On 4/23/07, Robert Bradshaw <[EMAIL PROTECTED]> wrote:
>
> This can't always be done in general, and even if it can it's usually
> really ugly. For example, sin(sin(y) + y) = x^5 - x + 1.
>
> There are other algorithms to do this kind of implicit plotting.
Maybe we should enumerate some. For example, there's this obvious one
for plotting F(x,y) == 0 on some rectangle R.
(1) Determine if F is a polynomial in either x or y. If not, either
(a) give up,
or (b) somehow find a polynomial approximation to F in x using
Taylor series.
Assume without loss that F is a polynomial in x (with coefficients that
are arbitrary functions of y).
(2) For each of a slightly randomly chosen list of sample points p_i in the y
direction, compute the polynomial F(x,p_i) numerically. Then use
the (very fast GSL) numerical polynomial solver to compute the
double precision roots of the polynomial F(x,p_i).
(3) Use some sort of clever sorting method to order the solutions found in (2)
along a curve. Then plot that curve.
The input parameters to the plot routine would influence the number of
sample points;
also one could adaptively refine in step (2) depending on the nature
of the solution
to F(x,p_i)=0. Hopefully there is a paper about this somewhere, which explains
the subtleties I'm missing.
William
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