It's nice to know someone actually understands it enough to work with
the math. All I've ever done is put some numbers in to these sort of
primitives. Got a hat shape I wanted that way though ;) pure luck and
then a little tweaking.


Nieminen Mika wrote:
> 
>   The formula of an elliptical cylinder along the z-axis is the following:
> 
> x^2/r1^2 + y^2/r2^2 = 1
> 
> where 'r1' and 'r2' are the radiuses of the ellipse.
>   What we want is this kind of cylinder which changes linearly from z=0 to
> z=1 so that in z=0 the ellipse has radiuses 'a' and 'b' and in z=1 radiuses
> 'c' and 'd'.
>   This means that when z=0, r1=a and r2=b. Then r1 and r2 should linearly
> change so that when z=1, r1=c and r2=d.
>   This can be expressed this way:
> 
> r1 = a*(1-z)+c*z
> r2 = b*(1-z)+d*z
> 
>   Substituting in the above formula we get:
> 
> x^2/(a*(1-z)+c*z)^2 + y^2/(b*(1-z)+d*z)^2 = 1
> 
>   That's it. Now we have to reduce that to polynomial form so that we can
> write the quartic. This is a quite laborious job. I will not type the
> polynom here because it's so long, but it begins this way:
> 
> b^2*x^2 + 2*(b*d-b^2)*x^2*z + (b^2-2*b*d+d^2)*x^2*z^2 + ... and so on
> 
>   Now, to write the polynomial, we look at the table in the page 212 of the
> povray manual (the "Poly, Cubic and Quartic"-section). We place all the
> terms of the polynomial in the right places in the vector. The result is:
> 
> quartic
> { <0, 0,  0,  0, 0,   0,   0,  b*b-2*b*d+d*d, 2*(b*d-b*b), b*b,
>    0,  0,   0,  0,   0,  0, 0,  0,  0, 0,
>    0, 0,  0, a*a-2*a*c+c*c, 2*(a*c-a*a), a*a, 0,  0,  0, 0,
>    -(a*a-2*a*c+c*c)*(b*b-2*b*d+d*d),
>    -(2*((b*d-b*b)*(a*a-2*a*c+c*c)+(a*c-a*a)*(b*b-2*b*d+d*d))),
>    -(b*b*(a*a-2*a*c+c*c)+4*(a*c-a*a)*(b*d-b*b)+a*a*(b*b-2*b*d+d*d)),
>    -(2*(b*b*(a*c-a*a)+a*a*(b*d-b*b))), -a*a*b*b>
> }
> 
>   Now we only have to cut the unwanted parts out (ie. everything that is
> at z<0 and z>1) and bound with a proper object for speedup.
> 
> --
> main(i,_){for(_?--i,main(i+2,"FhhQHFIJD|FQTITFN]zRFHhhTBFHhhTBFysdB"[i]
> ):5;i&&_>1;printf("%s",_-70?_&1?"[]":" ":(_=0,"\n")),_/=2);} /*- Warp -*/

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