POV-Ray : Newsgroups : povray.general : Bezier spline approximations to circles : Re: Bezier spline approximations to circles Server Time: 19 May 2019 11:02:34 GMT
  Re: Bezier spline approximations to circles  
From: JimT
Date: 20 Feb 2019 14:10:01
Cousin Ricky <ric### [at] yahoocom> wrote:
> On 2019-01-22 11:23 AM (-4), JimT wrote:
> > If you want a Bezier Spline approximation to an arc of a circle subtending angle
> > theta at the centre, the four control points are
> >
> > (1,0), (1,a), (cos(theta) + asin(theta),sin(theta) -
> > acos(theta)),(cos(theta),sin(theta))
> >
> > where a = (8/3)(sin(theta/2) - sin(theta)/2)/(1-cos(theta))
>
> Hmmm.  This is the formula for 'a' that I used for the ring shank cross
> section in GemCuts:
>
>    #declare Gem__fn_Bezier_arc = function (x)
>    { (8 * cos (x / 2) - 4 - 4 * cos (x)) / (3 * sin (x))
>    }
>
Sorry I'm a month late. I didn't spot your post. And you probably won't spot
this.

I plotted them in Matlab to convince myself they were the same. Then I
multiplied top and bottom by sin(x/2)/cos(x/2). 1+cos(x) is 2cos^2(x/2) and
1-cos(x) is 2sin^2(x/2).


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