Thank you!

The formatting of your message is a bit skewed on my end, but I will 
attempt to retrace your steps myself and check my work against yours.

  -Shay

Jaap Frank wrote:
> "Shay" <n@n.n> schreef in bericht news:4c13b889$1@news.povray.org...
>> Possible with an isosurface function?
> 
> Shay,
> 
> Years ago I needed a formula for the same shape you now request and, beleive 
> me, it took me several month to figure it out.
> 
> You start with:
> 
> x = (A + a.cos(alpha)).cos(phy)
> y = b.sin(alpha)
> z = (B + a.cos(alpha)).sin(phy)
> 
> A and B are the different main radii in x and z direction,
> and a and b are the different radii in de pipe of the torus.
> So the major radius R is now changed in A and B
> and the minor radius r is now changed in a and b.
> When you rotate alpha and phy over 360 degrees, you get the torus you want.
> If you want to translate this to the f(x,y,z) = .... you need much patience 
> and time and
> at the end (couple of days at least) you get your variables you can put in 
> the
> poly shape.
> In this range t = a/b so this is the ratio of the minor radii.
> If you want a circular pipe, then t = 1.
> Constant: 
> powers:
> +t^4 
> x^4y^4
> +2t^2(B^2-a^2) 
> x^4y^2
> +(B^2-a^2) ^2 
> x^4
> +2t^6 
> x^2y^6
> +2t^4 
> x^2y^4z^2
> -2t^4{(A^2+B^2)-3(B^2-a^2)}                                           x^2y^4
> -2t^2{(A^2-a^2)+(B^2-a^2)-4(AB-a^2)}                           x^2y^2z^2
> +2t^2{(B^2-a^2)^2-2(B^2-a^2)(A^2+a^2)}                       x^2y^2
> +2{(A^2-a^2)(B^2-a^2)+2(A-B)^2a^2                               x^2z^2
> -2(B^2-a^2)^2(A^2+a^2) 
> x^2
> +t^8 
> y^8
> +2t^6 
> y^6z^2
> +2t^6{(A^2-a^2)+(B^2-a^2)}                                             y^6
> +t^4 
> y^4z^4
> -2t^4{(A^2+B^2)-3(A^2-a^2)}                                           y^4z^2
> +t^4{(A^2-a^2)^2+(B^2-a^2)^2+4(A^2-a^2)(B^2-a^2)}  y^4
> +2t^2(A^2-a^2) 
> y^2z^4
> +2t^2{(A^2-a^2)^2+2(A^2-a^2)(B^2+a^2)}                      y^2z^2
> +2t^2{(A^2-a^2)^2(B^2-a^2)+(A^2-a^2)(B^2-a^2)^2}    y^2
> +(A^2-a^2) ^2 
> z^4
> -2(A^2-a^2)^2(B^2+a^2) 
> z^2
> (B^2-a^2)^2(A^2-a^2)^2 
> constant
> 
> This equation works fine if the poly shape power 8 is possible, but for some 
> reason today this is limited to power 7. There was a short period of time 
> that power 15 was alowed, but alas, not in this times. For me it would be 
> fine if this limitation is raised again.
> In the short period that power 8 was alowed, it traces fine and not too 
> slow. The slowing down happened with textures. Then it slowed down 
> considerable. But I think that with our modern fast computers this is no 
> problem anymore.
> 
> You can still use this with the parametric Object, but then the shape is 
> made out of triangles and is not the pure mathematic form.
> 
> Maybe this mail causes that the max power is raised to 8 so this shape can 
> be used in his pure mathematical form again.
> 
> Jaap Frank 
> 
>

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