POV-Ray : Newsgroups : povray.general : Bezier spline approximations to circles Server Time: 19 Feb 2019 23:58:55 GMT
  Bezier spline approximations to circles (Message 1 to 2 of 2)  
From: JimT
Subject: Bezier spline approximations to circles
Date: 22 Jan 2019 15:25:00
Message: <web.5c47356c36095c6fbe7517870@news.povray.org>
Just so it didn't get lost in an old post...

I was in a rush when I replied last time and didn't have this to hand. Second
installment.

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))

As mentioned before, this is tangent to the circle at theta/2 as well as the
endpoints and outside the circle otherwise.  For theta = pi/2 the defect is
0.03% of the radius. For theta = pi it is 1.8%, still not bad, but for theta =
5pi/4 it is an unusable 7.6%.

Tweaking a to have the spline cross the circle twice reduces the error only to
about 70% of the value given and is probably not worth it.

If anyone wants code for a single segment Bezier Spline sphere sweep, I will
post it again, since the last time was years ago.


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From: Cousin Ricky
Subject: Re: Bezier spline approximations to circles
Date: 25 Jan 2019 21:09:09
Message: <5c4b7af5$1@news.povray.org>
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))
   }

Presumably, they are somehow the same formula, though I haven't figured 
out how to reduce one to the other.

> As mentioned before, this is tangent to the circle at theta/2 as well as the
> endpoints and outside the circle otherwise.  For theta = pi/2 the defect is
> 0.03% of the radius. For theta = pi it is 1.8%, still not bad, but for theta =
> 5pi/4 it is an unusable 7.6%.

My notes say that the curve deviates markedly from a circle if x > 90 
degrees (pi / 2), though I did not quantify the error.

> Tweaking a to have the spline cross the circle twice reduces the error only to
> about 70% of the value given and is probably not worth it.

For large angles I just use 2 segments.

I've actually tried multiple crossings to approximate a quadrant of a 
superquadric ellipse, but it required an arbitrary amount of fudging.  I 
do not have a general formula for this.  I also tried using two 45 
degree segments, but the result was horrible.

> If anyone wants code for a single segment Bezier Spline sphere sweep, I will
> post it again, since the last time was years ago.

There are two Object Collection modules that already do this for 
multiple segments: PointArrays and SphereSweep.


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