William F Pokorny <ano### [at] anonymousorg> wrote:
> Ref:
>
> http://news.povray.org/povray.general/thread/%3C609d7f7f%241%40news.povray.org%3E/
>
> Web Message: 609a4569$1@news.povray.org
>
> As part of the effort referenced, added a new inbuilt functions called
> f_vangle(). A first pass of the help text follows.
>
> ---
> Returns angle of rotation between two vectors. Function mimics the
> traditional VAngle macro shipped with POV-Ray in two ways.  The angle is
> returned in radians. Wrap with degree() for answer in degrees.
>
> The first an implementation of the acos of the dot product - with -1 to
> 1 clamping of the dot product - which is the traditional one.
>
> Tor Olav Kristensen suggested a more accurate method to the povray
> general news group on May 06, 2021 which is the second method.   In
> isolation it's obviously slower - sometimes significantly so - (30-40%)
> over the traditional acos based method. However, in the blur of branch
> predictions supporting both methods and modern CPU speculative execution
> the form as implemented herein is often the fastest. When slower it's
> not that much slower in the context of typical parser time use.
>
> In targeted random vector experimentation the dot product was found to
> return max and min values of +1.00000000000000067 and
> -1.00000000000000067, respectively.  Meaning with the traditional method
> the -1..1 clipping is absolutely necessary to avoid acos domain errors.
>
> Method 0. By targeted random vector experimentation the traditional
> method provides for 7 digits of accuracy on when very near a zero angle
> or very near pi. In practice, with vectors not near coincident, or near
> exactly opposing the accuracy is much better.
>
> Method 1. By targeted random vector experimentation the traditional
> method provides for 12 significant digits of accuracy near 0.0 or pi in
> angle. It often provides a few more digits.
>
> The largest differences seen between the two methods, over tens of
> millions of random vector pairs, was always less than 1e-7.
>
> Using this inbuilt function directly is up to 2.5 times faster no matter
> the mode than using the macro VAngle. The macro may be more convenient
> in that vectors can be passed directly.
>
> Parameters:
>
> 1. First vector x value.
>
> 2. First vector y value.
>
> 3. First vector z value.
>
> 4. Second vector x value.
>
> 5. Second vector y value.
>
> 6. Second vector z value.
>
> 7. Method to use. If 0.0, the traditional acos(dot(v1,v2)) method is
> used. If 1.0, Tor Olav's suggested method is used.
>
> ---
> In math.inc now have:
>
> #macro VAngle(V1, V2) f_vangle(V1.x,V1.y,V1.z,V2.x,V2.y,V2.z,1) #end
>
> #macro VAngleD(V1, V2)
>    degrees(f_vangle(V1.x,V1.y,V1.z,V2.x,V2.y,V2.z,1))
> #end
>
> #macro VRotation(V1, V2, Axis)
>     (f_vangle(V1.x,V1.y,V1.z,V2.x,V2.y,V2.z,1)
>     *(vdot(Axis,vcross(V1,V2))<0?-1:1))
> #end
>
> #macro VRotationD(V1, V2, Axis)
>     (degrees(f_vangle(V1.x,V1.y,V1.z,V2.x,V2.y,V2.z,1))
>     *(vdot(Axis,vcross(V1,V2))<0?-1:1))
> #end


Nice work Bill!

It's good to be able to choose which formula to use for the angle calculation.

But I'm struggling to see the usefulness of the VRotation() and VRotation()
macros. I can not remember that I ever had the need to do the exact calculations
they do. And if one understand the intricate explanation for how to use them,
one is likely to do fine without them. Also there's to much work preparing for
them.

Perhaps someone can post code for a case where one of them is really useful ?

--
Tor Olav
http://subcube.com
https://github.com/t-o-k

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