James Tonkin wrote:

> In article <3b17c237@news.povray.org>, Warp  <war### [at] tagpovrayorg> wrote:
> 
>>Ron Parker <ron### [at] povrayorg> wrote:
>>: Not really.  Bezier patches/bicubic patches are neither non-uniform nor
>>: rational.
>>
> 
> Ok, there's about 4 types of curves being discussed here, from (mathematically)
> simplest to most complex, they are:
> 
> 	Bezier curves
> 	Bicubic patches
> 	B-Splines (uniform, non-rational)
> 	NURBS (which stands for non-uniform, rational, b-spline)
> 
> All of them are based on the idea of a set of control points, and a set of
> blending functions.
> 
> Starting in 2 dimensions, with Bezier curves:
> 
> A curve can be defined in terms of a single parameter, (typically 's' or 't'
> are used, depending on your mathematical background) which represents the
> normalized distance along the curve, from it's starting point (control point 0) 
> towards it's ending point (for a Bezier curve, this is control point 3, 
> although the algorithim could be generalized to any number of points).  By
> 'normalized distance' I mean that the parameter (I'll use 't') is 0 at the 
> starting point, and 1 and the ending point.
> 
> Each control point will have an x and y co-ordinate (<x0,y0> to <x3,y3>), and
> an associated blending function.  
> 
> For Bezier curves, the blending functions are just the standard expansion
> of ( t * (1-t) ) ^3, seperated according to the power of the 't' term.
> 
> (Historically, I don't think this was the motivation for development of
>  this set of blending functions, but this is how it works out)
> 
> So blending function 3 is t^3.
> 	b.f. 2 = 3 * (t ^2)*(1-t)
> 	b.f. 1 = 3 * t * ( (1-t)^2)
> 	b.f. 0 = (1-t)^3
> 



Could you please tell me what separation of a formule is? i.e., how do 
you get the blending functions from ( t*(1-t))^3?

Thanks in advance,

Remco Poelstra

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