can have any number of control points. In the special case where the B-spline has exactly 4 control points, it is the same as a cubic Bezier curve with those same 4 control points.

So yes, De Casteljau's algorithm on the given control points works great:

given  midpoint  midpoint  midpoint
(0, 0)
      \
       (1/2, 0)
      /        \
(1, 0)          (3/4, 1/4)
      \        /          \
       (1, 1/2)            (3/4, 1/2)
      /        \          /
(1, 1)          (3/4, 3/4)
      \        /
       (1/2, 1)
      /
(0, 1)

So the given spline

starts at (0,0) at t=0,
goes through (3/4, 1/2) at t=1/2, and
ends up at (0,1) at t=1.

Best Answer

Related Solutions

[Math] the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline

[Math] Finding the mid-point of a B-spline curve.

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