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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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